Monokinetic charged particle beams: qualitative behavior of the solutions of the cauchy problem and 2d time-periodic solutions of the vlasov-poisson system

Let (f,U) be a solution of the 2d Vlasov-Poisson system for charged particles f=f(t,x,v) is the distribution function, defined on the phase space (x represents the position and v the velocity of the particles) and U=U(t,x) is the self-consistent potential given by the Poisson equation. Uo is an external potential which depends only on |x|


U ( t , x ) = --lnlxl * p ( t , z )
and study the problem in a framework for which this formula makes sense.
For the same reason, we will also frequently assume throughout the paper (but in most cases it is not essential) that the external potentialwhen there is oneis harmonic : there exists some po €10, +w[ such that The paper is divided into two parts. The first one is concerned with the evolution problem (existence, regularity or uniqueness as well as more qualitative aspects, like dispersion results, growth of the support for compactly supported distribution functions or equipartition of the energy). The main estimate is a Lyapunov functional which is used to control the energy (and provides an existence result for non compactly supported distribution functions) and to give an estimate for the dispersion (N = 2 appears to be the critical dimensinq-and we have to use logarithmic estimates). In the second part, we focus on time-periodic (and stationary) solutions and present results in two directions: first, we give an (uncomplete) classification of the solutions that are time-periodic (but radially symmetric if they are averaged over one time period); then we study a special class of solutions which is the counterpart of the class of solutions of the Vlasov-Poisson system in dimension three that satisfy the so-called Ehlers & Rienstra ansatz.
Three sections, which are of general interest but rather technical, are rejected at the end of the paper: Appendix A deals with a formal derivation of the two dimensional model, which main interest is to show that the model is local, with two consequences: it is not restrictive to take the confining potential harmonic, and there is no a priori natural boundary condition for U. In Appendix B are stated two interpolation lemmas, with an explicit computation of the constants. Appendix C provides explicit and detailed statements for Jeans' theorem.
Following B. Perthame's definitions (see [36]), a weak solution is a solution in the sense of the distributions such that the energy is bounded but not necessarily constant. A strong solution (in dimension N = 2) is a solution that has moments of order 2 + E in v and z (at least when the external potential is harmonic) and such that the energy does not depend on t. In this paper, we will make use of an intermediate notion of solutions corresponding to the case f(Uo+lvlZ) in L1 with Uo growing at least logarithmically at infinity (confinement case) or such that f has a moment in z of order m E [1,2]. For such solutions, the self-consistent potential energy is continuous w.r.t. the time (in dimension N = 2), but the kinetic energy and the external potential energy are only bounded w.r.t. the time.
Since the number of references in this work is quite huge and concerns very different subjects, the references will be mentioned throughout the paper.

Introduction
The first part of this paper is devoted to various results on the initial value problem for the two dimensional Vlasov-Poisson system in the presence (or not) of a confining potential.
In section 1, we introduce two notions of Lyapunov functionals. In section 2 (see the complete list of references therein), an existence result is given, in a more general framework than what had been established by S. Ukai & T. Okabe in [40] and S. Wollman in [41,42] . This result essentially benefits of recent papers on the (more difficult) theory for the three dimensional problem. The questions of the regularity and of the uniqueness are treated with the approach developped by P.-L. Lions and B. Perthame in [33].
A dispersion result is given in section 3: if there is no confining potential, the solution is vanishing for large time. This result is obtained using the same methods as R. Illner & G. Rein in [31] or B. ~e'rtharne in [36] for the case of the dimension three. Dimension two corresponds to a limit case for this method (use of logarithmic estimates).
The question of the growth of the support of an initially compactly supported distribution function is studied in section 4. One has to mention that the method, which is strongly dependant of the potential when it is applied to the computation of the size of the support in the phase space, gives the growth in the velocity space even if there is no confining potential, as in the paper [38] by G. Rein, but with a different method.
An equipartition of the energy result is given in section 5. The estimate .' obtained there is in fact the keypoint of Part I since it allows the computations on the Lyapunov functional. Section 5 also contains the moment estimates that are needed to compute the Lyapunov functional as well as to define the notion of solutions.

A Lyapunov functional and a priori estimates
Following the same idea as in B. Perthame 1361 and R. Illner and G . Rein in [31], we first derive a Lyapunov functional for the Vlasov-Poisson system in dimension 2 with an external potential. In the following, we shall assume that f and U are smooth, and that f is compactly supported, in order to perform any integration by part that is needed. We shall see later how to handle the non smooth case.
Multiplying the Vlasov equation by lvI2, ( x . v ) and 1x1, we can obtain respectively / / f ( t , x RZ x Ra In dimension N = 3 (see [36] and [31]), it is enough to compute (for some a > 0) f (~, x , v )  (iii) If Uo(x) = Kln(1 + 1x1) for some K < g, for any a > 0, we can get the following dispersion-type estimate:
Case 2: Assume that z c, belongs to Lm(lRZ).' Let us compute (with a > 1 a n d t > 0 )

DOLBEAULT
The reason why such a quantity is decreasing if x . VUo(x) I 0 for almost all x E n2 and why one has to introduce a term J J f ( t , x , v ) 1~(~ dxdv is related to the notion of asymptotic dispersion profile. This is the subject of a paper in preparation with G. Rein [20].
The Lyapunov functional is finite for any t > 0: we have indeed Note that the energy estimate holds as an inequality V t > 0 in both cases.
These solutions are weak solutions but it is easy to prove that they are in fact strong as soon as fo satisfies the condition

o ( x , v ) (~x l~+ '
since the momenta of order 2 + e are finite for any t > 0 (see lemma 5.2). In that case (2.2) becomes an equality, while estimates like bounds on the energy are enough for weak solutions.
The conditions on f could be weakned using the notion of renormalized solutions (see [11.12]) while the notion of solutions in the sense of the characteristics still holds (in the renormalized sense) as soon as VU + VUo at least belongs to w,~,'(R+ x RZ) (see Remark 2.2 below). Since the proof of such results rely on now classical methods, let us only mention the main ingredients.

Sketch of the proof :
1 ) a priori estimates : Assume first that f is a classical solution, smooth enough to justify the integrations by parts. a) for any C 2 convex funtion s defined on 10, +w[ x, v)) dxdv (t,x,v). sr (f(t,x,v)) dxdv

R Z x R Z
(Of course the result also holds for a non convex function s, but the result does not pass to the limit: see below).
This proves for example that for any p E [I, +CO[ and For p = 1, this proves the conservation of the mass: b) conservation of the energy: 2) compactness: A simple method to pass to the limit in the equation is to notice that as soon as 1 fn(t,.,v)+(v) dv strongly converges in L~ for any L" with compact support function +, then the limit f is a solution of the Vlasov-Poisson system in the distribution sense. This is easily obtained using the averaging lemmas (see 1281, 1241, [25], [13]) since and since V(U + Uo) f (t, ., v)t+h(v) dv belongs to L1 for any Lw function ph (take for example p = and q = 3 in (2.7)). in the sense of R-J. DiPerna and P-L. Lions [ll] with where t I+ (Xo(s, t, X, u), Vo(s, t,x,v)) are the characteristics in the external potential U, defined by The solution f R of MONOKNETIC CHARGED PARTICLE BEAMS As in the proof of Theorem 2.1, a direct computation shows that, for any c > 0, 2*1lER(t, .) + VUOIIL- (R~; d z ) (split the integral J &p(t,y) dy into two parts corresponding respectively to 1 2y( > 1 and 1 2yl 5 1 and use Holder's inequality with p = and p' = 2 + E for the second one). Assumption (2.8) just asserts that IIER(t,.) + VUo(lt- (R:) does not depend on R for R large enough. Taking then f = f R proves that p belongs to Lm((O, T ) x R:).
2) There exist an a > 0 (a = e-CT for some C > 0) such that the characteristic curves X ( S , t , xi, V J = V ( S , t, xi, ~i ) , V ( s ) = E(s,Xi(s)) ,  T ] whenever E + VzUo = -VU is given by p through the Poisson equation.
The proof relies on the estimate for any x, y E R2 such that lxyl < 1/2, which is proved exactly in the same way as in (331: since  4) the uniqueness result can be shown in the same way as in 1331 (with the simplification that we dont have to care about the field term coming from the initial data since we only have to consider the difference between two solutionsthe argument is not used for getting estimates on "higher moments"). Assume that there exist two solutions f~ and f2 corresponding to the same initial data fo and define  fo (x-vt,v)

R2
Using the fact that div, (&) is the Dirac distribution, Applying the Cauchy-Schwarz inequality, we get where C is a constant which only depends on the energy : this computation is summarized in the identity Remark 5.8 : Other moment estimates are easily obtained (see also [34], [36], [lo]): for example, if Uo is harmonic, then which is easily proved by multiplying the Vlasov equation by and integrating with respect to t, x and v.
Part I1 : Since the problem for stationary solutions is easier than for time-periodic solutions, we present first some classification results for the solutions of the stationary Vlasov-Poisson system. We extend then the ideas developped for these solutions to the time-periodic case. A detailed version of these result is given in Appendix C.
Section 2 of this part is devoted to the study of a subclass of these timeperiodic solutions.

.I. A classification result for stationary solutions
We first explicit the special class of solutions satisfying the weak Ehlers & Rienstra ansatz (see [23], [6]), and then give a factorization result which proves that these solutions are in fact the generic ones that have a radial spatial density.
It is easy to realize that any distribution function depending on x and v only through the quantities 1 2

Time-periodic solutions
For time-periodic solutions, we may proceed exactly in the same way. Consider now ' 2 The same kind of factorization result as for stationary solutions (under regularity and non resonance assumptions) shows that a time-periodic solution of period T such that the average over one period of the self-consistent potential is radially symmetric satisfies a factorization property, which is global w.r.t. t and local in the phase space R2, x

all ( t , x , v ) E [O,T] x supp(f),
A detailed version of this result is given in Appendix C. In the following, we will make one more assumption and assume that g does not depend on t . The class of solutions we shall consider is therefore defind by: there exist constant T > o and a function g : R2 + R+ such that for almost all ( t , x, v ) E R x R2, x RZ,

f ( t , x , v ) = g ( E ( t , x , v ) , F ( x , v ) ) and f ( t + T , x , v ) = f ( t , x , v ) . (weak E R )
In the next section we will prove that under a technical but generic assumption, the solution satisfies the (strong) Ehlers & Ftienstra ansatz for some function g and for w = $?. In that case, exactly as in the paper by J. Batt where the linear operator A is such that v . Ax = x A v for any ( x , v ) E RZ x R2: f and U are in a solid motion of rotation around the z-axis with a constant angular velocity w and take the following form and the problem is reduced to the nonlinear Poisson equation for w with G(w) = x Jwf m g(s) ds (we assume here that Uo is an harmonic potential).
We may first mention that such a formulation provides a very simple way for constructing time-periodic solutions to the Vlasov-Poisson system: since the equation for w does not depend on x, for any solution w, w, defined by x ct w,(x) = W(X+T) is also a solution for any r E R2, which clearly does not have the same symmetry properties as w (see Remark 3.2). Thus the potential U and the distribution function f are time-periodic solutions which are not radially symmetric and depend therefore explicitely oft. We will also consider the case where the confining potential Uo is not radially symmetric, and exhibit a branch of solutions that have a logarithmic growth, starting from the solutions that are radially symmetric up to a translation. These solutions are time-periodic (and generically explicitely timedependent (i.e. non stationary) solutions. Adequate conditions on G ensure that they have a finite mass.
But this part of the paper will be mainly devoted to the class of nonisotropic solutions with quadratic growth, i.e. the solutions such that for some e E [O, 11, (x1,x2) beeing a system of cartesian coordinates of x E R2.
In [6], the solutions that were considered were 3d-solutions of the gravitational Vlasov-Poisson system corresponding in the 2d-case to solutions such that 0 = 0 or 1 (ld-solutions), or such that 0 = a (radially symmetric solutions). We will adapt their results to the ad-electrostatic Vlasov-Poisson system with a confining potential (Section 3, Proposition 3.4: existence of ld-solutions, and Proposition 3.5: existence of radially symmetric solutions), but also study the general case: 0 E (0, I), 0 # $.
The spatial density (t,x) I+ J, , f(t,z,v) dv is, up to a rotation of angle w t given by It belongs to L1(R2) if s I-, G(s) is sufficiently decreasing for s + +m. Note that the asymptotic behaviour of u for such solutions is for any 0 belonging to [@I], which corresponds to a non standard asymptotic behaviour (see Theorem 3.3: asymptotic behaviour, necessary conditions for the existence of time-periodic solutions and consequences).
For this study, our main mathematical reference is a paper by J. Batt, H. Berestycki, P. Degond & B. Perthame [6] (three-dimensional Vlasov-Poisson system in the gravitational case). Compared to it the main results of this paper are the following: the class of solutions is larger (6 E [o, 11 instead of 0 = 0, $ or 1) and the symmetry assumptions are weaker ("weak" Ehlers and Rienstra ansatz instead of "strong" Ehlers and Rienstra ansatz) -roughly spoken, we prove that it corresponds to the class of the solutions that are in a solid motion of rotation with a constant angular velocity and such that the self-consistent potential has an at most quadratic growth, since the confinement of the particles is due to an external potential and not to the self-consistent potential (the force between the particles is repulsive), it is possible to perturb it and build a branch of solutions starting from the radially symmetric solutions (up to a translation), -the asymptotic boundary conditions are systematically explored; choosing a quadratic growth for the self-consistent potential (like in BBDP]) is not absurd (one has to keep in mind that the model corresponds to the study of a beam locally near its axis, as shown in Appendix A).
The study of the stationary solutions has been neglected. Most of the results for timeperiodic solutions are easily extended to the stationary case by simply taking w = 0. One has also to refer to J. Batt, W. Faltenbacher, & E. Horst [8] for this point. An attempt of a general classification of the time periodic solutions by the mean of Jeans' theorem is presented in Appendix C.
Ideas , which are very popular among astrophysicists (see [14,15] for a review) have been introduced from a mathemati cal point of view in 181, from which some notations are taken.

The Ehler-Rienstra ansatz for time-dependant solutions
In the rest of the paper, we will consider the special class of solutions of the Vlasov-Poisson system that are such that (this ansatz will be referred as the Ehlers & Rienstra ansatz) for some w E R, and where E and F are defined by Such solutions are also called in the physical literature "locally isotropic sc+ lutions" (see [23]). Before studying these solutions in details, we will notice that this class of solutions corresponds to a apriori larger class of solutions, the class of solutions satisfying only the weak Ehlers & Rienstra ansatz i.e. such that f (4 2, V ) = h(E(t, 2 ,

v ) , F(x, v ) ) (weak ER)
provided they are explicitely time-dependant. This result will be important in view of an attempt of classification of all the time-periodic solutions of the Vlasov-Poisson system given in Appendix C.

MONOKINETIC CHARGED PARTICLE BEAMS
Moreover f and U may be written in the following form

(2.3)
where w is a solution of the nonlinear Poisson equation and A is the linear operator such that v . Ax = x A v for any ( x , v ) E R2 x R 2 .
Proof : 1) A simple computation shows that 2 ) Let ( X I , X Z ) be cartesian coordinates of x E R2, so that X A v = xlv2 -x2vl, and denote by A the linear operator such that Ax is represented by ( -x 2 , x 1 ) . Let us define F by :

F ( x , v ) = x A v = ( v . A x ) .
Then, with these notations 4) According to the assumptions on h, one may assume either that there locally exists a C 1 function E ct F ( E ) such that or that there locally exists a C' function F ct E ( F ) such that ax; which would imply that p E po on V, in contradiction with the assumptions on P.
In case 2, the proof is exactly the same except that one has to exchange the roles of E(t, x, v ) and F (x, v ) , E ( F ) and F(E). C is therefore contained in a finite union of manifolds of dimension 2: (R2 x R2)\C is connected. (x, v) H w(E(t, x,v), F(z,v)) is a constant. Obviously, w does not depend either on t: w = w(E, F) is therefore almost everywhere w.r.t. (E, F) equal to a constant, which we still denote by w, when E,F belong to the set so that there exists a function g : R H IR such that 6) Equation gives for f the expression

The nonrinear Poisson equation and time-periodic solutions
This section contains the main results of the paper. Theorem 3.3 (Asymp totic behaviour, necessary conditions for the existence of time-periodic solutions and consequences) and Theorem 3.7 (Existence of time-periodic anisotropic solutions with finite mass) are completely new results. They present a priori considerations on the asymptotic behaviour of the solutions, and existence results for a new class of solutions, which includes the solutions given by J . Batt, H. Berestycki, P. Degond & B. Perthame in [6]. These solutions have the property that they are explicitely time-dependant and may have a finite L'-norm (mass), which was not the case for the solutions given in [6].
The other results of this section (Proposition 3.1: equivalence with a nonlinear Poisson equation, Proposition 3.4: existence of radially symmetric s+ lutions and Proposition 3.5: existence of ld-solutions) are more or less an adaptation of some of the results given in [6] (for 3d-solutions of the gravitational Vlasov-Poisson system) to the 2d-solutions of the Vlasov-Poisson system with a confining potential.   ( R 2 ) of with G ( w ) = a J : " g(s) ds. The relation between ( f , U ) and w is given by Equations (2.3) and (2.4). x I-+ p(t,x) is locally Lipshitz, and U belongs to C 1 ( R x R2). Assume that t and x are fixed, such that x belongs to the support of p(t, .), i.e. the support of

MONOKINETIC CHARGED PARTICLE BEAMS
Because of the assumptions on g, 5 has a strictly positive measure The other side of the proof is obvious. follows that for any solution w , w, defined by is still a solution for any T E R2. This very simple fact allows us to exhibit explicitely time-dependant priodic solutions as soon as w is not constant, even in the case when w is spherically symmetric, which means that the corresponding solution (f, U ) of the Vlasov-Poisson system is rotationally symmetric and stationary (see Proposition 3.4). It is possible to consider such solutions because the domain R2 is of course translation invariant and because the boundary conditions are specified only at infinity. In the following, we will not consider such a cause of time-dependance (except for the study of 1-d solutions, see Proposition 3.5) since it is a consequence of the assumption that the confining potential takes the very special form but we will concentrate our attention on an other cause of time-dependance, which is much more fundamental: the anisotropy of the solutions (see Proposition 3.5 and Theorem 3.7) which is clearly related to the asymptotic boundary conditions (see Theorem 3.3) and to the asymptotic behaviour of Uo.

{ v c n 2 : g l (~-~~) ( t , x , v )
The asymptotic condition on the behaviour of the density lirn sup p(t, x) = 0 121-++w fives a lower bound on We will see in Proposition 3.5 (Existence of 1-d solutions) that this bound is optimal. Such an asymptotic boundary condition is far from the usual one lim U(t, x) = Constant , 121++w but we will see that there are no solutions satisfying such a condition.

1619
(iv) Assume moreover that for some E > 0, ( t , x )  Since G is positive decreasing and not identically equal to zero, and since Applying the Maximum Principle, we obtain for all n E m, which is impossible : which proves (i).
Assume now that liminflZl,+, > -% > 0. We will now study the problem of the existence for different classes of solutions characterized by their symmetries and the corresponding asymptotic behaviour, and give some of the properties of these solutions.
We begin with radially symmetric. These solutions are time-independant (except if they are translated: see Remark 3.2), but may have a finite total mass M = 11 f ( t , x, v ) dxdv Ra x Ra Exactly as in the paper by J. Batt, H. Berestycki, P. Degond & B. Perthame ([6]) we will also study 1-d solutions that may be non stationary time-periodic solutions (even when they are not translated), but are always of infinite mass (Proposition 3.5).
The last result of this section will be devoted to solutions intermediate between the ones of Proposition 3.4 (radially symmetric solutions) and the ones of Proposition 3.5 (1-d solutions), which appear to form (Theorem 3.7) a new class of solutions (as far as the author knows). These solutions have the interesting property that they are timeperiodic non stationary solutions, and that they may have a finite total mass. They moreover have a non standard asymptotic behaviour : in a well choosen system of cartesian coordinates. If g = -kG', then f belongs to c ( R , L1(R%, x R2), U belongs to C1 (R, C Z ( R 2 ) ) and f(t. z , v ) = ( l l vw h 1 2 + w(lewt*zl)) ,
The rest of the proof is a simple computation.

Remarks :
1) for radially symmetric solutions, U has a logarithmic growth: This property is characteristic of the radially symmetric solutions among the class of the solutions considered in this paper (see Theorem 3.7 and Remark 3.8).
2) Using the fact that the equation -A w + 26 = G ( w ) is obvoiusly translation invariant, i.e. that for any solution w and for any $0 E R2, x ct ( w ( x + x 0 ) is also a solution, it is therefore easy to find solutions which are not radially symmetric and therefore explicitely timedependent. But the translation invariance is of course strongly related to the fact Uo has been choosen to be an harmonic potential. When this is not the case, see Remark 3.8.
We will now consider 1-d solutions in the following sense: look for a func-  Assume that G is a Lipshitz decreasing function, such that lim G ( w ) = 0 and w ! % G ( w ) = G m < co .

w -t + m
Let us define by For any 6 > 0, Equation

d2w
-- has a unique solution for (w(O), g ( 0 ) ) E R2 given. For any 6 > 0, any solution of Equation (3.1) satisfies one of the two following properties : (i) There exists so E R such that %(so) = 0. If wo = w(so), then 3 cases may occur w is convex in Case 1.a and concave in case 1.c.
(ii) There exists so E R such that w(s0) = F and either In case 3, the same result occurs as s + + m , but as s + -co, s r-t g ( s ) is constant.
and concavity or convexity properties are enough to prove directly the results on w. The rest of the proof is a simple computation again.
Let us notice that the solutions corresponding to Case (i) of Proposition 3.5 are "radially symmetric" 1-d solutions, i.e. even, up to a translation such that so = 0. We will now exhibit a third class of solutions, that will be intermediate between 1-d solutions and 2-d radially symmetric solutions, so that they will be time-periodic (with period Z ) and have a finite total mass. Before giving the result, let us introduce some notations (that have already been used in the introduction): assume that ( x l , x 2 ) are cartesian coordinates of x E R2. Let us 1 1 and consider a solution v of with Q e ( x ) = 6(6'x: + (1 -6')~:) .

It immediately follows that
We can notice that if 8 = 0 or 8 = 1 these solutions correspond to the ldsolutions, and that if 8 = 112 = 1-8, then they correspond to radially symmetric 2d-solutions. Theorem  Proof : The first part of the proposition is obvious. The last part does not present any difficulty: the time-dependance property of f is easily derived from the fact that the asymptotic form of w = u+QB (as 1x1 -+ +co) is clearly not invariant under rotations. The solution is by construction 5 periodic. In fact the proof below also works on the subset of the functions which are symmetric with respect to the origin, and proves that ( f , U ) can be choosen $ time-periodic. We therefore only have to prove the existence of a solution. We will do it in four steps.

T w ) ( z ) = ( T~w ) ( z ) + w o -I ( w ) V z E R 2 ,
with I ( W ) = inf;sRa ( ( T~W ) (x)a ( x ) ) , are continous on Kwo ,c, which is a convex closed subset of X.

4~1,
which gives for & ln 1x1 * G(w(x)) the estimate For any C 2 C(wo), KwOsc is non-empty and stable under the action of T . In the following, we shall consider K = Kwo,c(wo) and get a fixed point result on T ( K ) c K using Schauder's theorem. According to its definition, K is bounded for the norm 11. /Ix. For the moment, let us forget the problem of the non-compactness of R2 and try to apply Ascoli's lemma. We have to prove that T ( K ) is uniformly equicontinous. As in [21] or in [9], it is a consequence of the inequality (with c = a(a -G(wo)) . min (8, ( 1 -8 ) ) ) if q = 1 , and Applying Ascoli's lemma, T ( K ) would be relatively compact if the functions of K had been defined on a compact set.
4th step : conclusion : Since the set K is a set of functions defined on R2, one has to be careful. Let R be a strictly positive real number.

1) The operator T~ : P(B(O, R ) ) + CO(R2; R) defined by with IR(w) = infz,B(o,R)
, is continous on the set K R defined by and applying Ascoli's lemma, one proves that its restriction T R defined by is such that T~( K~) is precompact : Schauder's theorem then implies that T R has a fixed-point on K R in the following sense: there exists a function wR of K such that

MONOKINETIC CHARGED PARTICLE BEAMS
2) We just have to compute the difference between wR and T W~: But on one side uniformly in wR E K~ as R   1) The solutions of Theorem 3.7 may have a finite total energy 1632 DOLBEAULT provided s * g(s) is sufficiently decreasing as a + +co. 2) Last part of the proof shows that T ( K ) is precompact. An alternative method would be to apply directly Schauder's fixed-point theorem to T and K .
3) The question of the uniqueness seems to be more technical than difficult. For example, assuming that g is decreasing and such that the operator T is then contracting on the set K . Of course, this only proves the uniqueness of the solution of the fixed point equation since the asymptotic boudary condition is not sufficient to determine the location of the minimum. It is for example clear that the solution of the following fixed point equation where & ( x ) = Q s ( z + x0) for some xo E RZ provides a solution to whivh gives a solution to -Au = G(u + q/xI2) satisfying the asymptotic boundary condition provided 4 ) Using a developpement of w ( z ) for lzl -i +co, and the techniques developped in [26] and adapted in [9], it is probably not very difficult to prove that up to a translation, the solution w with the anisotropic asymptotic boundary condition is, up to a translation, symmetric with respect to the origin and therefore periodic of period $.

)
Instead of qB(x) = a(26 -G(wO))(8z: for any E €10, G(wo)[, wo > a. This allows to give an existence result of a solution w such that Wo > inf,,,~ w ( x ) > m. There exists indeed a solution we for which wo -6 = i n t E R z (~( x ) , for any E > 0 small enough. But the passage to the limit E + O+ is not clear since is obviously not bounded from below. For the same reason, we cannot apply directly Lebesgue's theorem of dominated convergence. Note that the meaning of wo in Theorem 3.7 is the same as in Propositions

and 3.5 if e E [i(G(wo) + 26),G(wo)[. The proof is easy. If we denote by we
for any solution w in the sense of Proposition 3.4 or 3.5.
6) The method also applies when the confining otential is not radially symmetric (but has an asymptotic quadratic growth). For instance, we may consider the case where where Vo is a continuous compactly supported radially symmetric perturbation. When the solution (obtained by the fixed pont method) is unique, this allows to prove the existence of a branch of solutions (parametrized bythis can be seen as an implicit function therorem) and gives an existence result when Uo is not an harmonic potential. It is worth to notice that this provides a branch of solutions starting from the radial solutions (plus a translation) which may have finite mass, a logarithmic growth, and are generically timeperiodic non stationary solutions.

Conclusion and some open questions
First, let us summarize the results obtained in part 11. The Jeans' Theorem given in Proposition C.2 proves that generically any time-periodic solution such that its average is radially symmetric in fact satisfies the weak Ehlers & Kenstra ansatz, provided it does not explicitely depends on time.
But according to Theorem 2.1, any solution satisfying the weak Ehlers & Kenstra ansatz is time-periodic and obeys to the usual (strong) Ehlers & Rienstra ansatz. The dependance in time is obtained through a rotation with a constant angular velocity. Solving the Vlasov-Poisson system is completely equivalent to solving a nonlinear Poisson equation (Proposition 3.1), whose nonlinearity can be choosen in an arbitrary way.
Whatever .the nonlinearity is, if the spatial density belongs to L1, the angular velocity is bounded by a quantity depending on the confining potential. The asymptotic behaviour of the potential at infinity is such that the level curves are (asymptotically) ellipses, provided the potential is at most quadratic at infinity, which is a reasonable assumption (Theorem 3.3). It is also proved that the potential cannot be constant at infinity.
Special solutions have then been constructed (Proposition 3.4: radially symmetric solutions, and Proposition 3.5: ld-solutions) using the ideas of [6]. The most general class of solutions compatible with the natural asymptotic boundary conditions is studied in Theorem 3.7: these solutions are nonisotropic solutions and therefore naturally time-periodic; they may have a finite mass and are explicitely time-dependant, which was not the case for the radially symmetric solutions.
In view of simplifying the computations, the confining potential was supposed to be harmonic. This very special form allows to introduce an other method for giving time-dependant solutions: one can indeed use the invariance by translation of the equation for w (from which the potential is deduced) to get nonsymmetric solutions that will therefore be explicitely timedependant, but this is a very special property of the harmonic potential. However this allows to build a branch of non trivial solutions by perturbing the harmonic potential. On the other side, the nonisotropic solutions given in Theorem 3.7 are independant of the local form of the confining potential provided its asymptotic behaviour is quadratic.
A complete study of general confining potentials has still to be done. The question of the uniqueness of the solutions (up to a translation, and up to the addition of a constant to the potential) and of their stability would be of the greatest interest. This last question may not be out of reach in view of the characterization that has been given for the time-periodic solutions, but probably implies first the removal of the technical assumptions used in this paper, and a new approach for the timedependant Vlasov-Poisson system. It is also probably possible to prove symmetry results, as mentioned in Remark 3.8 (this is related to the questions on the uniqueness).
As a concluding remark, let us note that the Ehlers & Rienstra ansatz and the nonlinear Poisson equation have been (from a mathematical point of view) used in many. mathematical papers for the study of the 3d Vlasov-Poisson gravitational system and that the results given in the paper may easily be adapted to this case.  x R

Ix-2'12 '
Here w represents the asymptotic dispersion (of order -&) of the velocities around wo of the original distribution function, which justifies the word "monokinetic". w is a conserved quantity of the microscopic dynamics and the averaged density function is a solution of the 2d Vlasov-Poisson system It makes sense to ask that f belongs to L m ( R + ; L 1 ( R Z x R Z ) ) , which means that one looks for beams with a finite mass per unit length along the axis, but the behavior in x modelizes the local distribution function and there is therefore no a priori natural boundary condition for U . For the same reason, it is also assumed that the exact form of Uo is not important and that one can take an harmonic potential for modelizing it near its bottom.

Appendix B : two interpolation lemmas
In the two following lemmas, the relations between the norms and the exponents are easily recovered using scalings in x and v . The first lemma can be found for instance in [32,33]. The second one is a generalization of the first lemma to moments higher than one. These lemmas are related to the estimates used by B. Pertharne [36] or R. Illner & G . Rein [31] for the question of the dispersion in dimension three (see concluding remark). Detailed proofs are given here, including the explicit form of the constants which appear in the inequalities. Lemma B.l: ' k t f be a nonnegative function belonghg to Lp(RN x RN) for some belongs to L ' ( R~) for some (r, k) 6 [I, +w[x]O, +w [. Then the function belongs to L'J(RN) with and satisfies When r varies between 1 and +oo, q varies between 1 + and p + 9.
Proof : Assume to simplify that p < +oo. Let p be defined by We can split the integral defining p into two integrals and evaluate these integrals in different ways

DOLBEAULT
If we optimize on R, then we get with The Lq-norm of p is now bounded by and using Holder's inequality, we obtain  The interpolation method can also be used for any moment in (x -vtJ (as in [36], [31] for ) Xv t J 2 ) .
The integration with respect to v of f on { J xvtl < R) instead of the integration off on {Ivl< R ) provides an explicit dependance in t for the optimal R, which gives a decay for the interpolated quantity. The result is (formally) easily recovered by the change of variables v I+ (xv t ) ) .

of the orbit t H ( ( x ( t ) , % ( t ) ) .
(Global result) If r H r 3 ( 2 + 9) is monotone increasing, then the above factorization result is global: We have to notice that ( f , U ) is a solution of the stationary Vlasov-Poisson system (with the same spatial density) as soon as ( f , U ) is a solution such that U is radially symmetric.
For the proof of this proposition and for more realistic conditions on U than the (NR) condition, one has to refer to [14,15]. The weak formulation of Jeans' theorem was given in [8] (for the special class of distribution functions such that f = f in the three-dimensional Vlasov-Poisson casein that case, the result is automatically global

E (~o , v o ) , F ( x o , v~) ) , i = 1,2, then for almost all ( t , x , v ) E [O,T] x supp(f) f satisfies the factorization identity :
The factorization identity is not exactly the same as in the weak Ehlers & Ftienstra ansatz. In part 11, the study has been restricted to the subclass of g which do not depend on t .
Remark : The method applies in the same way when one assumes that for some O0 ~] 0 , 1 [ (but one has then to make some essentially technical modifications on the assumptions on g: see [8], [14] for similar problems in dimension 3). It also applies to the more realistic but also more technical case corresponding to the situation when Bo depends on t (and is timeperiodic of period T = $). The set of timeperiodic solutions we have found can be extended to the case when the angular velocity is equal to zero (infinite period) and provides a new class of stationary solutions which seems not to be known up to now. These solutions are such that the asymptotic behaviour (for large 1x1) of the potential is not isotropic. Among the distribution functions satisfying the Ehlers & Rienstra ansatz, and if we forget about the explicit dependance in t for the factorized distribution function g, the set of stationary solutions (i.e. the set of timedependant solutions of zero angular velocity) is much wider than the set of time-periodic solutions with strictly positive period, even in the limit when w goes to zero. It is not very difficult indeed to prove the existence of stationary solutions for large classes of h that do not satisfy the (strong) Ehlers & Ftienstra ansatz, in the same way as it has been done in [8] or [6] (see also references inside).

Introduction and the statement of results
The singular Yamabe problem can be stated as follows [SY]: Can one find a complete metric ij on the complement of a closed subset C of a compact FAKHI Riemannian manifold ( M , g ) , which is pointwise conformally equivalent to the metric g and whose scalar curvature is constant ?
If one looks for the metric j as 5 = u A g , where u is some positive function on M \ C, then the problem reduces to the existence of positive solutions of the following semilinear elliptic equation which blow up sufficiently fast at any point of C in order to ensure the completeness of the corresponding metric. In the above equation, Rg and Re are the scalar curvature of the metric g and j.
In the special case where M is the unit n-sphere with the canonical metric, one can use the conformal invariance of (2) to show that, when Re # 0, the study of t,his equation is equivalent to the study of one of the following semilinear elliptic equations with p = s. Depending on the sign of the scalar curvature there are two different equations which we will refer to as the "positive" or "negative" case.
Motivated by this geometric considerations, the existence and properties of singular solutions of ( 3 ) have been extensively studied over the past years.
In the negative case, many existence results are known since the pioneer work of Loewner arid Nirenberg [LN] where they have established the existence of McOwen has shown in [Mc] the existence of a P-1 solution to when C is a smooth submanifold without boundary of dimension k > ko and S is a continuous function on M , which satisfies Finn has established [Fnl] the existence of a solution of ( 4 ) which satisfies ( 5 ) when C is a smooth submanifold of dimension k > ko with boundary.
For recent progresses we refer to the work of Finn [Fn2] in which the author has recently narrowed the gap between the known necessary and sufficient conditions for the existence of solutions of ( 4 ) . In this work, Finn shows the existence of a solution of (4) when C possesses at every point a regular tangent cone of dimension greater than ko.
In the positive case, extending previous partial results of Mazzeo and Smale [MS], Pacard IPl], [P2], Rebai [R], Mazzeo and Pacard [MP] have proved the existence of infinitely many positive weak solutions of (1) which are singular on the union of finitely many disjoint smooth submanifolds without boundaries, provided the dimensions of the submanifolds belong to ( n -2 5 , n -3 1 .
In the remaining of the paper, we will focus our attention on the positive case. In this framework and in the light of both the work of Mazzeo and Pacard and the work of Finn, one can ask the following question : Do there exist solutions of (3), if the singular set C has a boundary or is the union of non disjoint submanifolds ?
In this paper we will partially answer this question. Our main result reads Theorem 1. Let R be a regular open bounded subset in Rn, with n 2 4, and let C a smooth k-dimensional submanifold of R with smooth boundary, 1 5 k < n -2. For all p greater than but close to -, there exists infinitely many positive weak solutions of in R \ C with u = 0 on do, whose singular set is C . In addition, for any such solution u , there exists a constant c > 0 such that as x tends to C.
Let us emphasize that, in the Theorem, the constant c does depend on the solution u. We now further comment the restriction " p close to $&".
This assumption is probably not necessary and might be removed by adapting the proof of Mazzeo and Pacard [MP], though this seems far frotn obvious since major technical difficulties have to be overcome. Moreover, with this assumption, one can avoid almost all of the technical difficulties and therefore, make the construction more transparent.
Organization of the paper : In Section 2 we recall some properties of radial singular solutions of (6). In Section 3, using the results from [MP] we define singular solutions of (6) whose singular set is a half line. In Section 4, we give the definition of appropriate spaces and describe the mapping properties of the Laplacian operator in these spaces, furthermore we construct approximate solutions and we obtain the existence of the desired solutions by using a fixed point argument.

FAKHI 2 Solution with one isolated singularity
In this section, we assume that N 2 3 and we recall some well known results concerning the existence and properties of radial, positive, singular solutions of We will restrict our attention to the case where p > A, since otherwise singular solutions do not extend to weak solutions in all R N . In the remaining Proof: The proof of this result is in [MP]. 0 We will also need the following estimates: R e m a r k : Let us notice that b(i) = -8, (+uO(i/t)) .

IE=l
Proof : The fact that uo(2) 5 cp 1 3 1 -3 when p is close to is proven in the work of Chen and Lin [CL]. The bound for 4 is proven in [MP]. This special form of singular solutions already appears in the negative case, in the recent work of Finn [Fn2]. If one can find a solution of this form, then the singular set of u will be given by Do = {reo : r 2 0) c RN+'. A simple computation shows that the equation Up to a rigid motion, we can assume that Qo is the north pole of SN and that, the upper hemisphere of SN is parameterized near 60 by In the remaining, 7r will denote the orthogonal projection from the upper We now define the weighted spaces we will work with in this section We should keep in mind that, for all w E C:(SN \ {00)) there exists a constant c > 0 such that lw(6)J 5 c lx(6)lW near 00. Now we set cP = N -P-1 ( p-1) and we investigate the mapping properties of the operator As,v -Ep, when acting on the above defined weighted spaces. Proof: We will denote for short p(0) = dist (0, do). To begin with, let us remark that in the coordinates defined above, the operator A,N has the following form in the neighborhood of do. For a proof of this expansion, we refer t o [FM]. Hence, we obtain where yl = inf(2v -2, N -5) > 0.

P-1
Proof : The existence part of the Theorem follows for example from [MP]. But since the estimates (12) which are not given by the result of [MP], are a key point of our proof of Theoiem 1, we felt the necessity to give a short proof of the Theorem 2. The idea is first to construct an approximate solution of (9) and then to perturb it.
To construct an approximate solution of (9), we proceed as follows. Let uo be the function defined in Proposition 1. Since the (7) is invariant under dilation, for all E > 0 the function u, defined by is also a solution of (7). Now, fix a parameter 6 > 0 small enough and choose a cut-off function q, defined on SN, such that q(0) = 0 if p(0) 2 2b and ~( 0 ) and we set f, = Asiv6, -G, + Gf. We make use of the expansions given in Proposition 1 as well as of the one in (11) in order to conclude that there exists a constant c > 0 such that, for all E E (0, I), we have Now, using a perturbation argument, we will construct an exact solution of (9) which is close to 6,.
We have to find a function v such that w = G, + v satisfies in SN \ (9,). Let us rewrite this equation as where f, is the error term, which has already been defined, and where q,(v) = 1656 FAKHI 1 6 , + vIP -6:. In order to solve (14), it is enough to find a fixed point for the nonlinear mapping in the space C:(SN \ {OO)).
Let us now fix E > 0 and, for all K > 0, we define where t,he constant yl is the one which has been defined before.

Now, Taylor's formula yields
In particular, there exists some constant c > 0 such that This is at this stage that the assumption p close to A, which we have not used so far, simplifies the proof of [MP]. Indeed, v being fixed greater than 2 -N, we have v > 3 for p close enough t,o A. Say p < p. The fact that v > 5 will be used in many estimates. For example we get readily the following estimate since v(p -1) > -2. Therefore, using (8) in proposition 2 we get Let us emphasize that a t this stage that we need the operator Gp t o be bounded uniformly on p but may be depending on v. This is why, instead of making the assumption v > ,,+, we have fixed v in the interval (2 -N, inf(0,4 -N)). Now let's C1 be an upper bound of Gp for p E (A, E), then we get from (15) Notice that c,, tends to 0 as p tends to A. Thus for p close enough to A , say p E (A, p) there exists KO > 0 and EO > 0 such that for all E E (0, EO] , NE is a contraction mapping from the ball &,, into itself and therefore, N, has a unique fixed point v, in this set. The fact that w, = 6, + v, is then a weak solution of (9) is standard (see for example the end of section 4.4).
This ends the proof of the existence of w, as well as (12) for k = 0. The estimate of (12) for k # 0 then follows from Schauder's estimates.
As an immediate Corollary, we get

Singular solutions with a submanifold with boundary as singular set
Given our preliminary results, we are now in position to give the proof of Theorem 1. We will pay a special attention to the novel features and to the most important points of the proof : the derivation of the estimates which ensure that the approximate solution is good enough to be perturbed into a solution of our problem and the mapping properties of the Laplacian when define between suitably chosen weighted spaces. These are the main contributions of our paper.

Notations and different coordinate systems
We set n = N + k.
From now on, C is assumed to be a closed smooth submanifold of R c Wn with smooth boundary dC. It is always possible to find C an open (at least C4) smooth submanifold of R c Wn which is an extension of C in the sense that the closure of C is included in 9.
For all b > 0 small enough, will denote a tubular neighborhood of radius 6 around C and $ a tubular neighborhood of radius b around 2. It is easy to see that, provided b > 0 is small enough, the set can be decomposed into two subsets, one of which is a disk bundle over C and will be denoted by ?;P, the other being a half of a tubular neighborhood of radius 6 around dC and will be is denoted Ba. A similar decomposition holds for $. We will finally assume that b is chosen small enough so that 5 c q. We define

Mapping properties of the Laplacian
Granted these definitions, we now define the weighted spaces we will work with, in this section.
for which the norm is finite.
In the following Proposition we study the mapping defined in the weighted space V;(R \ C).
properties of A when Now, let f E D:-,(R\C) be given. Since we have assumed that p > 2 -N , it is easy to see that f belongs to L1 (R). Moreover there exists c > 0 such that 139 f) In particular, this already guarantees the existence of w E i7qE[l,n/(n-1)) W ( ) solution of (18). Moreover, Green's representation formula (see for example [GT]) yields the existence of a constant c > 0 independent of f such that for all x E R such that dc(x) 2 6. In fact, we have that FAKHI where w, is the volume of the unit n-sphere. If zo E R\Q, the above estimate allows us to conclude that In particular lw(x)l is bounded by a constant times If lpW2 on d?.
Using the preliminary computation and the fact that the operator A satisfies the maximum principle, we see that, v = cpl f lp-2P' with c, = -> 0 is a supersolution for (18) in q. Similarly, -v is a subsolution for (18) in the same set. Therefore we conclude that Iw(z)l 5 cp I f l p-2 Pj' in Q . Combining this together with the first estimate, we obtain for some constant c independent of f and for all x E R \ Ba. Which, in particular implies that on 8Bb.
Finally, in Ba we use +multiplied by some suitable constant, more exactly 6 = E, l f l,-z~p as a supersolution for (18) and -6 as a subsolution to conclude that in Bs. Here

Construction of the approximate solutions
In this section, we construct a sequence of approximate solutions S, and derive some important estimates. To this aim, we will use both u, the radial singular solut,ion of A w~u + U P = 0 in RN \ (0) which has hccn defined in (13) and a, the sequence of solutions of Aw~+lu + U P = 0 in RN+' \ Do which has been defined in Corollary 1.
Throughout this section, q will denote a smooth cutoff function defined in If%, such that SEMILINEAR ELLIPTIC EQUATIONS and We will also need a smooth positive function a : C -R which is equal to the geodesic distance from jj E C to dC in a tubular neighborhood of radius 26 around dC and which is greated than b away from this neighborhood. This function a can be extended to 2 by setting for all g E 9 \ C, a to be equal to minus the geodesic distance from Q to dC. The remaining of the section is devoted to the careful derivation of estimates for AS, + S ! in wich we will use the inequality u > without explicit mention. For example in 11-2-3 and in 11-3 specially in the estimate of (111) and (IV).
Proposition 5. Assume that p < i n f ( s , u). Then there exists a constant c > 0 such that Let us before giving the proof, comment our choice of the parameter p. The main reason for choosing an other parameter than v, say p is that on one hand we need that 2 -p > 0 in the estimate of the error term /AS, + S:l,-2 but on an other hand the parameter v was be chosen, as we have seen, independent of p.

FAKHI
Proof: Let f, = AS, +SF. Since S, G 0 outside 5 , we may assume from now on that x E z.
Case I -First of all, let us assume that 6 / 2 < dc(x) 5 6. In this case, it follows directly from the second estimate of (12) as well as from Proposition 1, which provides the asymptotic behavior of uo near m, that Therefore, we always have when 612 < dc(x) < S and where c is some constant depending only on 6.
Case I1 -Next, we assume that dc(x) 5 6/2. In this case let us further distinguish three subcases, according to the value of a @ ) .
In order to estimate properly this expression, three subcases have to be distinguished. For convenience, we define With this definition, we see that Then we obtain from the first estimate in (12) [el . v2sEI + le2 . VS,I 5 cp-' 7 -3 + ~7 ' Therefore, we obtain (27) .?3 -P (~E )~-~~A s , in this case.
Case 11.3 -Let us finally assume that a @ ) E (6/2,6). In this case SE = 77 S,' + ( 1 -7 ) S:. We may now compute In this expression, the quantities (I) and (11) can be est'imated as before and one obtains It remains to estimate (111) and (IV). To this aim, we need some information concerning S,' -S:. Notice already that thanks to (12).
The estimates for (ii) follow simply from (12) and, without any difficulty, one finds that where the function C$ has been defined in Proposition 2. Using the estimates of Proposition 2 as well as the fact that if F / E < 1, while Thus, we conclude that

The fixed point argument and the proof of the main Theorem
For the time being, let us prove the existence of v E V i ( R \ C) solution of where f , = AS, + Sr, is the quantity which has been estimated in the previous section and where Q, = IS, +VIP -Sr. As in Section 3, the existence of v will be a consequence of the contraction mapping argument.
Let now define for all K > 0 In order to obtain a solution of (31) it is enough to find a fixed point for the nonlinear mapping We need the following Let us emphasize that the restriction p < < is necessary in our method since we use the estimation of the error AS, +Sf given in which itself uses the assumption p < <. Furthermore, as we will see in a while that the condition In particular, there exists some constant c > 0 such that It is then a simple exercise to check, using the fact that JS,IP-'(2) 5 c$-' ( d = (~) ) -~ for E small enough, and p > 3 that The conclusion follows a t once from the fact that the constant c, tends to 0 as p tends to A.
In order to complete the proof of Theorem 1, it remains to prove that U,  Now, assume that Ck1+k2 is a closed kl + k2-dimensional submanifold of Rn ( with n = N + kz), and assume that the boundary of Ckl+k2 is locally given by Ckl+' xlWk2-' (up to a diffeomorphism), can one find a solution of ARnu+up = 0 whose singular set is Cklfk2?
Even if we restrict our attention to p close to N!iF:,, the linear analysis of Section 4.2 is missing.

ACKNOWLEDGMENTS
It is my pleasure to express here my deep gratitude to my thesis advisor, Professor F. Pacard, for suggesting this problem and for his patient guidance and several helpful discussions for improvement.

INTRODUCTION
There has been considerable progress during the last ten years in the mathematical study of long-time existence properties of solutions of geometricallybased classical field theories. A significant portion of this work has focussed on the study of what are called wave maps in the mathematics literature and nonlinear sigma models (or, in certain special cases, chiral models) in the physics literature. These are defined as maps II, from a Lorentzian geometry (Mm+', q), e.g. Minkowski space, to a Riemannian geometry (Nn,g), e.g. a symmetric space or a compact Lie group, with $ being a critical point for the functional ' and hence satisfying the wave map equation here V is the (torsion-free) derivative operator determined by the metric 7, and I? are the (torsion-free) connection coefficients compatible with g.
Wave maps have a well-posed Cauchy problem, and it is known that for 1+1 dimensional base geometries (MI+', q), every choice of smooth initial data evolves into a global smooth solution [l, 2,3], while for 3+1 (or higher) dimensional base geometries, certain smooth initial data leads to solutions with singularities [3,4]. Not yet understood is what happens in general for 2+1 dimensional base geometries. This is the "critical dimension" (see [ 5 , 6 ] ) , where global smooth solutions are expected, at least, for all smooth initial data of sufficiently small energy. While global existence results are known to hold for certain classes of rotationally-symmetric wave maps in 2+1 dimensions (without restrictions on the energy) [7,4], not much is known otherwise for critical wave maps [ 6 ] .
An interesting modification to the wave map equations can be obtained by adding torsion.
This can be done in 2+1 dimensions, without adding extra dynamical fields, as follows. One fixes a pair of background fields: a closed one-form field v on the base manifold M2+' and a non-closed two-form field p on the target Nn. The field p serves as a "torsion potential" in the sense that the torsion tensor on N n is defined as where X is a coupling constant and c is the 2+1 volume tensor normalized with respect to q. The torsion wave map equation obtained from (4) is given by where are the connection coefficients compatible with g, with torsion Q. Note that the effect of the torsion is to add the nonlinear term to the wave map equation (2).
Wave maps without torsion have a conserved, symmetric stress-energy tensor [5]  The critical dimension for torsion wave maps, just as for standard wave maps, is 2+1. While we do not attempt here to investigate the general class of critical torsion wave maps, we are able to prove global existence for various reductions of critical wave maps, with and without torsion, where the base geometry is Minkowski space. These reductions are defined by the invariance or equivariance of the wave map II, under a one-dimensional group of translations acting on M2+ '. More specifically, choose Cartesian coordinates (x, y, t) for (M2+', 7) and denote the translation group action by In each case the 2+1 wave map equation for 111 yields a I f 1 reduced equation for 4, a L , @R, @c, respectively.
We establish global existence of solutions to the Cauchy problem for the class of translation-invariant wave maps with torsion in Section 2. While the proof for these wave maps is very similar to that for 1 + 1 wave maps with 110 torsion, the torsion terms do introduce some subtleties into the analysis, which we highlight.
In order to prove global existence of solutions to the Cauchy problem for the three classes of translation equivariant wave maps with torsion, we find it useful to work with a frame formulation for 2 + 1 wave maps . In Section 3 we introduce the frame formulation for general targets and then proceed to relate wave map equivariance for Lie group targets to frame invariance and equivariance. In particular, our global existence theorems for equivariant wave maps have a natural formulation and proof using frames.
The proof for the left equivariant, right equivariant, and conjugate equivariant wave maps with torsion is fairly similar in each case. We focus on the left equivariant case (which corresponds to invariant frames) and carry out the global existence proof in detail, in Section 4. We then briefly note in Section 5 the differences entailed in proving global existence for the other two cases. We make a few concluding remarks in Section 6.

INVARIANT WAVE M A P S WITH T O R S I O N
The translation invariance condition (10) is characterized by the wave map functions (15) being independent of y. Under this reduction the torsion wave map equation (5) becomes where is the 1 + 1 Minkowski metric (a, P run over x and t ) and cap is the 1 + 1 Levi-Civita tensor. We hereafter take vy to be constant, but we make no further restrictions: The target (Nn, g ) can be any Riemannian geometry, and the torsion potential p can be any non-closed two form on Nn.
Interestingly, while the torsion term appears in a nontrivial way in the reduced wave map equation (19), and while the stress-energy tensor (8) generally contains a torsion term, for translation invariant wave maps the torsion drops out of many of the stress-energy tensor components. We have all of which contain no torsion, along with Note that v, and vt do not appear in the reduced wave map equation (19); setting them to zero does not affect (19), but it does result in T,, and Tyt vanishing.
We now consider the Cauchy problem for translation invariant wave maps (15) with torsion. Initial data at t = to consists of a pair of maps (here C = R1 or S1 allowing for periodic boundary conditions). A solution to the Cauchy problem is then a map : C x R1 --. M~+' -+ N which satisfies (19) along with the initial conditions Note that there are no constraints on the choice of initial data (4,8). Global existence of initial value solutions is established by the following theorem.

Theorem 1.
For any smooth compact support initial data, the Cauchy problem (19) and (29) has a unique smooth global solution q5(x,t) for all t E R'.
Proof: The PDE system (19) is manifestly hyperbolic; hence, local existence and uniqueness are immediate [9]. To prove global existence, it is sufficient (by the usual open-closed arguments [9]) to show that if $(x, t) satisfies (19) on C x I, with I a bounded open interval in R1, then 4(x,t) and all its derivatives are bounded on C x I.
To show that and its first derivatives da4 are bounded, we use an argument based on stress-energy conservation (see 131). From the form of the stress-energy components (21) to (271, together with the conservation equations we find that It then follows from standard results (see [6]) for the wave equation on 1 + 1 Minkowski space that Ttt is bounded on I. Thus the first derivatives of 4 are bounded. As a consequence of the mean value theorem and the assumed compact support of the initial data, 4 is then bounded as well.
There are a number of ways of proceeding to argue that second and higher order derivatives of I#J are bounded. Here we use an argument which 34 is compactly supported if it is constant everywhere outside a compact region in C; 6 is compactly supported if it zero outside such a region.

1676
ANCO AND ISENBERG is adapted from Shatah [6] based on bounding successive kth order energies Note that the ordinary energy is bounded and independent of t , Eo(t) = fo(to), for smooth compact support initial data.
We start by rewriting the torsion wave map equation (19) in the form where V : = 8,4A, Do = rQpDp, and Do = ap + rA~cvg + XQABC~pa~: defines a covariant derivative operator which includes the connection with torsion. If we now apply Dp to equation (34) and commute Dp past the derivative operators, keeping track of the various curvature and torsion terms which arise, then we obtain a nonlinear wave equation for V:: where P(V, V, V) denotes an expression which is trilinear in VpA and involves no higher derivatives of @". By multiplying (35) by r o P g~~~t~~, we straightforwardly derive the conservation equation I;), trilinear in V with no higher derivatives of 4". Now, integrating (36) cver C, we obtain for the 1st order energy defined in (32). Estimating the right hand side of (37), we find and hence It follows from Sobolev inequalities that which bounds El(t), and therefore bounds the L2 norm of DV. Hence Ild;$llLz is bounded.
To bound E2(t), we start from the wave equation ( Hence we have that E2(t) is bounded and therefore so is the L2 norm of DW. Thus, since DW = DDV, it follows that ll@)#~11~z is bounded.
The argument proceeds to all successively higher orders and we thereby determine that all derivatives of 4 are L2 bounded. It follows from Sobolev embedding that all derivatives of 4 are pointwise bounded, which completes the proof of Theorem 1. a

AND THE FRAME FORMULATION
We begin by setting up a frame formulation for wave maps with and without torsion. (See also [7]). We first choose a frame basis {et)(a = 1, . . . , n) for the target geometry (Nn, g) and let e"A(11) denote the frame associated to +.
We now define the "frame fields" where {e", are the components of the dual basis to {e!). These frame fields Up to this point in setting up the frame formulation, we have made no restrictions on the choice of the target or on the nature of the wave maps. We now focus on equivariant wave maps (12) to (14) and their corresponding frame formulations, so we assume the target geometry to be a Lie group G. While K can be defined for any frame basis on G, the frame field equations are simplest if we require that {et) be a left-invariant basis for G. It then follows that the commutator coefficients Cbca are independent of + and are constant. If we make the further restrictions that the metric g be a leftinvariant tensor on G, GLOBAL EXISTENCE FOR WAVE MAPS 1679 and the torsion potential p be a left invariant two-form on G, so that the components gab and p, b are constant, then the coefficients Cak($) are independent of 11, and constant while so are the frame components @a($) as well; in particular, we have with Remark 1: Every nonabelian Lie group admits both a left-invariant metric g and a left-invariant two-form p. However, for semi-simple Lie groups G, if G has dimension three then all left-invariant tweforms p are necessarily closed, and consequently Q = 0 so there is no torsion. This is not the case if G has larger dimension. In particular, a non-closed left-invariant two-form p and hence non-zero torsion Q is admitted by all nonabelian semi-simple Lie groups G other than the three-dimensional ones (namely SU(2) and its real forms S0(3), S0(1,2), SO (2,l)). See Proposition A in the appendix.
We now find that, assuming the restrictions just noted, we can write equations (48), (50) and (51) strictly in terms of the frame fields K , with no explicit 11, dependence: The field equations (56) and(57) together are a self-contained PDE system for K which is equivalent to the wave map equation (2); the field equations (56) and(58) likewise are a self-contained PDE system for K which is equivalent to the wave map equation with torsion (5). Note that the system with torsion reduces to the system without torsion when X = 0.

Suppose that K," is a solution of the field equations (56) and (58). If
M2+' ' is simply connected, then there exists a torsion wave map qA, satisfying equation (5), which is related to K," by (47).
Proof: To prove part (I), we first note that for Ka, given by (47), the field equation (56) is an identity. We then verify that, through the torsion wave map equation (5), the substitution of (47) for K," satisfies the field equation For the converse, to prove part (2), we note the field equation (56)  We note that, independent of their usefulness for the study of wave maps, these field theories in terms of K viewed as a Lie-algebra valued one-form field on M2+' have some interest as a nonlinear generalization of Maxwell's equations. Indeed, for the abelian case CabC = 0, the field equations (56) and (57)  Since we will use frame fields to study translation equivariant wave maps, we now characterize frame fields which correspond to the three classes of equivariant wave maps (12), (13), (14). We begin with the following definitions of invariant and equivariant frame fields under a translation group action. Invariant Frame Field: Equivariant Frame Field: Here A is an element of the Lie algebra of the target Lie group G , and (x, y, t ) are standard coordinates for the Minkowski space base geometry (M2+', 7).
Geometrically, the translation equivariant group action (62) on K arises via the pull-back of the dual frame components { e l ) under right multiplication in G by the one-parameter exponential subgroup generated from the Lie algebra element R. When R = 0 this group action reduces to the translation invariant group action (61) on K. (Alternatively, note that the translation invariant group action arises directly by left multiplication in G since the dual frame is left-invariant.) Based on Proposition 1, the correspondence between invariantlequivariant frame fields and wave maps is summarized by the following two results. Proposition 2.

If $ is conjugate equivariant (Id), then the corresponding frame field K is equivariant (62).
The proof of these correspondences amounts to a direct calculation using a matrix representation for 11, and K. There are straightforward converse correspondences as well.

If K is equivariant (62) with the components K, O constant, then the corresponding wave map 11, is right equivariant (13).
Proof: Let U denote a the matrix representation of the wave map 11, cor-responding to K. We first prove part (1). It follows from the definition of frame field invariance, together with relation (59), that U satisfies Integrating the y component of this equation, and then multiplying both sides by U, we obtain the linear matrix ordinary differential equation where f is an arbitrary Lie-algebra matrix valued function (independent of y). The general solution to (64) is where V is an arbitrary Lie-algebra nonsingular matrix valued function (independent of y), and A : We now impose the x, t components of equation (63). Calculating U-'dtU with U from (65), we find which implies that Similarly, working with U-'8,U and imposing 0 = a,(U-'a,U) we determine that Thus A must be a constant Lie-algebra valued matrix, which we denote L; then (65) becomes Condition (12) immediately follows, so the wave map corresponding to K is left equivariant. We now prove part (2). From the definition of frame equivariance, there is a y-independent Lie-algebra matrix valued field f,(x,t) and a constant Lie-algebra matrix A such that

1683
Hence, from relation (59), ,U(x, y, t) must satisfy The y-component of this equation yields and after some manipulation we obtain the linear matrix ODE where V is an arbitrary Lie-algebra matrix valued function, and B is defined as Working with the other components of equation (72) we derive and Then rearranging (77) and using (75), we obtain Since both ft and V are independent of y, if we take dy of both sides of equation (79) we have which implies that a t B = 0. Similarly, using (78), we find that &B = 0. Hence B is a constant Lie-algebra matrix, which we denote L. Thus, after combining (75) with the definition W = U exp(-yA), we see that Finally, we prove part (3). From (81) we have (82) which is assumed to be constant. By differentiating with respect to y, we obtain [V-'LV, A] = 0, and hence (81) becomes K, = V-'LV + A. Thus, it follows that B := V-'LV defines a constant Lie-algebra matrix which commutes with A. We then have As a consequence of Propositions 2 and 3, we can prove global existence of solutions to the Cauchy problem for the three classes of translation equivariant wave maps (with or without torsion) by using invariant or equivariant frame fields. We do this first for the invariant frame fields in the next section.
Our analysis makes essential use of the wave map stress-energy tensor (8). Through the relation (47) for K in terms of $J, we obtain Tp, = f l Y~z~: g , b One verifies that, for solutions K of (56) and (58) in which ( M 2 + ' ,~) is Minkowski space, this non-symmetric stress-energy tensor satisfies the conservation equation Hereafter we specialize to the situation where v is constant on M2+'. This makes the analysis of the field equations considerably simpler. In particular, the stress-energy is strictly conserved, apTi', = 0.

EQUATIONS WITH TORSION
By definition (61) of translation invariance for frame fields, the component functions K," are independent of y. Then, adopting the convenient notation we find that the translation-invariant frame field equations take the form for the functions {Ea(x, t), Ha(x, t), Ba(x, t)). Note that, in this system of field equations, (87) is a constraint equation while (88) to (90) are evolution equations.
Initial data at t = to for the Cauchy problem is specified by choosing (on Ba(x, t)) satisfying (88) to (90) and the initial conditions To show that the Cauchy problem is well-posed, we note that well-posedness is known for the wave map equation without torsion [6], which is equivalent to the system (87) to (90) up to the addition of the torsion terms involving A. These terms do not involve any derivatives of the fields and hence do not effect the well-posedness. Alternatively, we note that, up to such terms, the system is equivalent to the Maxwell equations in 2+1 dimensions, which constitute a well-posed system. It follows that the system (87) to (90) is well-posed and, moreover, is first-order hyperbolic.
In this section we prove global existence of smooth solutions to the Cauchy problem for the 1+1 field equations (87) to (90). The proof relies on the use of the stress-energy tensor (84) along with light cone estimates.
To proceed we write out the components of the stressenergy tensor (8) in terms of Ea, Ha, Ba. Using the coordinates (x, y, t) for M2+' we have where E2 := EaEbgab and E . B = E"Bbgab, etc..
For derivation of light cone estimates, it is useful to work with null components of the stress-energy tensor. We introduce null coordinates which mix t and x (but not y): Then we find (for the components we will need): T e e = 1(,2 + XU! Ha ~;p,b For these components the stress-energy conservation equation (85) has the null component form These equations are essential for the derivation of the light cone estimates we will need. Also important for our analysis is the energy function We note that for certain values of the coupling constant A, the energy E(t) can be negative, and it therefore does not in general control the L2 norm of Ea, Ha, or Ba. However, for sufficiently small A, there is a constant k > 0 such that and hence the energy is positive, so that E(t) does consequentIy control 11 E11p,11 H1ILz , and 11 B~( L Z .
We assume henceforth that X is sufficiently small for this to be the case.
We now state our main results. Let C denote R1 or S1, and introduce coordinates (x, t ) for C x R1 2~ M2+l. Fix constants vt, v,, v,. Let G be a Lie group with CbCa denoting the Lie-algebra commutator structure tensor. Fix on the Lie algebra of G a positive definite metric tensor gab (it need not necessarily be compatible with the commutator) and a skew-tensor pa&. Let QDbc be the tensor defined by (54).

Theorem 2.
Let X be a small c~n s t a n t .~ For any smooth compact support initial data (92) satisfying (91), the Cauchy problem (87) to (90) has a unique smooth global solution {Ea(x, t), Ha(x, t), Ba(x, t)) for all t E R1 Combining this result with Propositions 2 and 3 from Section 3, we have a corresponding result for wave maps.
Theorem 3. The Cauchy problem for left-translation equivan'ant Lie group wave maps (I!?), with or without torsion, has a unique smooth global solution for all smooth compact support initial data.

Proof of Theorem 2:
Local existence and uniqueness of smooth solutions of the PDE system (87) to (90) follows from standard results (see, for example, [9]) for first-order hyperbolic systems in 1+1 dimensions. In order to prove global existence, it is sufficient by the usual "open-closed" arguments [9] to establish the following: is the flux. If we are working on C = S1, then aC is empty, so 3 ( t ) = 0. If instead C = R1, then we note that as a consequence of hyperbolicity of the system (87) to (go), the fields {Ea, Ha, Ba) have compact support on C for all t E I , and hence F ( t ) = 0. Thus, the energy is conserved, for all t E I .
As we noted earlier, the energy controls the L2 norm of the fields {Ea, Ha, Ba), so long as A is sufficiently small (as assumed in the theorem). Hence we have for some constant k (depending on &(to)), for all t E I.
S t e p 2: Bounded H" In the system (87) to (go), the field Ha enters in a different way from Ea and B", since the evolution equation (89) for Ha involves no spatial derivative terms, and the constraint equation (87) (1 12) and (1 13) together with the bounds (111) that

GLOBAL EXISTENCE FOR WAVE MAPS
for a constant k2. Combining (114) with the mean value theorem, we obtain controls on the spatial variation of E,(x, t) for any fixed time t. In particular, for any xl, x2 E C with fixed t, we have If we are working on C = R1, we can choose xl outside the support of Ha(x, t) for all t E I , and therefore it follows from (115) that IHa( If instead we are working on C = S', we need to do more to bound Ha(x, t). Consider Jsl Ha(x, t)dx, which is the spatial average of H a on S1. From the fundamental theorem of calculus, and from the evolution equation (89), we obtain (for t E I ) Next, using standard quadratic algebraic inequalities, we note that Jsl CkaBbHcdx is bounded in terms of the energy, for some constant k3. Hence, L : Jsl CbcaBb(x, s)HC(x, s)dxds is bounded above and below, I 1; C&OB~(X, S)H'(X, s)dxdsl 6 (tto)k3&(t0) 6 4 (118) for some constant k4, for all t E I. Then since Jsl Ha(x, t0)dx involves initial data only, it also is bounded above and below. Therefore, from (116) we have that (119) and so the average of Ha over S' is bounded above and below, for all t E I.

ANCO AND ISENBERG
Combining this result with the spatial variance control (115), we conclude that Ha(z, t) is bounded (above and below) on C x I.

Step 3: Bounded Ea and Ba
While standard 1+1 light cone arguments do not directly apply to the system (87) to (go), a modified argument can be used with the pointwise bounds on Ha achieved in Step 2.
Using the null form of the stress-energy conservation laws (106)-(107), along with the expressions (102)-(105) for the stress-energy components, we have We use the field equations (87) to (90) to remove all of the derivatives which appear on the right-side of these equations. Thus It is convenient here to let aa := Ba + Ea and p" := -Ba + Ea, and so we have Since Cak, Q h , A, vt and v, are constant, and since Ha is bounded on C x I, we immediately have the following estimates for the right-sides of (124) and (125): with some constants ks, k7, kg, and kg.
We now apply a light cone argument to the differential inequalities (126) and ( Similarly, from (129), we derive We want to show a(t) and b(t) are bounded functions of t by applying a Gronwall type argument to the coupled inequalities (133),(134). It is useful first to divide by ~' /~( t ) in (133) and by b1/2(t) in (134), yielding We estimate the term a ( t~) a -' /~( t ) by using the fact that a(t) is a monotonic increasing function of t , due to positivity of G2 in (131). Thus, a(to)a-'I2(t) is bounded by to). In addition, we note the term klo(t is bounded since t E I is bounded. We thereby obtain Similarly, we obtain Adding (137) and (138)' and defining c(t) := a1I2(t) + b1I2(t), we derive Gronwall's inequality immediately applies to (139), and so we determine that c ( t ) is bounded for all t E I. Then a1I2(t) and b1/2(t), which are positive, are bounded.
Returning to the inequalities (133)-(134), it follows that $ a ( t ) and $b(t) are each bounded. Hence, from the definitions of a and b, we obtain that supc a2 and supc P2 are bounded for all t E I. Since a2 = (Ba + Ea)' and P2 = (-Ba + Ea)', we conclude that Ba(x, t ) and Ea(x, t) are bounded on C x I .

Step 4: Bounded Derivatives
Now that we have determined that Ea, Ha, and Ba are bounded on C x I, we proceed to show that the first derivatives of these functions, and subsequently all higher order derivatives, are bounded on C x I.
We start with Ha. From (87)  For Ea and Ba, we use light cone arguments much like step 3, but involving a "derivative stress-energy" tensor. Specifically, let as defined analogously to the stress-energy components (102) where Yl and Y2 are homogeneous quadratic in &E and &B, with bounded coefficients. Although (144) and (145) are not strict conservation equations, we can nevertheless proceed similarly to step 3.

ANCO AND ISENBERG
Through use of the field equations, we can express (144)  with and p2 of the same nature as Yl and Y2. It then follows from standard algebraic inequalities that We now apply the light cone arguments of step 3 to the differential inequalities (148) and (149)

FRAME FIELD EQUATIONS WITH TORSION
As discussed in Section 3, while left-equivariant wave maps (12) correspond to invariant frame fields (61), conjugate-equivariant wave maps (14) and rightequivariant wave maps (13) correspond to equivariant frame fields (62). In this section we show that global existence holds for smooth solutions to the Cauchy problem for translation equivariant frame fields.
We first note that by definition of translation equivariance, K,"(x, y, t) can be expressed as in terms of some Lie-algebra valued fields {Ea(x, t), Ha(x, t), Ba(x, t)) which do not depend on y. Here R is a fixed (constant) elenlent in the Lie algebra; the left multiplication by exp(-yR) combined with right multiplication by exp(yR) denotes the adjoint action of a one-parameter Lie subgroup on the Lie algebra.
Substituting expressions (151) to (153) into the frame field equations with torsion (56) and (58) on Minkowski space, we obtain the following 1+1 reduced PDE system provided that Ca& and Qak are invariant under the adjoint action of exp(yR).
We note that the only difference between these equations for translation equivariant frame fields and equations (87) to (90) for translation invariant frame fields is the presence of the commutator terms involving R.
While the expressions for the field equations are changed somewhat in passing from invariant to equivariant frame fields, the expressions for the stress-energy components (93) to (100)  The global existence result, and its corollary, are stated as follows. Let C denote R1 or S1, and introduce coordinates (2, t) for C x R1. Fix constants vt,v,,v,. Let G be a Lie group and let Ra be a fixed (constant) vector in the Lie algebra of G. Assume G admits on its Lie algebra a positive definite metric tensor gab and a skew tensor pab which are each invariant under the adjoint action of the Lie subgroup generated by Ra: where CbCa denotes the Lie-algebra commutator structure tensor. Let QObc be the tensor defined by (54).
Theorem 4. Let X be a small ~o n s t a n t .~ For any smooth compact support initial data (92) satisfying (158), the Cauchy problem (154) to (157) has a unique smooth global solution {Ea(z, t), Ha(x, t), Ba(x, t)) for all t E R1.
&om Propositions 2 and 3 we obtain a corresponding result for wave maps.
Theorem 5. The Cauchy problems for conjugate-translation equivariant wave maps (14) and for right-translation equivariant wave maps (13), with or without torsion, have unique smooth global solutions for all smooth compact support initial data.

Remark 2:
Under the translation invariant form (61) for frame fields, which corresponds to left-translation equivariant (12) or translation invariant (10) wave maps, the reduction of the frame field equations and corresponding wave map equation is consistent for any Lie group target. However, this is not the case under the translation equivariant form (62) for frame fields, which corresponds to conjugate-translation equivariant (14) or right-translation equivariant (12) wave maps. The translation equivariance ansatz gives a consistent reduction of the frame field equations and corresponding wave map equation only if the target geometry (GI g,p) is invariant under right multiplication by the translation group generated by the Lie algebra element R appearing in (12) to (14) for wave maps and (62) for frame fields. We refer to this condition, given by (159) and (160), as translation invariance of the target. As shown in Proposition A in the appendix, every compact semi-simple Lie group G admits a translation invariant geometry (G,g,p), except that the dimension of G must be greater than three to support a non-zero torsion Q (see Remark 1).

Proof of Theorem 4:
The proof of Theorem 4 is very similar to that of Theorem 2. We summarize the differences (if any) in each step.

S t e p 1: Conserved Energy
Since the expression for the energy is unchanged and since it is conserved, there are no changes in obtaining L2 bounds for E4(x, t), Ha(x, t), B4(x, t).
S t e p 2: Bounded H4 Instead of (112), we have The first of the two terms on the right hand side of (161) may be handled as in (113). As for the second term, we have where the second inequality uses the compact support of Eb together with the Holder inequality, and the last inequality follows from the L2 bound on E4. Hence we obtain l a ,~~l d x 5 kzr (163) analogous to (114).
If C = R', the argument leading to a pointwise bound on H4(x, t) for t E I proceeds exactly as in the proof of Theorem 2. If C = S1, then we need to modify the argument which begins with (116). We have The term J,' , Jsl C4aBb(x, s)R"dxds can be bounded from above using the same quadratic inequality that is used in (162), and so we obtain 1698 ANCO AND ISENBERG analogous to (119). The argument for pointwise bounds on Ha(x,t) for C = S1 can then be completed as in the proof of Theorem 2.
Step 3: Bounded Ea and Ba From inequalities (126) and (127) onward, the light-cone arguments used to bound Ea(x,t) and Ba (x, t) in the proof of Theorem 2 work identically to bound Ea(x, t ) and Ba(x, t) here. To arrive at (126) and (127) we use the following equations, analogous to (122) and (123), Adopting the notation a " := Ba + Ea, ,P := -Ba + Ea, these equations becorne Then, noting that Caa, Qabc, A, vt, vz and R are constant, and recalling that Ha is bounded on C x I, we obtain and a t p 2 5 bl@ + k 3 2 G @ (I71] which are identical to (126) and (127).

S t e p 4: Bounded Derivatives
One can see in Step 2 and Step 3 that the presence of the commutator terms involving R in the field equations (154) to (157) changes little in the arguments for boundedness, since these extra terms are easily controlled by the analogous quadratic terms appearing in the equations. The same holds true for Step 4. We can define the derivative stress-energy components just as in (140) to (143) and then obtain conservation equations similar to (144) to (145), with small modifications in the expressions Yl and Y2 which appear there. These modifications are readily handled in deriving the estimate (148) and (149). The rest of the argument proceeds unchanged.
Hence we obtain global existence. I

C O N C L U D I N G REMARKS
The wave map global existence results we have obtained here extend previous work in two significant ways. First, our study of translation equivariant critical wave maps for Lie group targets (Theorems 3 and 5) provides a counterpart to work on rotationally equivariant critical wave maps for symmetricspace targets (see [4,6]). Second, our inclusion of torsion gives an interesting generalization of critical wave maps for arbitrary targets, which ties into current work on integrable chiral models in 2+1 dimensions in the case of Lie group targets [gl.
Furthermore, our results demonstrate the utility of the frame formulation of wave maps for Lie group targets (Proposition 1). The translationequivariant reduction of critical wave maps studied here is motivated by this formulation and the analysis is especially straightforward in terms of frames. An important question to investigate for future work is how the frame formulation might help in understanding the unreduced critical wave map equation for general Lie group targets and symmetric-space targets.
A P P E N D I X Proposition A. Let G be a semi-simple Lie group with commutator structure tensor Cbca.

The Lie algebm of G admits a translation invariant (159) positivedefinite metric gab if G is compact.
pab with non-zero torsion (54) if G is compact and has dimension greater than three.

3.
If G has dimension three then the torsion (54) is zero for every skewtensor pab on the Lie algebra of G.

Proof of 1:
If G is compact then its Lie algebra admits an invariant positivedefinite metric gab (see, e.g. [13]), which satisfies (In particular, the Cartan-Killing metric given by gab := -CaecCke is both invariant and positive-definite.) Hence condition (159) holds.

Proof of 2 and 3:
Hereafter gab denotes the Cartan-Killing metric. We first remark that, for any G, the natural construction is easily seen to yield a translation invariant skew-tensor. But the resulting torsion tensor (54) is always zero, since by the Jacobi identity.
In three dimensions it is easy to show that Cabegec must be proportional to the totally-skew Levi-Civita tensor €,a, while any skew-tensor pab can be expressed in the form for some vector pc in the Lie algebra of G. Thus, it follows that pab must have the form (173) where Re is proportional to pe, and hence from (173) and (174) we have that the torsion tensor (54) is zero. This shows that there is no torsion for any three-dimensional G (and hence none in particular with pab being translation invariant). Now suppose G has dimension greater than three. In this case, G must have rank greater than one and hence the Lie algebra of G possesses an abelian subalgebra of dimension at least two (see, e.g. [13]). This allows the explicit construction of a translation invariant skew-tensor pab as follows. Let pa, qa be any two (linearly independent) commuting vectors in the Lie algebra of G, sopaqbCabc = 0, and let pe := geapa, qe := geaqa. Set Ra := apa+pqa # 0

1701
with constants a, p. Then it is straightforward to show that the skew-tensor defined by is translation invariant as a consequence of p and q commuting with R. Now it remains to show that the torsion tensor given by (54) and (176) is non-zero.
We have To show that the tensor (177) is non-zero when G is compact, we contract (177) with the vector sa = paqeqeqapeqe satisfying saga = 0. This yields with sapa = pap,qdqd -(paqa)2 # 0 due to positive-definiteness of gab.
Moreover, since G is semi-simple, its Lie algebra has empty center and so Ckeqe = Cebaqegca is non-zero (that is, there exists a vector vb so that Cebaqevb # 0). Therefore, sagdQdk is non-zero and thus so is the torsion tensor (177). 1

Introduction and main results
In this paper we prove several non-existence theorems for the boundary value LANCONELLI AND UGUZZONI \Ye directly refer to section 2 for lnore details about the notation and definitions we need.
Througliout the paper we shall assume the function f locally lipschitzcoritiriuous on R and satisfying the following conditions: ( H ) for suitable positive constants m, p > 1.
Our aim is to prove non-existence results for problem (1.1), when a belongs to certain classes of open subsets of Wn, by means of some techniques first introduced in [26] and [28]. For example, when f (u) = urn and fl is a halfspace of W" we obtain the following result. Theorem 1.1 Let 11 be a ~lonnegatiue weak solution of 9 (1) If In > % and u7"-' E L 2 (n), then u -0.
111 T1ic.or~n1 1.1 and in wliat follows, we agree to let v E Lq-(0) if there exists 1) < q such that 21 E L" (R). \\t would like to remark that in the critical case This special case has bee11 st,udictl in tlie previous papers [26] and [28].
In the classical contest of tlie Laplace operator, results like that of Theorem 1.1 are well-known and have been established in [13] and [18, 191. However, we stress that novel aud significant difficulties arise in the setting of tlie operator AH.. Tliese difficulties are rnainly due to tlie lack of good a priori estimates for the derivative &IL of a n.eak solution u to (1.1). Indeed such a derivative, in the Heisenberg group setting, should be considered a second order derivative, si~iue at = -f [Sj, I > ] being ,Ti,, I > , j = 1, .... n, a basis for the Lie algebra of W". The key point of our approach is to deduce the needed estimate of dtu 9 froni the hypotlicsis urn-' E L 2 .

1705
Tlicortm 1.1 liolcls for a niore general class of domains R and nonlinearities f .

Definition 1.2 ' W e shall say that a smooth open set R is of class A if
We also suppose the boundary afi regular enough to assure that every bounded weak solution u to (1. \\.e explicitly remark that every open set of classlB satisfies the regularity assun~ption (1.2). Actually, we shall prove our results for a class B of domains slightly niore general than this one. \lie refer to section 4 for more details.
In order to state our main results, next Theorems 1.    Findly, let n+ in > 1 + i.
The paper is orgauized as follows. In section 2 we complete the list of the notation and establish some preliminary results.
In section 3 we study the case when R is a domain of class A. IZt

Notation and preliminary results
The Heisenberg group Wn, whose points will be denoted by < = (2, t) = (x, y, t), is the Lie group (R2"+l, 0) with conlposition law defined by w l~e~e BQ is a positive constant whose best value has been determined by .Jc~ison a r d L w in [24]. If R is an open subset of Wn, we shall denote by S1(R) the Sobolev space of the functions u E LQ'(Q) such that Vanu E L*(R). The norm in S1 (12) (R)) (see [17]). LYe remark that these definitions agree with the ones given in section 1 when R is of class A.
\\*e recall the follon.ing representation formula which can be found in (161: where ill, is the nleari value operator defined by I11 section 3 and section 4 we sliall need the following result. where AIT is the mean value operator introduced in (2.13) and LVe explicitly remark that Bd(<,r(<)) C R for every < E R, r E C(Q,R+) and r ( z , t ) = r ( z , 0 ) for any ( 2 , t ) E 0, since dR is parallel to the t-axis. The follo~ving proposition states all the properties of the operator T that me will use in this section.

Proposition 3.1 (1) T is a linear operator from L~,,(R) into C(R).
(2) The operators T and at commute. More precisely if dtw E C ( R ) then also dt(Tw) E C ( R ) and dt(Tw) = T(dtw). (R) and Awnw < 0 then Tw < w.
(5) \Ye first remark that. being Awnw 5 0 and w + 0 at any point of dS1 U {m), b! . the niaxirnum principle w > 0. hloreover, since Tw < w and T is increasing (see tlie previous assertions (3) and (4)   Proof Let u be a solution to (1.1) satisfying the hypothesis of the theorem.
(see [15]). hloreover, since R is of class A, dR has no characteristic points and then u E r2(n) (see [25], [21]). By Theorem 3.3, we now only need to is a classical solutzon to (3.10)   From now on (in this section) we will then assume (1)- (2)   In the sequel we shall always assume (4.1) when looking for estimates at infinity (respectively (4.2) when looking for estimates at the origin). The first step of our approach consists in finding an estimate of atu at the boundary of Q.
\Ye will use cylindrically syrri~netric barrier functio~ls and split the proof in a remark and two lemmas. For every R 2 1 we set and for every n > 0 and (4.16) Since u = 0 in dR and u -+ 0 at infinity, we only need to prove (4.16) in On the other h a~~t l , Hence, the masinluln pri~iciple for Aan yields [ Lemma 4.11 For every E no we have Proof \Ye fix [ E Q0 and for sake of brevity we set r = r ( < ) , Q = e(<). Let <' E Bd(<, e) and let us denote \ \ e have (see (2.1) and (2.4)) Q " Hence ((1 < Q and Is1 5 1s + 2(2, (k, -h))l + 2(51/</ < e2 + 2121~. On the other hand (4.18) yields g2 + 212(Z = r . Therefore ( tt'l = Is1 < r and \Ye also set p = i, cu = 2 + ,!3 and define 1726 LANCONELLI AND UGUZZONI Lemma 4.14 There exist A1 > 0 and Ro > ro such that for every R > Ro we h u e Proof C'sing (4.11) and (4.1) it is not difficult to verify that, for a large Ro, Then, if we prove that latvl 5 i l i (~i + R~G~) in a& (4.24) (4.23) will follow from the ~naxirnum principle. From (4.8) and Proposition 4.7 0 1 1 the other l~w l i t l (4.21) yields anti (4.22) gives w h r~c ,I1 is a constant not depending on R. Moreover ldtvl is a continuous Proof From (1.23) it follows that, for R large enough, T l~c n , if ( = ( 2 , t ) E D and ( z / is sufficiently large, we have Since also (4.22) holds, the corollary is lxoved. Then we define From the Hiilder continuity of u (see (4.7)) we obtain i.e. ( P ) + holds. Therefore it is sufficient to show that, setting = t, and we will get ( P ) , and prove the lemma. Let us then fix p €]O,l] and assume (P),. \\'e set F : Bd(<. Q(<)) C 0 for every E E (I0 := n n a small neighborhood of the origin.
The only thing which is to notice is that, for small lzl > 0, ll~I-e,lzl+e1 2 Hence we get Lemma 4. 18 There exists c > 0 such that for every < E no we have Proof See the poof of Lemma 4.12. Hence sup lvol I cp. Since vo is Awn-harmonic as well as v, (4.25) holds also B replacing v with vo. Therefore we obtain C IVwnv(to)l = lVwnvo(to)l <_ -sup lvol I c, e l 3 where c is a positive constant not depending on to. Since w E r2(Wn) we finally get (4.26).

0
We now fix , D ~] 0 , 1 [ and set cu = 2 + p. Lemma 4.20 There exist EO > 0, y > 0 and A l > 0 such that, setting for every and 9, :  Proof of Proposition 4.6 Tl~anks to (4.26) n7e only need to prove that The proof follows from Theorem A.3 in the Appendix, Theorem 1.9 and the next lernma.
This proves the first part of the lemma. We omit, for brevity, the proof of the second part (see e.g. [26]).
Next lenima allows to obtain u E LP for some p < Q*. The proof is essentially contained in (261. However we inc!ude it below, since this is a crucial st,ep of our tecllnique. Proof \Ye have and \ k + 1.. Exactly in the same way as in the proof of Lemma A.1, we can see that for every k E N problem (A.3) admits a weak solution uk such that a d u k -+ u weakly in SA(Q). \Ye now want to prove that \\e fix k E N and, for sake of brevity, we set v = uk. Then we define, for every 0 [12] G . C    In this paper we shall study global well-posedness of initial value problem (IVP) associated to the semi-linear wave equation Thus for data the conservation law (1.4) allows to extend the local solution of (1.1) to any time interval.
Our aim is to establish global well-posedness results for data with less regularity that the one described in (1.5). To illustrate our results first we consider the particular case of (1.1) which is locally well posed for data and globally well posed for data Theorem 1.1. [0, T ] . Moreover,

Let s E ( 3 / 4 , 1 ) . Then for any T > 0 and ( u O , u l ) E H S ( I R~) n L~( R~) x H~-' ( R~) the.IVP (1.6) has a unique solution u ( t ) defined i n time interval
with Above we have introduced the notation In the general case our results can be stated as follows. In this case the results in (1.10) and (1.13) will be in local Sobolev spaces in the space variables.
Our proof follows Bourgain's ideas. In [l] he introduced a general scheme to establish global well-posedness of nonlinear evolution equations in H S for the values of s between those determined by conservation laws. It was I first used for the 2-D Schrodinger equation, see [I]. It was also applied in [2] to the PBVP for the semi-linear Klein-Gordon equation in one and two space dimensions, and later in other models, see [3], [4], [9].
Roughly speaking, one splits the data into two pieces: high and low fre- We observe that the inhomogeneous part z(t) of y(t), see (2.12), is in I?'.
Thus, we add r(AT) to v(AT), and repeat the argument in the time interval [AT, 2ATl. In each step of this process the estimates for the involved norms grow. This has to be taken into account to make the process uniform. It is here where the restriction on s appears.
In the next section we shall prove Theorem 1.1. It will be clear from our exposition how the argument extends to the general case in Theorem 1.2.
We start by recalling the homogeneous version of Strichartz' estimate, [lo], for the associated linear problem Theorem 2.1.
The solution w = w(x, t) of the I V P (2.1) satisfies For the proof of (2.2) we refer to [8]. It was proven in [5] We need an estimate for the lower derivatives of y.
Using ( To reach the time T we have to apply the above argument rn Then, the total added to the expression in the right hand side of (2.7) will be (see (2.33)) To make the above computations uniform we just need (see (2.7)) that  BEN-ARTZI, DERMENJIAN, AND GUlLLOT results imply "low energy" estimates for this operator, as well as the validity of the "limiting absorption principle".

Introduction
Consider the operator H = -c2(y) p(y) V . (h V) , where V = V,,,, (x, y) E Rn x R, n 2 1. This operator is known as the "acoustic propagator1' in (unperturbed) stratified media ( [4,7,9.19]). In what follows we impose the following assumptions on the real functions p(y), c(y) (y E R): (1.1) p(y) E C1(R), c(y) is piecewise continuous with finitely many jump discontinuities, and there exist positive constants pml phf, cm, CM S U C~ that (1.2) There exist positive constants y,, p*, c* such that c+ < c-and P(Y) = P*, C(Y) = c* for f y > y,.
It is well-known that the operator H can be defined as a self-adjoint operator in Taking the (partial) Fourier transform wit,h respect to t'he x-coordinates, we have the unitary equivalence.
The spectral multiplicity of Hp jumps at c?p2 from 1 to 2.
It is the purpose of this paper to study the behavior of the resolvent kernels associated with the family {H,), specifically near the points {c:p2).
We obtain estimates on the kernels and their derivatives (in terms of the spectral parameter), uniformly in p in compact intervals of [0, m). In particular, these est inintes hold at "thresholds" (see Remark 2.6).
Such estimates constitute the main technical tools needed in the study of the resolvent of H near the continuous spectrum, and in particular the "limiting absorption principle" ([4]) and "low energy estimates".
To illustrate the situation at hand we take the Laplacian -A in W t l and let T be its self-adjoint realization in L~(IW~+'). Then, parallel to (1. In particular, this applies to estimates near z = 0, the "low energy estimates". We shall derive such estimates in the Appendix, as a motivation for our treatment of H in terms of (1.3).
As we show in this work, the kernels associated with (H, -2)-I share some of the main properties of the kernels L,, thus permitting analogous results for H .
We note that in particular we give here new proofs to Propositions A.  [6,11,161. It is expected that the techniques developed here will serve in the spectral study of those cases as well.
Our method of proof is based on a study of the analyticity properties of the kernels in terms of the variable k = ( z c~~ -p2)1f2 (see next section for details). This is closely related to the work of Cohen and Kappeler [8]. In particular, their treatment of the Wronskian of Jost functions (Proposition 2.4 in [8]) corresponds to our conclusion (2.12). However. since we are dealing with faniilics of operators. we need to study the way in which the zeros of the Wronskian "move around" (see (2.10)) and get uniform estimates on a family of kernels. Furthermore, in order to get estimates on the z-derivatives of the kernels, we need to study analytic extensions of the kernels to sectors in the lower (k-)half-plane. We refer the reader to (1, 181 for works concerning such extensions. To formulate our results, fix L > 0, a > 0 and 0 < /3 < T . For every p E [0, L] set (1.9) It follows from (1. lo), (1.8), that the "blow-up" rade for the kernel a, a t the bottom of the essential spectrum is no worse than that of L,, associated with -A. In particular, repeating verbatim the proofs of Lemmas A.l, A.2 in the Appendix, we obtain a "low energy estimate" for the resolvent (Hz)-'.
In it's formulation, we use the weighted-L2 norm introduced in the Appendix, and also the "limiting absorption principle", which allows the extension of (Hz)-' to positive z (continuously in the same operator space).  .9). Note that z E 0; implies k E @+ n {Re k > 0). However, using the asymptotic behavior (2.3) for cp, q!I, it is obvious that both functions can be extended to k E @.f, so that they solve (2.2) with z = c:(k2 + p2). In view of standard theorems about solutions of ordinary differential equations we can summarize the properties of p, @ and their y-derivatives vt, as follows. The resolvent kernel (Green's function), @,(El q; k), k E @+, is given by [12], I11 view of Claim 2.1, the kernel 9, is analytic in k E @+ and can be extended continuously to k E @f except for zeros of [p, $1. It follows from our next rlairil that in fact [ ; , $1 does not vanish for k E R\{O). , we see that they are all simple, but we shall not need this fact). We arrange these zeros such that The cont,inuity of [p, $1 (k,p) in p and standard analyticity arguments imply that the integer function J(p) is constant in intervals where [cp, $1 (0,p) # 0.
However. the following claim shows that this is the case except possibly for a finit,e number of values. We now turn to the study of [p.$] (k,p) for real k near k = 0. Our next claim will show, in particular. that k = 0 is at most a simple zpro for p = p,, 1% $1 ( k P ) = 2 i p ; ' k m PI.
On the other hand, using (2.7) once again to evaluate [$J, $1 by (2.13), We conclude that which proves (2.12) in view of (2.14). 0 We may now conclude the proof of Theorem A.
Proof of Theorem A. Take 6 2 2 ( L + i (~) ) , and let r6 = @+ f l {lkl L 6 ) . In what follows we use C > 0 to denote positive constants not depending on k, p.
so that in view of (2.12), (2.24) and the uniform boundedness of @ ( I ) ; k, p) @(<; k, p), Note that by Claims 2.4 and 2.7 the function k a,(<, I); k), for fixed <, q,p, is analytic near k = 0, so that the estimates (2.21), (2.25) are valid also at k = 0. Furthermore, the function 0 (see (2.10) for Cj(p)) is analytic in r6 and continuous in r 6 . It therefore attains its maximum (absolute) value on d r a Take 0 < E < 5. Due to (2.11) We also note that, as before, the exponent of $I , and dl decays exponentially as y --+ -m, uniformly in k E Bi. We have the representation (2.13) for k E Bi\{O), p E [e, L], hence also (2.14). By (2.29), (2.14),

PROPERTIES AND ESTIMATES OF KERNELS
Evaluating (2.13) at y = yc, we get for k # 0, and in view of (2.31) and Icp(yc;k,p)l = (cpl(yc; k,p)J = 1, we conclude We can now establish the key estimate Turning back to the proof of the theorem, we take ko E (0, $) and let A0 E Bj be the circle centered at ko with radius equal to (2 + (+ + q+)-'ko sin E . By (2.33) the analytic function k a,([, q; I;) ((J, q) E R x R) is bounded on A0 with a bound which is independent of (J, q), so that the Cauchy formula implies Changing back to X = (k2 + p2) c : E I,, X < c?p2 7 6, and noting (2.33), we obtain (1.13) away from c:p2.
To prove (1.13) for X near c?p2, we need to study & a,((, q; k) near k = i(p). We observe first that while cp,cpl can be extended analytically to a full neighborhood of i ( p ) , the functions $ , $ I and ${ (see (2.22)) can be extended analytically to sectors for 6, E > 0 sufficiently small (uniformly in p E [t, L]). In order to maintain continuity we need t,o take now Im(k * i)'J2 < 0 for I; E Bi;,, Im I; < 0.
The denominator is greater than IRe(k2 -,&2)11/2 and this last one greater than For a E R we denote by L2."(Rn) the weighted-L2 space normed by Since IL,(E, r]; z)l 5 Clz -p21-'I2, we get We shall now use this estimate to establish an estimate'for M(z) near z = 0. So, using the notation in the Introduction, let f (x,  Certainly (A.7) is not "optimal", at least for n > 2 (see [3,5,13,141). However, the simplicity of its derivation makes it easy to generalize to cases such as those mentioned in the Introduction. Also, if n 2 3 we can improve the trace

PROPERTIES AND ESTIMATES OF KERNELS
wherein N denotes the Newtonian potential on Rn so that A ( F * N) = F, and u satisfies the homogeneous Neumann problem The results contained within this article appeared in the author's Ph.D.
thesis [16]. We note that the regularity results described here overlap considerably with those in recent work of Fabes, Mendez, and Mitrea [3].

Statement of the Estimates
We begin with some basic definitions. We say that a bounded, open, con-    Of course the dualized statement also holds and will be very useful. (1'4 It is proven first for elements of C,"(Rn) and then extended to all of LP, via density.
We are now ready to describe precisely the sense in which the Neumann problem will be solved. Suppose that 1 < p < m , $ < a < 1 f i, and Notice that when w E L: (R),v E L~( R ) , and F E L"(R), the standard definitions of elementary distribution theory show that (13) may be written in a simpler, more transparent form: By density and continuity, this implies, among other things, that the bilinear form on the right-hand side in (13)  We present, at last, our principal result.

Some Boundary Function Spaces and Estimates for the Dirichlet Problem
Denote by F, SyQ = FllQ(Rn-') the inhomogeneous Triebel-Lizorkin spaces on euclidean space. When 1 < p < co and q = 2 these are just the standard Sobolev spaces that we have already encountered, when q = p they become the familiar Besov spaces B i = BiJ' , and when 0 < p < 1, q = 2, and s = 0 they are quasi-normed spaces identical to the local Hardy spaces hp. We will not explicitly define them here but refer the reader to lliebel [13], [14] or Frazier-Jawerth (41. First consider a region R above the graph of a Lipschitz function +. Let $j be an isometry of Rn such that When 1 < p < w , q = p , O < s < 1 or else when 1 < p < w , q = 2 , 0 5 We may also define dual spaces to our positivC boundary Triebel-Lizorkin spaces. When 1 < p < w , q = p, -1 < s < 0 or else when 1 < p < oo,q = 2,-1 < s 5 0, we set FpS1q(dSZ) = (Fp;S1q(dR)) * where 1 /p + 1 /pl = 1. LJ'(8R) lies in F,Svq(dR) under the pairing < g, f >= i g f do, f E F;slq(dR), g E Lp(dfl).
In other words the corresponding norm is ZANGER Since, on euclidean space, F,9" = (FPTSlq)* for the index ranges specified, it follows that, employing coordinate mappings, we may redefine the negative Triebel-Lizorkin spaces in terms of corresponding euclidean space norms.
When 1 < p < m , q = p , -1 < s < Oorelse when 1 < p < m , q = 2,-1 5 s 5 0, F'lQ(aR) is the completion of the space of all g E L1(dfl) such that for all j = 1, ..., N with respect to the norm As noted within the introductory remarks, the observation that surface area density is included as a weight factor in the equivalent definition (17)  As we have already noted, the euclidean Triebel-Lizorkin spaces coincide with euclidean Sobolev and Besov spaces for appropriate values of the indices. Acknowledging this fact, we will also call the space F , s~~(~R ) , 1 < p < m, -1 5 s 5 1, a b o u n d a r y Sobolev space and interchangeably employ the notation Lz(aR) for it. Similarly, when 1 < p < CQ, -1 < s < 1, we will often write B,P(aR) for F,"J' (aR), and call F,"J' (dR) a b o u n d a r y Besov space .
It will in addition be important that Holder's inequality implies that LP(i3R) acts on B<,(~R), -1 < s < 0, via integration: and thus IP(i3R) is naturally embedded in B,P(dR) for this range of s.
Since we are interested in the solution to the Neumann problem we will consider mean-value-0 versions of these boundary function spaces. Recall that any f E B[(aR),l < p I oo, -1 < s < 1 , s # 0, acts on the real line either via integration over i3R (i.e., < f , r >= J, , r f do for all real numbers r ) if s > 0 or by virtue of the identification of B,P(BR) as a space of linear functionals on an appropriate function space if s < 0. Thus we set and similarly for our boundary Sobolev spaces (as well as for the spaces L1(i3R), Lm(i3R)) for the ranges for which they have been defined.
We remark that if 3 C L1(aR) is a dense subset of some space Bf(aS2) (resp. @(an)) then Fli {f -& Jan f dal f E 3) is a dense subset of B,P(aR)li (resp. @(i3R)1~). One easily shows this by using the fact that LP(i3R)-convergence implies L1(8R)-convergence (in the case for which s > 0) or that norm convergence implies weak-* convergence (in the case for which s < 0).

Estimates for the Inverse Calder6n Operator
Let R be a bounded Lipschitz domain and suppose that Q E an. We define the (interi0r)nontangential cone r,(Q) for a 3 0 via If u is a function on dR we may define its nontangential maximal function

M(u) by setting
We say that u has a nontangential limit at Q E dR if there is a finite, well-defined. limit (which we will call) u(Q) as P + Q from within I',(Q) for all cr > 0.
Next recall the classical method of layer potentials to solve Laplace's equation with Neumann boundary conditions. Given g E L1(dR), its single layer potential (SLP) is the function defined via where wn is the surface area of the unit sphere in R n . We will occasionally write N ( X , Q) for the kernel IX -QI2-" -which is the Newtonian potential.
The integral in (20)

. Assume that f E L P (~R ) , I and let u = ST-'f be the single layer potential of T-'f considered as a function on R. Then u E L!(R) and for all v E L~( R ) .
Given Theorem 3.1, the proof of this is quite standard, so we do not tarry here to address details.
Remark 3.6.Suppose that 1 < p < co, and assume that the function u E Lf(R) whose maximal function M ( u ) is bounded in LP(dR) has nontangential boundary values g in LP(dR), i.e., that g is the nontangential limit of u on aR. Then g = n u a.e. do.
The proof of this fact is rather similar in spirit to that of Theorem 3.5 and is left to the reader.

ZANGER
As its title suggests, the principal goal of this section it to state and prove a theorem yielding estimates for the inverse Calder6n operator for a range sufficiently large to prove Thm. 1.6. Using interpolation, the estimates (28) of Cor. 3.4 are nearly all we need. However, to state the complete theorem, we will require an additional estimate, which is in the nature of an "Eimprovement" over what would have been possible w.ith the estimates (28) alone. This estimate involves the Neumann problem for data in Hardy spaces on Lipschitz hypersurfaces and will require some preliminary definitions.
Let A(Qo,r) = {P E dRllP -Qol < T ) , and assume that r is less than diam(dR). Also let d = n -1 denote the dimension of dR. Following Brown [I], we say that the function a is an atom for HP(dR), with ( n -l ) / n < p < 1 if for some Qo and r we have We denote by HP(dR) the Hardy space on aCl (with index p) and define it as the collection for some sequence of atoms a j and complex numbers X j . The quasi-norm for

HP(dR) is given by
Recall that for 0 < p < 1 the local Hardy space h p = hP(Rn-' (see Goldberg [5] or Stein [ll]) is identical to the Triebel-Lizorkin space F, 092 = F,012(Rn-I) (see Triebel [13], (14]). In light of this, for (nl ) / n < p < 1 we may identify HP(dR) with the space F ,~I~(~R )~L , itself defined as the closure In analogy with the situation for HP(aR), we may define Hf(dR) in terms of the Triebel-Lizorkin space Fi>2 = F1p2(Rn-') P on euclidean space (once again see Triebel [13], [14] or hazier-Jawerth [4] for detailed definitions). In fact for (nl ) / n < p < 1 we have Hf(dR) = Fd12(,aR), where this latter space is the completion of the space of all g E L1(aR) such that We shall also need R. Brown's corresponding uniqueness result.

(32)
with $ vanishing in the HP-sense, then u is a constant.
We have not defined here what it means to "vanish in the HP-sense" (though of course the interested reader may find a definition for this in Brown's paper). Suffice it to say that the condition that = 0 nontangentially a.e, on dR, which we will have, will be strong enough to imply it.
We will also shortly find occasion to exploit the space (HP(dR))*, (nl ) / n < p < 1, which we define as the vector space dual of the space Hp  We can also consider, if we wish, the mean-value-0 counterparts of each of the spaces Fd12(dR), F$)vm(dR), and F~p)-'~"(dR), denoted F,'>2 ( d o ) F$)-'!" (do) and F$')-'lW ( a n ) ,L respectively, which are defined in the same fashion except that one takes only g E L1(dR). with mean-value-0 in (31), (36), and (37).
The final ingredients required prior to stating our estimates for the inverse Calder6n operator are the necessary interpolation results. These we collect within Corollaries 3.10 and 3.13, after introducing the requisite interpolation spaces. The first interpolation method for the euclidean Triebel-Lizorkin spaces that we will invoke to obtain interpolation results for our boundary spaces is $rnply the familiar real interpolation method for Sobolev spaces.
We can define boundary interpolation spaces corresponding to this method in a familiar fashion. For 1 < p < oo, 0 5 so # sl 5 1, s = (1-Q)so+6s1, and 0 < 6 < 1, we define (F,"012(dR), F,"l~2(dR))o,p as the space of all functions g on dR such that j = 1, ..., N, with norm For the negative range of spaces, the surface area density weight factor must as usual be included. For 1 < p < oo, -1 5 so # sl 5 0, s = (1 -6)so + 6sl, and 0 < 0 < 1, we define (F,"0*2(dR), F , "~Y~(~R ) )~, as the completion of the space of all g E L1(dR) such that ZANGER j = 1, ..., N, with respect to the norm The interpolation property (which says of course that a linear operator bounded on two indexed function spaces should be bounded on the spaces in between) for the spaces (F,S012(a1;2), F,S"2(8R))e,p is directly inherited from that for the spaces (F,S012, FpS1j2)e,p. The relevant interpolation theorem is Theorem 3.9.Let 1 < p < m, -w < so # sl < m, s = (1 -B)so + Bsl, and For this result, see Theorem 2.4.2 in Triebel [13]. Note that for this range of indices, the spaces on the left in (38) are Sobolev spaces and that on the right is a Besov space. We can now define boundary interpolation spaces in the natural way. For the positive range of spaces, we define < F;;lq~(dR), F ; ; l q l (ail), 0 > as the completion of the space of all g E L1(dil) such that j = 1, ..., N, with respect to the norm whenever the indices involved remain within one of the following ranges: As discussed in the introductory paragraphs to this article, interpolation is possible between spaces whose indices lie within the stated ranges   LP1(afl) for 2 -6 < p' < oo. We will obtain our theorem by interpolating between a rather wide selection of pairs of these operators. Thus we wish to prove that, wherever their domains of definition intersect, any two of these operators coincide.

ZANGER
Our strategy will be to show that all of these maps agree with Y : it follows that, whenever 1 < p < 2 + d and 1 -6, < p < 1, we have that where E > 0 is chosen small enough that, setting po B,pl -PO, go r ql 2, so -1, sl 0 and po = p,pl -PO, go r ql r 2, so -0, sl = -1, these two sets of indices lie within the ranges described in (c) and (c') respectively.
To obtain estimates for the range pb 5 p < oo of case (c), we first observe that Y satisfies  To accomplish this, we shall also require another integration-by-parts formula on Lipschitz domains. For direction to its proof consult Grisvard [6], < Tr(Vh), n > (vlaR)do = /n FvdV + /n < Vh, Vv > dV On the other hand, setting g = vlaR, E Cw (Ti) JaR c L$-,,,, (XI), we recall that there exists a unique harmonic function vg in L~ ( R ) with vg(dR = g. Consequently, employment of Lemma 4.2 once again gives so that comparison of (60) and (61) yields the result, as long as F E P ( R ) and v E C w (n). Since P ( R ) is dense in (~' ( 0 ) ) ' for s > 0 and it is wellknown that C W ( n ) is dense in Q 1 ( R ) for s > 0, a simple density-continuity argument now establishes (59).
To complete our proof of the existence assertion of Theorem 1.6 we seek to apply our estimates for the inverse Calderon operator (Theorem 3. for all v E L;-, ( R ) .
To extend the results just obtained to the range pb < p < oo, fix such a p and once more select a from within the range of Theorem 1.6. Note that

L w ( B R ) l~ is a dense subset of BE-,-lIp(dR)l~ that is also a subset of all of
the spaces L f i ( a R ) ,~, 2 < 6 < pb. But, as before, for each f E L " ( a R ) l~, there exists a unique harmonic function up E ~f ( d R ) with nontangential Neumann show that V(En(l))JR = 0, it follows from the definition that < Lh, 1 >= 0.

Proof of Uniqueness
Recall that the well-known methods of Lax-Milgram ensure that that there exists a sequence of eigenfunctions uk E CW ( 0 ) (? Ly(R), k = 0,1,2, ..., with corresponding eigenvalues Xk satisfying the (weak) Neumann eigenvalue problem for the Laplacian for all v E L: (R).
To establish uniqueness for the inhomogeneous Neumann problem, we first observe that for all k = 0,1,2, ... we have uk E LP, (R) n L$-,(R) with ZANGER 1 < p < co, ru chosen from within the admissible range for Theorem 1.  where uo is the constant (normalized) eigenfunction. Therefore w itself is constant, and uniqueness in Theorem 1.6 is estab1ishe.d.

ACKNOWLEDGEMENT
The author wishes to gratefully acknowledge the invaluable guidance of his thesis advisor, Professor David Jerison, regarding all aspects of this work.

Introduction
Consider the stationary Schrodinger operator in L2(Rn) (n = 2 or 3), where the real potential q(x) E Lm(Rn) is periodic with respect to the integer lattice Zn: The spectrum of this operator is absolutely continuous and has the well known band-gap structure (see [lo], [13], (221, [31], [34], [38], where v(x) is compactly supported or sufficiently fast decaying at infinity (the exact conditions will be introduced later). It is known (see 151, Section 18 in [16], and references therein) that the continuous spectrum of H coincides with the spectrum of Ho, and only some additional "impurity" point spectrum {Aj) can arise. A general understanding is that eigenvalues X j should normally arise only in the gaps of the continuous spectrum (i.e. in the gaps of n(Ho)). In other words, the case of embedded eigenvalues when one of the eigenvalues X j belongs to one of the open segments (a,, 6;) is prohibited (at least for a sufficiently fast decaying impurity potential v(z)). Physically speaking, the existence of an embedded eigenvalue means a strange situation when an electron is confined, in spite of'having enough energy for propagation. There are known examples of embedded eigenvalues, first of which was suggested by J. von Neumann and E. Wigner (see [ll] and section XIII.13 in [34] for discussion of this topic and related references). The problem of the absence of embedded eigenvalues has been intensively studied. There are many known results on non-existence of embedded eigenvalues for the case of zero underlying potential q(x) (see, for instance, books [ll], [I91 Section 14.7, and [34] Section XIII.13). The case of a periodic potential q(x) is much less studied. In [ l l ] a one-dimensional example is provided where a not very fast decaying perturbation of a periodic potential creates embedded eigenvalues. There are many papers devoted to studying the behavior of the point spec-trum in the.gaps of the continuous spectrum (see, for instance, [I], [2], [4] -(61, [9], [15], [18], [25] - [27], [33], and [35] - [37]). Apparently the only known result on the absence of embedded eigenvalues relates to the one-dimensional case of the Hill's operator (see [35], [36]). However, it was shown in [14] and [20] that in dimension higher than 1 the set of embedded eigenvalues must be discrete. The purpose of this paper is to address the problem of absence of embedded eigenvalues in the multi-dimensional situation. Section 2 contains some necessary notions, auxiliary information, and our main condition on the periodic potential. It is conjectured that the condition is always satisfied. According to [3], this condition is satisfied at least when the potential is separable in 2D or of the form ql(xl) + qz(x2, x3) in 3D. In Section 3 we prove a conditional statement (in dimension less than four) on the absence of embedded eigenvalues for a periodic potential satisfying this condition. Finally, the Section 4 contains the main unconditional result. In our considerations we follow an approach that was suggested long ago by the second author for the case of zero underlying potential q(x) (see [39]). Some recent developments made it possible to adjust this method to the periodic case. In particular, results on allowed rate of decay of solutions obtained in [12] and [29] play a crucial role. A more limited version of the main result was announced by the authors without proof in [23].

Bloch and Fermi varieties
Let us introduce some notions and notations., We assume that q(x) E LbO(Rn) is a periodic potential. When c = 0 we will use the notation F(q) for Fo(q). Let us notice the following obvious relation between the Bloch and Fermi varieties:

Definition 1 The (complex) Bloch v a r i e t y B(q)
We will use the following notations for the real parts of the above varieties: The varieties BR(q) and F R ,~( q ) are called the real Bloch variety and the real Fermi variety respectively. The following statement is contained in  examples of what is called in complex analysis analytic sets (see for instance 181, [17], and [30]). Moreover, these are principal analytic sets in the sense that they are sets of all zeros of single analytic functions, while general analytic sets might require several analytic equations for their (local) description. Analyticity of these varieties (without estimates on the grows of the defining function) was obtained in [40].

Definition 4 An analytic set A c Cm is said to be irreducible, if it cannot be represented as the union of two non-trivial analytic subsets.
Irreducibility of the zero set of an analytic function can be understood as absence of non-trivial factorizations of this function (i.e., of a factorization into analytic factors that have smaller zero sets).

Definition 5 A point of an analytic set A C Cm is said to be regular, if in a neighborhood of this point the set A can be represented as an analytic submanifold of Cm. The set of all regular points of A is denoted by regA.
We collect in the following lemma several basic facts about analytic sets that we will need later. The reader can find them in many books on several complex variables. In particular, all these statements are proven in sections 2.3, 5.3, 5.4, and 5.5 of Chapter 1 of [8].
L e m m a 6 Let A be an analytic set.

a) The set regA is dense in A. Its complement in A is closed and nowhere dense in A. b) The set A can be represented as a (maybe infinite) locally finite union of irreducible subsets
c) Irreducible components are closures of connected components of regA. We will need the following simple corollary from this lemma. After this brief excursion into complex analysis we return now to Bloch and Fermi varieties. The next statement follows by inspection of (4): L e m m a 8 .The sets B(q) and Fx(q) are periodic with respect to the quasimomentum k with the lattice of periods 2nZn c C n (ie., the dual lattice to Z").

Corollary 7 Let
We choose as a fundamental domain of the group 2nZn acting on Rn the following set called the (first) Brillouin zone: It has been known for a long time (though not always formulated in these terms) that one can tell where the spectrum a(H0) lies by looking at the Fermi variety. We will now briefly remind the reader this classical result and, at the same time, the definition of spectral bands.
Consider the problem (3) -(4). It has a discrete real spectrum {X,(k)), where Xj + oo as j + oo. We number the eigenvalues in the increasing order. This way we obtain a sequence of continuous functions Xj(k) on the Brillouin zone B. They are usually called b a n d functions, or branches of the dispersion relation. The values of the function Xj(k) for a fixed j span the j t h band [ a j , b,] of the spectrum a(Ho). A reformulation of this statement is the following theorem, which can be found in equivalent forms in [lo] (Section 6.6), [13], [22] (Theorem 4.1.1), [31], and [34] (Theorem XII.98).

T h e o r e m 9 A point X belongs to the spectrum a ( H o ) of the operator Ho = -A + q(x) if and only if the real Fermi uariety F~, x ( q ) is non-empty.
In other words, by changing X one observes the Fermi variety Fx(q) and notices the moments when it touches the real subspace. This set of values of X is the spectrum. Now the question arises how can one distinguish the interiors of the spectral bands. The natural idea is that dhen X is in the interior of a spectral band, then the real Fermi variety will be massive. "Massive" means here "of dimension n -I", i.e. of maximal possible dimension for a proper analytic subset in R n . This is confirmed by the following statement. Proof of the lemma follows from the stratification of the real Fermi variety FR,x(q) into smooth manifolds (see for instance propositions 17 and 18 of the Chapter V in [30]) and from the obvious remark that when X belongs to the interior of a spectral band, then FR,x(q) must separate R n . These two observations imply existence of a smooth piece in FR,x(q) of dimension at least ( n -1). Now we introduce our basic condition: In fact, the Lemma 10 says that for X in the interior of a spectral band the Fermi variety Fx(q) does intersect the real space Rn over a subset of dimension n -1. We, however, need more, that every irreducible component of Fx(q) does the same. In other words, there are no "hidden" components that do not show up on the real subspace in any significant way. Thus, Lemma 10 arid irreducibility of Fx(q)/2xZn would imply validity of the Condition 11. In fact, we believe that (modulo the action of the dual lateice) the Fermi surface is irreducible. The following conjecture formulated in [3] is probably correct:

Fx(q)/2xZn is irreducible.
It looks like this conjecture is very hard to prove (see related discussion in [3] and [21]). As the rest of the paper shows, proving it would lead to a result on the absence of embedded eigenvalues. The following weaker conjecture must be easier to prove:

ABSENCE OF EMBEDDED EIGENVALUES 1817
The word "generic" could mean "from a residual set", or something of this sort.
One can easily prove using separation of variables and simple facts about the Hill's equation irreducibility of Fx(q)/2rZn for separable periodic potentials q(x) = xi qi(xi) in any dimension [3]. The paper [3], however, also contains a stronger non-trivial result: The proof of this lemma relies on the deep study of the Bloch variety done in [21].

The conditional result
We are ready to prove our main conditional statement.  where B(k) is a bounded operator from L2(Tn) into H2(Tn) and B(k) and C(k) are correspondingly an operator and a scalar entire functions of order n in Cn. Besides, the zeros of ((k) constitute exactly the Fermi variety F x ( q ) We conclude now that the following representation holds on the set R n \ F~, x ( q ) : Proof of the lemma. Applying linear functionals, one can reduce the problem to the case of scalar functions g, so we will assume that g(k) E C .

Theorem 15 If a real periodic potential q(x) E LCo(Rn) (n 5 3) satisfies the Condition 11 and an impurity potential v ( x ) is measurable and satisfies the estiinate
According to Lemma 6, the sets of regular points of components Zj are disjoint. Hence, the traces of these sets on R n are also disjoint. Consider one component Zj. The intersection of regZ, with the real space Rn contains a smooth manifold of dimension ( n -1). Namely, we know that Z i ,~ contains such a manifold, which we will denote Mi. The only alternative to our conclusion would be that the whole Mj sits inside the singular set of Zj. The proof of Corollary 7 shows that this is impossible, since lower dimensional strata cannot contain any open pieces of Mi.
Let us denote by m, the minimal order of zero of function ((k) on Zj. (We remind the reader that the order of zero ot an analytic function at a point is determined by the order of the first non-zero term of the function's expansion at this point into homogeneous polynomials.) Since the condition that an analytic function has a zero of order higher than a given number can be written down as a finite number of analytic equations, one can conclude that the order of zero of ((k) equals m j on a dense open subset of Zj, whose complement is an analytic subset of lower dimension. As it was explained in the proof of Corollary 7, lower dimensional strata cannot contain ( n -1)dimensional submanifolds of Rn. This means that one can find a point k' E Zj,n such that Z j ,~ is a smooth hypersurface in a neighborhood U of k' in R n and such that the order of zero of C(k) on Z,,R equals m, in this neighborhood. Let us now prove that g(k) has zeros on ZjtR n U of at least the same order as ((k). If this were not so, then in appropriate local coordinates k = (kl, ..., k,) the ratio g/( would have a singularity of at least the order k;', which implies local square non-integrability of the function .ii(k). This contradicts our assumption, so we conclude that g(k) has zeros on Zj,R n U of at least the same order as [(k). Consider the (analytic) set A of all points in C n where g(k) has zeros of order at least nzj. We have just proven that the intersection of A with Zj contains an (n -1)-dimensional smooth submanifold of R n . Then Corollary 7 implies that Zj c A, or g(k) has zeros on Zj of at least the order mj. Hence, according to Proposition 3 in section 1.5 of Chapter 1 in [8] the ratio g/[ is analytic everywhere in C n except maybe at the union of subsets of Z j , where the function ( has zeros of order higher than mj. This subset, however, is of dimension not higher than n -2. Now a standard analytic continuation theorem (see for instance Proposition 3 in Section 1.3 of the Appendix in [8]) guarantees that g/[ is an entire function. This concludes the proof of the lemma.
Returning to the proof of the theorem and using the result of the Lemma 10 and the assumption that the Condition 11 is satisfied for the potential q ( x ) , we conclude that G(k) is an entire function and that it is the ratio of two entire functions of order at most w = max(n,s) < 4, where s is defined in Lemma 16. Using (in the radial directions) the estimate of entire functions from below contained in section 8 of Chapter 1 in [28] (see also Theorem 1.5.6 and Corollary 1.5.7 in [22] or similar results in [7]), we conclude that G(k) is itself an entire function of order w with values in H2(Tn). Now Theorem 2.2.2 of [22] claims that the solution u(x) satisfies the decay estimate: for any p < W/(W -1)) where IC is an arbitrary compact in R n , and a E R n .
Using standard embedding theorems, we conclude that for any p < w/(w -1). Since w < 4, one can choose a value p such that However, Remark 2.6 in [12] and Theorem 1 in [29] state that there is no non-trivial solution of the equation with the rate of decay KUCHMENT AND VAINBERG Iu(x)I 5 Cexp(-c 1xIP), p > 4 / 3 . This contradiction concludes the proof of the theorem. [12] and [29], so this is the place where our argument breaks down for dimensions four and higher even if we require compactness of support of the perturbation potential v(x). The rest of the arguments stays intact (the embedding theorem argument, which also depends on dimension, is not really necessary).

Separable potentials
The result of the previous section leads to the problem of finding classes of periodic potentials that satisfy the Condition 11. As we stated in conjectures 12 and 13, we believe that all (or almost all) of periodic potentials satisfy this condition. Although we were not able to prove these conjectures, as an immediate corollary of the Lemma 14 and Theorem 15 we get the following result: T h e o r e m 18 If for 12 < 4 the background periodic potential q(x) E L"(Rn) is separable for n = 2 or q(x) = ql(xl) + qz(xz, x3) for n = 3, and the perturbation potential v(x) satisfies the estimate 6, then there are no eigenualues of the operator H in the interior of the bands oftthe continuous spectrum.

Comments
1. We have only proven the absence of eigenvalues embedded into the interior of a spectral band. It is likely that eigenvalues cannot occur at the ends of the bands either (maybe except the bottom of the spectrum), if the perturbation potential decays fast enough. This was shown in the one-dimensional case in [35] under the condition on the perturbation potential. On the other hand, if v(x) only belongs to L1, then the eigenvalues at the endpoints of spectral bands can occur [35].
2. Most of the proof of the conditional Theorem 15 does not require the unperturbed operator to be a Schrodinger operator. One can treat general selfadjoint periodic elliptic operators as well. The only obstacle occurs at the last step, when one needs to conclude the absence of fast decaying solutions to the equation. Here we applied the results of [12] and [29], which are applicable only to the operators of the Schrodinger type. Carrying over these results to more general operators would automatically generalize Theorem 15. The restriction that the dimension n is less than four also comes from the allowed rate of decay stated in the result of (121 and [29].

3.
It might seem that the irreducibility condition 11 arises only due to the way the proof is done. We believe that this is not true, and that the validity of the condition is essentially equivalent to the absence of embedded eigenvalues. To be more precise, we conjecture that existence for some X in the interior of a spectral band of an irreducible component A of the Fermi surface such that A n Rn = 0 implies existence of a localized perturbation of the operator that creates an eigenvalue at A. As a supporting evidence of this one can consider fourth order periodic differential operators, where the Fermi surface contains four points. In this case one can have X in the continuous spectrum, while some points of the Fermi surface being complex. Then one can use these components of the Fermi surface "hidden" in the colriplex domain to cook up a localized perturbation that does create an eigenvalue at X [32]. One well known example is the Bose-Einstein condensation, where the nonlinear potential is usually assumed to be cubic. It is natural to think of (0.4) as describing the evolution of a fluid with density p := 1zI2/2 and momenta p, := i(Zz,, -z2,,)/2, a = 1 , 2 , . . . , n.
In order to set the problem let me mention that, by standard arguments, one can show  T h e o r e m 1.1 : There exists a finite constant K such that Once the apriori bound in Theorem 1.1 is known, further regularity can be obtained in the manner explained in the introduction. The first step is to write the solution of ( l . l , a , b ) in integral form. The Green's function for the Schrodinger equation is For convenience let me call the nonlinear term by N [ z ] and the phase factor in S by O i.e. := lz(t,x)14z(t,x) ; i ( t , x )

lx12 N [ z ] ( t , x )
. (1.2, b) The solution of ( l . l , a , b ) can be written implicitly in integral form as follows, Let me give a name to the two terms in the expression above, The key ingredients in the argument are, first the conservation of energy i.e.

+ (1.3, a ) R 3 x t
The second ingredient is a Hardy-type inequality, namely The third ingredient is the identity, where 0 < la1 < 114. I would like to separate the inner and outer regions of integration, for this reason I want to introduce a smooth cut-off function, Now I can use the cut off function +(es), where e is a small parameter, t o write where

F,(t,x) := / $ ( r s ) N [ z ] ( t , x ) S ( t o -t , x o -x ) d x d t
The term Fo can be integrated by parts using ( 1 . 3 ,~) to obtain, (l.4, g ) Ix -xoI where, recall that ua := ( z axg)/lxxol and Next I want to separate the domain of integration in the region near to and the region far from to. Pick tl such that 0 < tl < to and a parameter a such that 0 < a < 114, for example take a = 116.  Assumption : I will assume in what follows that I would like to introduce first the technical tools needed for the intended estimates.
Consider a cut-off function $ such that, and let me use the notation,

NONLINEAR SCHR~DINGER EQUATIONS
Also, I will use the sets defined below,  A corrolary of the above theorem is the following. In order to simplify the presentation, let me call  [0, w2] or (ii) anti-periodic boundary conditions on one pair of sides of R and we may impose d u l d n = 0 = dAuldn on the remaining sides of R.
With these choices the set of equilibria is certainly a t least one-dimensional.
For, if u is a steady state, then a shift u(. + T , .) yields another steady state.
Two reasons motivated us to exploit boundary data of the above type: (a) Micro-graphs of real microstructure look periodic a t least away from grain boundary or boundaries of incompatible phases (see e.g. [27]). (I)) The minimizers of the one-dimensional energy functional are periodic, moreover the period is of order b1I3 (see [21]) for small values of 5. We stress that W is a double well potential here. Our prototype W is smoothed out version of w(() = min()ca)', 15 + a]') a # 0.

We have in mind the following examples
The feature of Wl is that it has just two minima, while W2 has a circle of minima. Our analysis will be applicable to both of these examples but not for the same set of boundary conditions. We collect our assumption on W as well as on the boundary conditions in Section 2 below.
One of our results presented here is existence of solutions t o (1.1) for to some extent permitted by the nonlinear term. Our basic method is a modified version of a transformation due to Andrews-Ball and Pego (see [I] and [23]) which was successfully applied not only to one-dimensional equation of ~iscoelasticit~ and its modifications (see 141, [ll], 1121, [18], [19], [23]) but also to multidimensional system of viscoelasticity, (see (251, [29]). In our problem global in time existence is obtained due to the energy identity The identity (1.4) is also a source of further a priori estimates, which lead to existence of the w-limit set in the C4J'(R) topology. They are proved by a patient boot-strapping argument. They are collected in Theorem 3.5 of Section 3.2.
Our main result, however, is Theorem 4.1 stating the convergence of each solution to an equilibrium point as t goes to infinity. The main hypothesis is that W is analytic. We also impose on W some non-degeneracy conditions.
We require that uo belong to some special neighborhood of a connected component of the set of local minimizers of E. The neighborhood in question is defined in terms of E alone. We also need that the kinetic energy 3 $ , u: dx is small. The analyticity assumptions is used to guarantee that critical values of E do not have an accumulation point. In this way we circumvent one of t l~e difficulties here which is the lack of knowledge of the structure of the equilibria of (1.1). This problem seems to be much less investigated than its counterpart for the Cahn-Hilliard equation, see [7], [8], [33], [34], [35].
It seems rather certain that the set of equilibria is large. This is true for periodic boundary conditions. Let us note that boundary condition (i) (rcspectively (ii)) imply that the set M of equilibria of (1.1) is at least twodimensional (respectively, one-dimensional), because shifts of u in X I , 2 2 (respectively X I ) yield new equilibria. Thus, if we fix u E M and set Lh = tliv(Da(Vu)Vh) + d2A2h, then we note that dim ker L 2 d i m M , But we do not know if the equality holds. If we knew that dim ker L = dim M, then we could apply the theory of Hale-Raugel [16] to deduce convergcncc. However, we do not know this. What we do here is we circumvent this problem. We analyze the behavior of the flow on the local center manifold W c ( u ) , where u is a local minimizer of E. Section 4.2 is devoted to establishing existence of Wc(u). The assumption that u is a minimizer permits us to deduce that it is stable with respect to the flow restricted to the local center manifold WC(u). The idea that it is sufficient to study behavior of the flow on Wc(u) has been present in the theory of dynamical systems for some time, e.g. we refer to [5]. Our proof of stability of the restricted flow requires some non-degeneracy condition on W, meaning D4W must be positive. The analysis is performed in Section 4.3. It is facilitated by a special scaling of the flow which is explained in Section 4.1. The proof of our convergence result is carried out in Section 4.4.
We present our analysis for n = 2. This restriction comes from the e~nbedding theorem which guarantees that for two space dimensions we have W112 (R) c Lp(R), for all p < co. This makes the energy finite for any polynomial growth of W provided that u E w2v2 (R). One may wonder, if for higher space dimensions, working with LP(R), where p = n, is an option. However, this is not the case since the available energy estimates yield houndedness of solutions only in W2v2(R). This is not sufficient to pol.c global-in-time existence of solutions for higher space dimensions and W of arbitrary polynomial growth.
Let us finally comment on convergence results for closely related problems. Most of them are one-dimensional.  [12]. In the multi-dimensional case the numerical simulations of [29], [20] and [14] suggest that convergence may hold, however we are not aware of any analytical result. The difficulty comes from the fact that the equation tends to have a big set of solutions for interesting boundary conditions (see e.g. [lo]).
(b) The case p = 0, S = 0, v > 0 , i.e. the equation of quasi-steady ;ipp~oxin~ation to (1.1) is better understood. If the number of the space clirnensions is one, then this problem is a differentiated in space reactiondiffusion equation for u,, for which we have convergence results due to Chen-AIatano [6] for a variety of boundary conditions and little smoothness of W. On the other hand the convergence result of [26] in multidimensional case requires that W is analytic. It is so, because the main tool of [6], i.e. the maximum principle is not applicable to the quasi-steady approximation of (1. 1). It is not clear to us if the analyticity assumption on W is essential.

Preliminaries and existence
We present here our assumptions on the domain R and on the boundary co~iditions. We provide here a list of boundary conditions for which the subsequent analysis applies. They all have one common ingredient, namely the hi-harmonic operator is the square of a sectorial one (actually, a selfadjoint one). We use this assumption in the process, of transforming our problem to a simpler one.
We also state here our assumption on the nonlinear term, a = DtW. In particular they allow W to have quite an arbitrary set of minimizers.
We will indicate that the energy functional, when considered on functions on a torus has infinite number of families of critical points. This is implied by symmetries of T2, however we do not pursue the task of studying the structure of the set of critical points of E.
We assume that u : R c 1Ft2 -+ IR. Our specific hypotheses on W : IR2 -+ IR, are: for all J E EL2, i = 0,1,2,3, where 1 . 1 is the Euclidean norm in EL2. 111 piuticular we rna.ke no assumptions on the set of minima of W , it might Itc it wrve (as in case (1.3)) as well as a discrete set of points (case (1.2)). 'rlrosv two conditions along with appropriate smoothness of W (say, W is C5) are sufficient to prove existence of solutions. However, our analysis requires niore, namely: (H3) for all R > 0 there exist CR > 0 and BR > 0 such that for all < E B(0, R) we have It is crucial for our studies to assume that W is analytic, meaning that for any u E C44'(R), W satisfies: uniformly in z E R for sufficiently small E E lR2, i.e., I(' \ < P(u), P(u) > 0.
Let us now turn our attention to the boundary conditions to which our suhseqlient analysis applies. We stress that some of them induce additional restrictions on W. Here is a partial list of some admissible boundary data: (Bl) 0 = [o, wI2, and u is periodic (i.e. u is well defined on torus T 2 ) and u dx = 0.
(B2) dR is smooth and u satisfies (B3) dR is smooth, v is the outer normal to dR and u satisfies This condition, however, requires an additional constraint on W , namely we need (H5) below to be able to perform integration by parts in the proof of the energy identity (see Theorem 3.7 (b) further in the text) and to prove that operator L in $4.2 is self-adjoint: We may also consider yet another boundary condition: (134) 71 : f2 -+ I!, where R = (0, wl) x ( 0 ,~~) and u is anti-periodic in x l , namely it satisfies and u also satisfies The same reason as in the case of boundary condition (B3) forces us to adopt the following restriction on W: Let us note here that anti-periodicity of u where u E wl+' (R), p 2 1, determines its average over R = (O,wl) x (0, w2) Namely, we have It is also easy to see that if u E C4 satisfies (Bl) and it is an equilibrium point, i.e.
-diva(Vu) + d 2 a 2 u = 0, then for all r E IR, u(. + r , .) and u(., . + r ) are also equilibria. However, if u satisfies (2.2) and (B4), then u(. + r, .) will also be ,an equilibrium if we assume that W satisfies in addition: We remark that other combinations of boundary conditions in the spirit of (B4) are possible, but we leave the details to the interested reader. We will concentrate our attention on (Bl) (or (B4)) since this assumption implies that the set of equilibria is at least two-dimensional (respectively, onedimensional). We stress, that (B4) compared to (Bl) requires additional conditions, namely (H6) and (H7).
Let us now state the fundamental property of all these boundary conditions Then for all p E [2, m), APyi are sectorial on Xp,i, i = 1,2,3,4. Moreover, if p = 2, then A2,i is self-adjoint and positive definite, i.e. 0 The proof is not difficult (it follows basically the lines of [17, 51.61) and it is left to the reader.
We also note here that the above Lemma implies that the fractional powers of A,,i are well defined, so are X t i the fractional powers of XPpi. We apply the convention: X,Ogi = Xp,i, X,',i = D(ApVi). We also have Proof. (a) If p = 2, then these statements are easy and left to the reader. The general case is shown in the course of proof of Lemma 3.6 in Section 3.3. Part (b) follows immediately from the first one and from the observation that 0 Subsequently for the sake of the simplicity of notation we drop the subscript i since the particular form of the boundary conditions is unimportant.
We  0 We leave an easy proof to the reader. It is also not difficult to notice that the periodic boundary conditions iniply that the set of families of equilibria is large. What we have in mind in particular is that each finite subgroup of a torus generates a family of solutions to (2.2). Namely, if G is a subgroup of T~, then it defines in a natural way an action on T2 which leaves E invariant. By the Symmetric Ckiticality Principle of Palais (see [22]) it follows that the critical points of E arc invariant under G. Since the number of finite subgroups of T2 is infinite, then we expect existence of an infinite number families of solutions t o (2.2).

Existence a n d s m o o t h n e s s of solutions
We first establish existence and uniqueness of strong solutions t o utt = divcr(Vu) + Aut -6 2~2 u , (3.1) boundary conditions: any of (Bl), (B2), (B3) or (B4).
Comparing (3.1) with (1.1) we see that p and u are set to 1. This is possible after an appropriate scaling. We will describe this scaling in Section 4.1, and we will see that 8 = 6u-1p1/2, but for the development of the present section the particular transformation of coordinates and the value of 8 is ~~niniportant (except for 6 > 0). From now on in this Section we will drop the bar over 6 and we will write 6 instead of 8.
In order to achieve the main goal of the Section we will use a version of the Andrews-Ball-Pego transformation. It had been applied to onedimensional equation of viscoelasticity but later it was adapted to a multidimensional setting (see [25], [29]). This transformation was also used to study one-dimensional viscoelasticity with capillarity, see [18], [19]. We will describe this transformation in Section 3.1. We will see that the problem is rctluced then to two coupled parabolic equations, to which the theory of analytic semigroups may be applied yielding existence and uniqueness of so-111tions to (3.1). We use the semigroup theory as exposed in the book by Henry, (171, with the updates made in the Russian translations. Since we intend to make rather straightforward applications of this theory the arguments we present are sometimes sketchy.
The second subsection is devoted to establishing a priori estimates on solutions. In particular, we need to establish higher regularity of solutions. These facts are required not only for a proof of the existence of the w-limit set but also we need them in our analysis of the flow on the center manifold. However, studying higher regularity leads to questions on the behavior of elements of XF for small a > 0. This is so because we avoid using the standard argument with flattening out portions of the boundary of R. We relegate our remarks on XF to the Appendix in 53.3.

Existence and uniqueness of solutions
By a strong solution to (3.1) we mean u belonging to aud satisfying (3.1) in the L2(R) sense.
We will prove global-in-time existence utilizing a version of the transformation due to Andrews-Ball-Pego. We will describe it momentarily.
If u is a strong solution to (3.1) we define P E D(Az) as a solution to by Lemma 2.1 P is well-defined and unique. We introduce the second variable Q E D('42) by If we insert the new variables into (3.1), then we discover that it turns into By Lemma 2.1 and the definition of strong solutions we may apply A-I to both sides of this equation. This yields where P ( z , 0) = Po(z) = A -' U~( X ) . For the sake of convenience let us set The eq~iation for Q is obtained by differentiation in time of (3.2b) . We thus have to deal with the system The second part of the Lemma follows from the first one if we notice that HOFFMANN AND RYBKA 0 R e m a r k . Only minor changes are required for other boundary conditions. We leave them to the reader.
We need two sorts of existence theorems: (i) for W satisfying (H1)-(H2), implying polynomial growth of DEW a t infinity; (ii) for cut-off systems. We use the latter in the construction of the local center manifold. (2) The Proposition requires only minor changes for the other boundary conditions. We leave them to the reader. Proof. We will apply the theory of analytic semigroups of [17]. We have ahcady noted in Lemma 2.1 that Bp is sectorial. We now claim that is locally Lipschitz continuous from YF to L2 (R). The Embedding Theorem [17,Theorem 1.6.11 and (H2) imply that a ( V ( P + Q ) ) E L' (R) Remark. Let us note that if zo E Y?, a < 1, then the initial data do not have enough regularity to guarantee that E(uo) is finite. However, we shall see that the data get smoothed out to some extent. Hence subsequently, we considcr much smoother solutions. (c) u is global in time; Proof. We carry it out in a few steps. Since the argument we use is stantlard, we present only a sketch while asking the reader to provide missing calculations. In the sequel, for the sake of simplicity of notation we write a ( t ) for a ( V P ( t ) + VQ(t)).   (6) We now define u = P + Q. It is easy to see from steps (3), (4) and (5) that u enjoy the regularity required a strong solution. One can also check if u satisfies our equation We are in a position to prove uniqueness. Suppose now we have two strong solutions ui, i = 1,2. If we set v = u1 -u2, then the difference v We now preceed in a usual way: we multiply the above equation by vt, we integrate the result over R x [to, t], (to > 0) and we integrate by parts.
Smoothness of a and Young's inequality imply We note that for all considered boundary conditions Poincarh's inequality is applicable (for (B4) this is guaranteed by (2.1)), which yields So, Gronwall inequality implies that v r 0, or u1 = u2, as desired. (8) We shall show the energy identity. In step (6) we established enough smoothness of solutions for differentiation of the left-hand-side of (b) to be permitted on (0,T). We leave to the reader checking that the derivative is zero. Continuity of (uo, ul) H E(uO) + S U: implies (b). (9) In order to show (c) it is enough to prove that P and Q are defined glohally. By part (b) we know E(u(t)) < E(uo) < m. This implies in

A priori estimates
We have not yet established higher smoothness of solutions. We will do this now. We essentially use a bootstrapping type argument here. While the ide is sirnple its implementation requires some work and rather lengthy calculations For the sake of readability we break our presentation in a number of steps. We also define Theorem 3.5. We assume that the assumptions of Theorem 3.4 hold. We assume that to > 0, then for all p E ( 0 , l ) there exist K = K(to) > 0 such that do this in two steps. We show first that 11 ~; /~z l l~~(~) 5 K, for t > to > 0.
We apply B;'~ to (3.9) (with B2 replaced by Bp), we see that Inequality (3.14) guarantees that the factor in front of the integral is finite, hence l l~~/~z ( t ) l l L v ( n ) < ~s ( l l~~'~z ( t o ) I l~~( n ) , to, E(UO), 6) 1.00 for t 2 to > 0 (3.15) The estimate for IIBEzJ(LP(n) requires a bit more work since a priori we do not know if F ( z ) belongs t o y:I2, i.e. if it satisfies the appropriate boundary ~ondit~ion. We note here that F ( z ) E D(B;/') automatically for periodic boundary conditions (Bl).
Formula (3.11), where B2 is replaced by B, yields for to > 0 where 11 > 0 is sufficiently small given by Lemma 3.6. So, we have and finally by the Sobolev Embedding Theorem we obtain Stcp (c). We claim that t

CONVERGENCE OF SOLUTIONS 1863
We now recall that D~U~ = D~A P , so by (3.17) and (3.18) our claim follows.
Step (d) We are now in a position to show that z E c~+ ( R ) , p E ( 0 , l ) and that Ilzllp,P(n) < K , for t L 1 .

Finally we look at J3. It is sufficient to estimate 11 J3(t)llwz,p(n):
We note that because of (3.17) and (3.18) we have a bound and the bound is independent of t > 1.
Step (e). The argument in step (d) shows also that for 6 > 0 given by Lemma 3.6 in the Appendix we have that for small E > 0. Moreover, the reasoning in step (d) shows that we have a bound Ilzlly,2+c 5 K8 < 0 3 1 (3.22) and the constant K8 is independent of t 2 1.
Our goal now is to show that z E W6J' (R), zt E W4J'(S2), hence we could perform differentiation with respect to time of equation (3.1).
We now examine 12, due to [17, Theorem 1.4.31, for a < 1 and (3.19) above we have for t > to > 0 Finally we look at 14, by (3.19) we obtain: Hence, we may conclude that for some small positive E: (

3.23)
Step (g). We improve here the estimates in step (d) by showing that I& notice that if we set q5 = DtaDw (resp. q5 = D p ) and v = V A P ( s ) (resp 1 1 = D2AP(s)) then they satisfy the assumptions of Lemma 3.7 of $3.3. Hence, we can conclude that Stcp (h). We may now show that rt E C([l, co); W4J'(R)). By formula (3.12) we have

CONVERGENCE OF SOLUTIONS
The argument presented in step (g) applied to the above formula shows that indeed zt E Y;.
Step (i). We conclude from step (g)  at Hence we may differentiate in time the RHS of (3.1) for t > 0. After a series of integration by parts where we used the specific form of our boundary conditions (and (H5) or (H6) if necessary depending on the boundary conditions) we can easily see The last term is easy to estimate By (3.14) and (3.18) the RHS is bounded. So, after integrating dEo/dt on [to + E , t], E > 0 and passing with E to 0, we come to the desired result.
As an immediate corollary to this Theorem we obtain low IVAP, l2 d~d t < m .

(3.24)
Remark. It is apparent from the proof of Theorem 3.5 that if zo E W6J' (R) and zo E D(B;) for p > 2, then z E C([O, m ) ; w~J ) ( O ) n D(BZ)). We will use this fact in 54.4.

Appendix
Our objective is to make clear that the spaces X : are independent of the boundary conditions for small a > 0. This fact is well-know for domains with siiiooth boundary, but in (Bl) and (B4) the boundary is not smooth. We also state a proposition on multiplication of element of X," by Holder continuous functions. This fact may be found in [24] but in a slightly different context. We will use the notation introduced in Lemma 2.1. Let us recall that F,",, stands for Triebel-Lizorkin spaces, (see [24], [30], [31]). At last we consider i = 4. We define El : LP(R) + LP((0, 2wl) x (0, w2)) by formula and we also set E2 : LP((0, 2wl) x (0, w2)) -t Lp(T2), where T2 = ( 0 , 2~1 ) x We set E := E2E1. We note that E u E Z. We also define two projections A, S : Lp(T2) n Z -+ Lp(T2) n Z: 1 Su(x1,x2) = -(u(x1,52) + ~(~1 1 -52)). 2 It is easy to establish that AS = SA, hence P := AS is a projection, moreover It is also easy to show that

Hence we reach
Our conclusion is that the spaces XE4 may be identified with complemented subspaces of X i , . Thus, (a) and (b) follow for i = 4 from the results for i = 1.  The boundary conditions (Bl) (respectively, (B4)) imply that d i m M 2 2 (respectively, d i m M > 1). The Theorem above implies that if the initial conditions are close to a minimum of E, then solutions must converge to a possibly different global minimum (this is in stark contrast to viscoelasticity in one-dimension, see [23], [Ill). We note that in some circumstances the theory of [16] seems applicable, e.g. for W given by (1.2). We have seen in Proposition 2.3 that the global minimizers are independent of 2 2 , so it is relatively easy to analyze the resulting problem and to show that actually dimker L = 1 = d i m M , where Lh = -div(D:W(VU)Vh) + d2A2h) and U is a global minimizer of E. But it is not clear if this kind of reasoning may be carried out for any local minimizer of E . What we do here is we circumvent this problem. Namely, we look at the behavior of the flow of (3.1) on a local center manifold WC(U) for a fixed U E M . Our main goal is to show that U is stable with respect to the flow restricted to WC(U). The general theory (e.g. [17,Chapter 91) implies then that U is stable, hence the w-limit set of (vo, u l ) must be a singleton. Of course, it is not true in general that a trajectory starting close to a manifold of equilibria will converge to a unique equilibrium point.
The difficulty in carrying out our program is the presence of the third ordcr terms in the expansion of E around a critical point U E M. We show that we can make this terms negligible by properly scaling the equations. We introduce those scalings in Section 4.1. Next, we prove existence of the local center manifold in Section 4.2 and in Section 4.3 we show that if U is a local minimum and U + w belongs to the local center manifold and the appropriate norm of w is sufficiently small. We will show that inequality
It is now easy to see that with these definitions equation (  2 BR, (4.5) with the hound independent of a. We also notice that the boundary conditions do not change under these transformations. Let us note that the following generalized eigenvalue problem will be important in our considerations -div,(Dp(V,u)Vh) + d2aZh = XA,h, in R We need to establish their relationship. It turns out that we have We note that +, is differentiable and its derivative is given by We want to look at the dynamics of the cut-off system introduced below.

CONVERGENCE OF SOLUTIONS
Otherwise we may assume that and r > 4.
It is not difficult to see that Thus, (b') follows immediately.
If we keep in mind all these definition, then the system, (4.9) may be rewritten as 2' + G.r = f (x, y), Our present task is to check that the assumptions of [17, Theorem 9.1.11 are fulfilled. We first show that G is sectorial on X .
It is a well-known fact that since Bz is sectorial on Yz, then it is also sectorial on Y: " .
On in -m < t < $00, then for all t , T . We shall see that the proof of this fact requires Re spectrum ( 6 ) > Xo > 0. (4.13) We postpone the proof of (4.13) for a while and we pick any 0 < p < Xo, N = 1, then we choose q such that for all v E V, p = l )~l l l a~i~~. q , where K is such that ~,-~IIvllx; 5 llvllv 5 ~I I v l l~; .

Equation (4.12) yields
By Lemma 4.3 (b) we have that Furthermore by Gronwall inequality we obtain for t > T, as well as for T > t: Now we shall show (4.13), it will follow that (ii) of [17,Theorem 9.1.11 holds. In the proof of E being sectorial we have actually shown that 6 is a compact perturbation of B2 restricted to a smaller space, thus their essential spectra agree. Hence the essential spectrum of G is empty. It remains t o clicck that the real parts of all eigenvalues are positive. Let us suppose to the contrary, i.e. (x1,x2) is an eigenvector such that the corresponding cigerivalue X has Re X 5 0. Then After adding them up we obtain ant1 it also follows that x l is smooth. We can solve this equation for 2 2 and feed the result into the first equation of the above system: We now take the inner product in ~' ( f l ) with X I , after rearranging the terms we see 1 Hence, WC(U)' is indeed a center manifold, i.e. it is tangent to the flow at (0,O) in the x, y coordinates. Moreover, by the same Theorem, because of (4.13), the set WC(U) is asymptotically stable, meaning that if Po < XI , 11x(~)11~; is sufficiently small, then there exists a solution (2(t), ~( t ) ) in WC(U) such that Remark. The membership of h in C1?l and (4.14) imply that

Behavior of 6E on the center manifold
We prove here an inequality which is central to our proof of stability of the flow restricted to the local center manifold. We are able to show this only for certain scalings of (3.1). On the other hand it is sufficient to show stability of the flow for some scaling parameter a, because if the flow is stable for some a it is stable for all positive a. Let us recall that V, h : V -+ YF have the same meaning as in the previous subsection and tilde over a letter denotes the transformation defined in (4.3). To be precise we should distinguish between arid V, however (4.3) defines an isomorphism between them. For our convenience we also introduce more notation: then the above i~iequality is not obvious. Proof. In our calculations we use that u(t) E W4+'(S2). This degree of smoothness was established in the proof of Theorem 3.5. We write a Taylor expansion of E, where we use that u is an equilibrium. It turns out that 121 is the most difficult term to estimate. One may think that this term scales in a as a fourth order term because h ( v ) is quadratic in V (see (4.15)). However it is not quite true, i.e. we will show that its scaling CONVERGENCE OF SOLUTIONS 1881 bcliavior is rather different from the behavior of I3 which is of degree four.
Our calculations are not straightforward. We have to exploit in an essential way the fact that dim V < oo. We note that by Young inequality we obtain where E > 0 is to be specified later. Now, due to (4.8) we see Subsequently, by (4.15) 24 4' We next choose 1 > p > 0, such that for IIVyCllL4(n) < p inequalities (4.18) and the following ones

Proof of the convergence result
We are now ready for the proof of our main Theorem. Let us explain the itlea. We first show that the w-limit set exists, this is guaranteed by the a priori bounds of $3.2. We pick a point (0, U) E w(uo, ul). Analyticity of W guarantees that in a neighborhood of M there are no other critical points of E. We analyze the flow restricted to the local center manifold Wc(U)for this we have to consider the cut-off system (4.9). We want to show that the flow restricted to WC(U) is such that all points in the w(uO, ul)-limit set are stable, hence they are all stable. This implies that w(uo, ul)-set must be a singleton. The proof of stability of the restricted flow depends very much o i l the inequality from Lemma 4.6. We stress that our analysis does not go through if U is just a saddle point of Ewe would have to deal with "bad" third order terms in the Taylor expansion of E near U. Proof of Theorem 4.1. We divide our reasoning in a number of steps. We stress that we work with t,he (P, Q) variables which are easier to deal with than the original ones.
(1) We first establish existence of the w-limit set. By Theorem 3.5 (a) the set {(P, Q)(t) : t 2 1) is bounded in c~+'(Q), p' > 0. But the embedding c4>/"(iI) c C4+(fl), for p' > p is compact. Thus existence of a connected conipact w-limit set follows from thc standard theory of dynamical systems (c.g. scc [17,Theorem 4.3.31 or [15,Theorem 3.8.21). We also know that We note that E is a Liapunov function of the system, hence w(uo, ul) must bc a subset of the set of equilibria, say (0) x M. We also notice that  We are now in a position to define the set U. Namely, we choose U := V,.
If uo E WGvp(R) is such that uo E U , then m -E(uo) > 0, so if we take any such that Sn U: < 2(m -E(uo)), then by Theorem 3.5 (see also the remark at the end of $3.2) P(t), Q(t) belong to W6J' (S1) for a11 t 2 0 and Thus E(u(t)) < m and u(t) E U for all t 2 0.
(4) Let us fix a point (0, U ) in w (uO, ul)-limit set. We subsequently choose 77 > 0 provided by Theorem 4.5. We conclude existence of the center manifold WC(U) for the cut-off system (4.9).
(5) We prove stability of the flow scaled first then restricted to the Wc(U). The scaling parameter is given by Lemma 4.6. For the sake of simplicity of notation we suppress the dependence upon 77 of variables p, Q. We set Q = Q -0, and we look at the equation for Q if + Qllx; < 7.
We now write p + Q = u + a + h(v), where 6 E 8. Thus, A, Q_= AY(U + i; + h 7 ) -P )~ and we take inner product of (4.21) with a + h(v We note that Qt = Pt -A,P, so T l~c sccond integral is finite due to Theorem 3.4 (b). The first one is finite t)ccalse of Poincark inequality and (3.24) Hcnce, for sufficiently large to Tlic above inequality shows that for some a > 0, the local minimizer U is stable with respect to the flow restricted to the center manifold. But stability is independent of the scaling, so in particular U is stable with respect to the unscaled flow restricted to the local center manifold.
We define a short range potential to be a real-valued function which is a classical symbol of order -2. Recall that the class, S$(Rn), of classical symbols of order m is the set of smooth functions f(z) such that there exist a sequence of smooth, homogeneous functions, fm-j, of order mj such that for lzl large where (2) = (1 + l~1~) ) .
Let A be the positive Laplacian on Euclidean space, R: . We are interested in studying solutions of with X a fixed positive number. It is well-known (see for example [6]) that given a function fo E Cm(S"-I), then there exists a unique solution u of (1.1) of the form, with f , g classical symbols of order -*, with the lead term of f equal to the function 1 2 1 -9 fo(z/lzl). The scattering matrix, S(X), is then defined to be the map taking fo to the lead term of g multiplied by lzl*. The scattering matrix is then a map on Cm(Sn-') and extends to a unitary map on LZ(Sn-I). We define the Poisson operator, P(X), to be the map taking fo to u. Throughout, we shall think of s n -1 as being the sphere at infinity and we can think of fo as being incoming data on this sphere and S(X)fo as being corresponding outgoing data. We will reserve x for 121-' which should be thought of as the boundary defining function of infinity.
Note that if we put w = z/lzl, then and so the compactified Rn is a special case of the scattering metrics discussed in 171.
We prove, Theorem 1.1. Let V1,V2 be short range potentials and let Sj(X) be the associated scattering matrices. Then Sj = Pja8, with Pj a zeroth order classical pseudodifferential operator and a* pull-back by the antipodal map and i j i n addition Pl -P2 E @DO-m(Sn-l) then V, -V2 E 5'-"(Rn).
and so i f f is an integrable function on Sn-', we can define and we have (A -X2)u = 0. If f is smooth we can apply stationary phase and conclude that u is an eigenfunction of the form and in fact we have a complete asymptotic expansion. This show that eiX'." is the Schwartz kernel of the Poisson operator. It also shows that the scattering matrix for the free problem is just pull-back by the antipodal map.
As in [7], our approach here is to construct the Poisson operator for the scattering problem for A + V -X2 and then the properties of the scattering matrix can be read off directly. As in 141, we shall see that in showing the recovery result it is easier to examine the difference of the Poisson operators associated to two potentials than to compute lower order terms of a single onehowever we shall indicate how the lower order terms can be computed if one so desired.
We construct the kernel of the Poisson operator, P(X), as a perturbation of the one for the free problem; in particular we look for it in the form eiXLW(1 + a ( z , w)) with a E S,;'(Rz) and depending smoothly on w. We successively obtain transport equations on the asymptotic expansion of a which are along the great circles running from w to -w. The transport equations degenerate on approach to -w and following [7], we need a different ansatz there which is really a micro-local blow-up. In particular, we look for P(X) in the form, with a E Cr([O,e) x [0,e) x Sn-'), where we have rotated z so that w is the north pole and z = (zl,z,).
Once we have constructed P(X) we see easily that the difference of the scattering matrices associated to two potentials which agree so some order is a lower order pseudo-differential operator (pulled-back by the antipodal map) and that the principal symbol of this operator contains precisely the data obtained by integrating the lead term of the difference of the potentials with a sine function weighting, over geodesics of length n. We then give a simple argument showing the injectivity of this transformation. An induction on the order of the difference then completes the argument.
We refer the reader to [4] for a brief history of this problem. The recovery result of [4] was extended to cover potentials of the form C/(z( + V with V short range in [3] by using complex orders of homogeneity and that could be done by the techniques here also. Vasy, [a], has shown that the techniques of [7] can be applied to realvalued potentials in S,;' (IW:) by considering complex orders of homogeneity which vary from point to point. It is therefore likely that a recovery result cdn be proved in that case also by a combination of those techniques with those of this paper and [4]. We also remark that the techniques developed here could also be applied to reprove the result of [5] with little change but we leave this to the enthusiastic reader.
The work here bears some similarities to that of Agmon and Hormander, [I].
They constructed solutions of higher order constant coefficient partial differential equations by using oscillatory integral techniques.
I would like to thank Maciej Zworski for pointing out the connections between (7) and the problem of recovering asymptotics. I also thank the referee for his alacrity and care.

THE POISSON OPERATOR AWAY FROM THE ANTIPODAL POINT.
In this section, we construct the Poisson operator away from the antipodal point and look at how a perturbation affects it.
Suppose V E Si2(R:) then we have, So we look for a Poisson operator of the form eiA"~"(l + a ) with a a classical symbol of order -1 in z depending parametrically in w. Applying tlie operator we obtain a a e'xz~w(-2iXw.
We wish to pick the asymptotic expansion so that this is Schwartz. This fact is what makes the inverse problem argument work as this essentially is the principal symbol of the difference of the scattering matrices.

THE ANSATZ AT THE ANTIPODAL POINT
We need a different ansatz to deal with the singularity at s = x -. Consider

-Jm'
We deduce that at stationary points, and thus that 2 = 0 if and only if 1 z'

(3.2)
We therefore compute that the phase at critical points is -Iznl and so is z, in z , < 0.
This is w.2 if we take w to be the north pole. The Hessian of q5 is ~ orthogonal to 0 is an eigenvector we have that the determinant is where 0 = z l / ( z J . Observing that 0 is an eigenvector and also that any vector So provided we are in z, < 0 we can use stationary phase to transform from form (3.1) to the original ansatz. Its important to realize that we can also do the opposite. To see this observe that if we have a function eixz.wa(z, w) with lead term with c of order m -1 so repeating and asymptotically summing we can achieve equality modulo an error of order -m. Note also that the amplitudes in each form are unique modulo terms of order -m and that there is a simple relation between their lead terms or principal symbols. Whilst this argument was carried out for w the north pole, we can always, at least locally, in w redefine our z coordinate system to make this true and patch together. So integrating by parts we obtain the same form but now with the amplitude, with b now a symbol of order (7 + 1, a + 2). Differentiating this yields a symbol of order (7 + 1, a + 1).
The second piece is obtained by applying the Laplacian to the phase. This multiplies the symbol by -, / m ( nl)(z(-' and so again we get a term in p+l,a+l.
The third piece is obtained by applying the Laplacian to the amplitude, it is straightforward to see that this is in 17+2,"+2.
The final piece and the most troublesome is obtain by differentiating the phase and the amplitude once each. This yields an amplitude: The second term is a symbol of order (7+2, a+3). The first term is more troublesome as we have a zj which is not allowed for in our definition. Before multiplication by zj the first term is in 17+3va+3. So the proof is complete subject to the proof of If $ denotes the phase function, we have So we can replace z, by 2 + -f+IzI. Returning to polar coordinates, the second l+I4 term will decrease the 7 by one. Integrating by parts in u j and returning to polar JOSH1 coordinates we see that the second term decreases the (Y order by one. The lemma follows. Of course we also need the mapping properties under multiplication by short range potentials. Away from the antipodal point, these are immediate as we can just convert to the form eix"."a(z, w ) , with a symbolic in z. We will also need a standard asymptotic summation result.  So bringing w back in, we conclude that the lead term is a pseudo-differential operator (after pull-back by antipodal map) of order a -7 -( n -2). Note that the principal symbol of this operator is the lead term of a as t goes to 0.
So to summarize, we have This result on asymptotics also allows us to prove a slightly surprising but very useful fact about the mapping properties under the Laplacian which we will need later. This is what makes this ansatz work, for example one could have blown-up the symbol at z/lz( = -w,lzl-' = 0 instead but then one would not have this property and the argument would not go through. Proof. Note that the Laplacian in polar coordinates is Now if we apply, (K, f ), and K is a pulled-back pseudo-differential operator of order --(7 + j), and that the principal symbol is equivalent to the lead term of the amplitude of u as S -+ 0.
Now we have with T = lzl that RECOVERING ASYMPTOTICS OF POTENTIALS 1919 so we conclude that J(A + V -X2)u(lz18, w) f (8, w)d8dw decays to order -9 -2 rather than -* -1 as predicted by Prop 3.4. This means that the lead term of the distributional asymptotics is zero and that the lead term of the amplitude of (A + V -X2)u vanishes as S -+ 0. This means that (A + V -X2)u E 17+j+1-0+1 + 17+j,"+'. If repeat our argument on the first piece, our result follows.
As we are attempting to solve by getting improvement in the 7 filtration we note that Proposition 3.5. If u E I"," = n 173" then u = e-"lzl f (z), with f a classical 7 symbol of OT&T -a -1.
After this one gets an error of the form e-"lZlc with c a symbol of order -?, this can then be removed by iteration. (Indicia1 equation argument.) Note that this term makes a smooth contribution to the scattering matrix in any case. The Schwartz error is then removed by applying the resolvent and also gives a smooth contribution.
So first we need to solve the transport equations right up to S = 0. As we have behaviour of the form & in the t variable the order of singularity allowed in s will increase by one as the order of lzl decreases by one. In particular, the lead term will have a singularity of order a -7 as s -+ 0 + . Now as we have to convert from our previous ansatz, using the stationary phase computation above, we must take 7 = -9, and a = 9, as the principal term should look like SnV2 as S goes to 0. We then have from Prop 3.4 that the distributional asymptotics have lead term e-"l"llz~-*~, with K the kernel of a zeroth order, pseudo-differential operator composed with the pull-back of the antipodal map. The term, K , will be the Schwartz kernel of the scattering matrix.
Now we carry out our construction. We look for our solution to be in I-+,+, close to the south pole and we know from above that uo = ei'z.W will solve for the leading term so applying A + V -X2, we obtain an error in el E I -* +~~~+~, using Prop 3.4. Now the symbol of this will blow-up to order n -2 as S -+ 0 + .
Converting back to the form of the first ansatz the symbol blows up to order 0 as we see that the quantity for k -2 is determined also. So if k is even we can reduce to k -2 = 0 and if k is odd to k -2 = 1. Now if we have no sin weighting and we differentiate, we obtain the difference of the function at the endpoints of the geodesic that is the odd part of the function. If we have a sin1 weighting differentiating a couple of times we see that we can similarly recover the odd part of the function.
Now if k is even, we can join together two half geodesics to obtain the integral of W-k around the full closed geodesic. Now it is a theorem of Funk that the integral of a smooth function around every closed geodesic on Sn-l, is zero if and only if the function is odd for n 2 3, see for example 121, so we conclude that if the principal symbol is identically zero for k even then W-k is identically zero.
If k is odd, we observe that if a geodesic starts on z, = 0 then the z, coordinate will be a constant multiple of sins, with s the geodesic distance. So can then apply Helgasson's result to z~W -~ and conclude that if all the integrals vanish then W-k is even and as we already knew it had to be odd, we conclude that it is zero.
Hence inducting on k we deduce that if the difference of the scattering matrices for one energy is smoothing then the difference of the potentials is Schwartz. We denote by fl the real representation of R, and by dfl its boundary.
We assume the following conditions Here indl o (resp. ind2 a ) is defined by indlo = -4 dcl log4R1Gl 4 1 2 , f). (R1, R2) E 8 6 . We also set R1 = R2 = 1 for the sake of simplicity. for P. We note that because the Toeplitz symbol is independent of z, we can apply Theorem 2.1 as well as Theorem 2.2 if k > 4.
By simple computations of the characteristic polynomials, we see that M(u) at u = is elliptic outside the set xy = 0 if and only if 4 < k < 12. Note that if k < 4 or k > 12 the equation is weakly hyperbolic and degenerates on the lines x = 0 and y = 0.
Next we will estimate the integer N in Theorem 2.2 when k > 4. Because P preserves homogeneous polynomials we study the injectivity of P on the set of homogeneous polynomials of degree greater than 5. By definition, a monomial x"yp ( v + p > 5) is in the kernel of P if and only if Since k > 4, we easily see that (2.10)  verified if t2 # f 1. In the case t2 = f 1, (2.13) implies that k $ 4731% f 6. It follows that k 9 [-6, -41 and k @ [6,8] since 0 5 731% 5 112. Therefore ( A . l ) is equivalent to k < -6 or k > 8. We can easily verify (A.2) for q1 = 0, ph = 1 under these conditions.
On the other hand, if we want to apply Theorem 2.1, it is necessary to verify (A.1). By simple computations this is equivalent to verify (2.13) for It( = p, where-p varies in 0 < 3pl 5 p I 3p2 < oo. Hence the condition is valid if k is sufficiently large. We will show that the type of M changes near the origin. Indeed, if k > 8 fo changes sign on the four lines fo(x, y) = 0 in R2 intersecting at the origin.
Hence the equation is hyperbolicelliptic near the origin. If k < -6 the origin x = y = 0 is the only zero of fo in R2, i.e., P is degenerate hyperbolic.
Because b > 6 and 0 5 ~1~2 5 112, (2.14) holds if c -8b > 0. The condition (A.2) for 171 = 1,712 = 0 is easily checked by definition. It follows from Theorem 2.2 that (2.7) has a solution if the order of g is sufficiently large. If we want to apply Theorem 2.1 we have to verify (2.14) for It1 = p with 0 < 3pl 5 p 5 3p2 < oo. Clearly, this condition is valid for sufficiently large b depending on P1 and P2.

Reduction to the boundary
In this section we obtain a crucial estimate for the linearized operator P in For cr E N2, p E N2 and a smooth function u we have x a g u = xu-Pxaatu.
Because the normal derivative r&, can be expressed by a tangential derivative 89, , we can restrict P to the torus (1x1 1 = R1) x {1x21 = R 2 ) Namely, we can regard P as the operator on W R ( T i ) . By definition we have aP = P on WR(Ti).
Let < Do > be a pseudodifferential operator on T i with symbol < q >:= ( 1 + 1q(2)1/2, where q is the covariable of 8. Then the operator .rrP <Do >-"= P < D g >-m on W R ( T i ) is called a Toeplitz operator on W R ( T i ) . Note that the Fredholmness of P <x& >-" on WR is equivalent to that of the Toeplitz operator .rrP <DO >-" on W R ( T i ) . For the proof we prepare

1935
The proof is done by exactly the same method as that of Theorem 3.1 in [6].

Proof of Theorems
Proof of Thwrem 2.2. We note that WR(Ti) is a Banach algebra by the usual multiplication of functions. We restrict the equation (2.2) on WR to WR(Ti).
Hence there exists a solution UJ E X to (4.2).
As to the uniqueness, let IIwlllR < p, llw211R < p be the solutions of (4.3).
Then we see that v = wl -w2 satisfies v + K(swl, Sw2)u = 0, where K(swl, Sw2) := K(Swl) -K(Sw2) is a polynomial of Swl and Sw2. Because 11 K(SWI, SW~)I(R < 1 for sufficiently small p, we have llvlIR 5 11 KvIIR < llKlIR11~11R < llvllR. Hence v = 0. Let w (w = Sv) be the anaytic extension of w into the polydisk DR. The analytic function M(Q+w)fo(x)-g (x) in DR vanishes on the Silov boundary of DR by the construction of w. Hence the maximal principle implies that it vanishes in DR. Hence lii is the solution of (2.2).
Suppose that there exist two solutions wl and w2 of (2.2). By the unique solvability of the restricted equation we see that w1 = w2 on the boundary.
In the following we denote by Q the pseudodifferential operator with symbol given by Q(0, <). For u = Ct G(t)@e"f E WR(TE), we have By changing the summation, the right-hand side is equal to   Proof. We make the change of variables x j = y j l in (6.1) to obtain for some Q. By simple calculations the Topelitz symbol of the linearized o p erator of the left-hand side of (6.4) at u = u0 is given by (6.2) and (6.3).
Moreover, we may solve (6.4) in a bounded complete Reinhardt domain b e cause 0' cc R,. By the similar argument as in Theorem 5.3, we can prove the existence of a solution for (6.4).
In the above theorem the existence of a linear part is crucial in actual applications. In the following, we suppose that the linear part in (6.1) is zero, 1944 YOSHINO namely a, p = 0. Furthermore, we assume n = 2, and we shall study regular and singular solutions to the equation where x = (x1,x2) E R2. Let a = (al,a2) E Z2. Let X j be the linear space of all formal power series in x, given by Xj = { u ; u = xiaJ Ego v k x j k ) ( j = 1,2). Then we have Theorem 6.2 Suppose that f = 0 and Ial = a1 + a 2 # -1. Then every solution of (6.5) of the form u = x-" CqEzt uqx-'J is contained either in X1 or i n X2. Hence they depend only on one variable. We first note that the change of variables xj = zy1, ( j = 1,2) in (6.5) yields where bj = zj(a/azj) ( j = 1,2) and g(z) = z;2z~2 f (z;', 2;').
By comparing the homogeneous part of degree 2m + k + 1 in (6.6) we obtain (6; + 62)uk+l = 0, to yield uk+l = c k + l z~+~+~. This proves that u E X I . We can similarly prove that u E X 2 in case a = (0, m ) . In view of la1 # -1, the other case m 2 0 is contained in the following proposition. Proposition 6.4 Let u = Cp=, u, be the homogeneow expansion of u in (6.6). Suppose that uj = 0 for any j < k and that uk # 0. Then we have the followings a) Either uk = czt or uk = cz,k holds for some c # 0. b)Either u, = c,zr ( v > k ) or u, = c,,z," ( v > k ) holds for some c,,.
Proof. If we prove a), b) follows by the same argument as in Theorem 6.2.
In order to prove a) we note that the comparison of terms of homogeneous order 2k in (6.6) implies that uk satisfies (6.6) with g = 0. We substitute the expansion of uk, uk = J $o ajziz:-j into (6.6). Since the term zfk does not appear we compare the coefficients of zfk-'z2, to obtain akak-1 = 0.
If ak # 0 it follows that ak-1 = 0. Next, by comparing the terms of we obtain akak-2 = 0, that is ak-2 = 0. In the same way we can show that uk depends only on zl.
We consider the case ak = 0. Because the assertion is trivial for k = 1 we assume k 2 2. By comparing the coefficients of z:k-2z,2 in M ( u k ) = 0 we have that a:-, = 0. Suppose that ak-, = 0 for v I: j L k -2. It follows that uk is of degree greater than j + 1 in 22. Hence, by comparing the coefficients of 1946 YOSHMO degree 2 ( j + 1) in 2 2 , we obtain ag-,-, = 0. It follows that ak = . = a1 = 0.
Hence uk depends only on 22. We will study the solutions of the form u = CVGz: u,,x-q of (6.5) or (6.6). By writing g ( z ) = C z 2 g, with g, being homogeneous degree j, we seek u = C,"=, u, where u1 = azl + bz2 and uj is homogeneous degree j. We assume ab # 0. If there exists a formal solution of (6.6), we can easily see that g ( z ) satisfies the condition g ( z ) = O(zlz2). By comparing terms of homogeneous degree 2 in (6.6) we have g2(z) = 4abzlz2. We assume these compatibility conditions. By the scale change of variables we may assume a = b = 112. Proof. We use the same notations as above. In order to determine u2 we compare the terms of homogeneous degree 3 in (6.6). We then have P1u2 = g3(z), where PI = zl(6; + 62) + ~~ ( 6 : + 61). If we substitute the expansions u 2 ( z ) = Q Z ,~ + c1z1z2 + c2z? and g3(z) = dlzlz; + d2z?z2 into the equation P1u2 = g3(z) we obtain that 6% + 2c1 = dl and 2cl + 6c2 = d2. These equations have a unique solution once we give q, or c2, which is a kernel element of M ( v ) = 0 in view of Theorem 6.2.
Suppose that we have determined u j for j < k . By comparing ( k + 1)-th homogeneous part of (6.6) we see that uk satisfies Pluk + (. ..) = gk+l(z), where the dots denotes the terms determined by u, with j < k. Because we can easily see, from the definition of M that these term are divisable by zlz2 the recurrence relations has the same structures as for u2 and we can determine u k if we assign the kernel element in uk. The rest of Theorem 6.5 is clear. 0 Corollary 6.6 Let u = ul + v + S v be a fonnal power series solution of (6.6 We define the space WR as in Theorem 2.2 with respect to the variable z. Now, we drop the assumption ab # 0 in Theorem 6.5. For simplicity we assume ul = 0 and we look for the solution of (6.6) in the form u = xZ2 uj, where u 2 ( z ) = az? + bzlz2 + czi, uj being homogeneous degree j . We define 94 = M(u2).