A variational proof of Nash's inequality

This paper is intended to give a characterization of the optimality case in Nash's inequality, based on methods of nonlinear analysis for elliptic equations and techniques of the calculus of variations. By embedding the problem into a family of Gagliardo-Nirenberg inequalities, this approach reveals why optimal functions have compact support and also why optimal constants are determined by a simple spectral problem.


INTRODUCTION AND MAIN RESULT
Nash's inequality [17] states that, for any u ∈ H 1 (R d ), d ≥ 1, where we use the notation v q = R d |v| q d x 1/q for any q ≥ 1. The optimal constant C Nash in (1) has been determined by E. Carlen and M. Loss. To state their result, let us introduce ω d , the volume of the unit ball B 1 in R d , λ 1 , the principal eigenvalue of the Laplacian with homogeneous Neumann boundary conditions, and x → ϕ 1 (|x|) an eigenfunction associated with λ 1 , normalized by ϕ 1 (1) = 1.
The compactness of the support of the optimizers in (1) can be understood by deriving Nash's inequality as a limiting case of a family of Gagliardo-Nirenberg inequalities [14,18] for all u ∈ H 1 ∩ L p (R d ), where 1 < p < 2, a = a(p) = d (2 − p), and b = b(p) = 2 p. Nash's inequality corresponds to the limit case as p → 1. The inequality (3) is equivalent to the minimization of u → ∇u 2 2 + u 2 p under the constraint that u 2 2 is a given positive number: any minimizer solves, up to a scaling and a multiplication by a constant, the Euler-Lagrange equation In our main result, which follows, we use the notation of Theorem 1.  (3), written with an optimal constant, is obtained after a possible translation and scaling of x, and multiplication by a constant, by the nonnegative radial solution u p of (4). The support of u p is a ball of radius R p > 0, such that lim p→1 + R p = R 1 := λ 1 , and u p converges, as p → 1 + , to Our result is in the spirit of [9]. In their paper, E. Carlen and M. Loss establish the optimality case by direct estimates as chain of inequalities which become equalities in the case of the optimal functions. Our contribution is to establish first why the problem can be reduced to a problem on a ball involving only radial functions, using the theory of nonlinear elliptic PDEs and methods of the calculus of variations. Identifying the optimal case is then an issue of spectral theory.
Let us highlight a little bit why we insist on the compactness of the support of the optimal function. The first application of Nash's inequality by Nash himself in [17] is the computation of the decay rate for parabolic equations. We consider a solution u of the heat equation with initial datum u 0 ∈ L 1 ∩ L 2 (R d ). If u 0 is nonnegative, so does u(t , ·), and mass is conserved : u(t , ·) 1 = u 0 1 , for any t ≥ 0. For a general initial datum, u(t , ·) 1 ≤ u 0 1 for any t ≥ 0. By the estimate and Nash's inequality (1), y(t ) := u(t , ·) 2 2 can be estimated by which, after integration, yields that the solution u of (5) satisfies the estimate for all t ≥ 0. This estimate is optimal in the following sense: if we take u 0 to be an optimal function in (1) and differentiate (6) at t = 0, it is clear that C Nash is the best possible constant in the decay estimate (6).
With this result at hand, it comes a little bit as a surprise that the optimal function has nothing to do with the heat kernel and is even compactly supported. This can be explained by noting that (6) is an optimal result for small times, but it can be improved concerning the long time behavior. Estimation of the solution u(t , ·) = G(t , ·) * u 0 of the heat equation by Young's convolution inequality gives The sharpness of this result (take u 0 = G(ε, ·) for an arbitrary small ε > 0) and a comparison with (6) imply An estimate, which is optimal for both small and large times, is now obviously obtained by taking the minimum of the right hand sides of (6) and of (7). The reader interested in further consideration on Nash's inequality and the heat kernel is invited to refer to [12].
This paper is organized as follows. Theorem 1.1 can be seen as a consequence of Theorem 1.2: see Section 2. Section 3 is devoted to the proof of Theorem 1.2. In Section 4 we adopt a broader perspective and sketch the proof by E. Carlen and M. Loss along with a review of several other methods for proving Nash's inequality.
The optimal constant C GN (p) in (3) is obtained by minimizing the quotient Except when d = 1, no explicit expression of C GN (p) is available to our knowledge. Theorem 1.2 implies As a consequence, the support of u 1 is the ball B R 1 and on ∂B R 1 x, u 1 (x) = 0 and x · ∇u 1 (x) = 0. We also deduce from Theorem 1.2 that u p uniformly converges to u 1 using standard elliptic results of [15] or elementary ODE estimates. Moreover, since lim p→1 + u p−1 p = 1, the optimal function u 1 solves the Euler-Lagrange equation This means that v 1 : . We recall that λ 1 is defined by where the infimum is taken on ϕ ∈ H 1 (B 1 ) : As observed in [5], λ 1 can be computed in terms of the smallest positive zero z d /2 of the Bessel function of the first kind J d /2 using, for instance, [11, page 492, Chapter VII, Section 8]. Indeed, solving the radial eigenvalue problem Hence J α is a Bessel function of the first kind and the boundary condition of Bessel functions of the first kind (see, e.g., [27]), we obtain λ 1 = z 2 α+1 = z 2 d /2 . Looking at the Euler-Lagrange equation was precisely the way how E. Carlen and M. Loss realized that the optimizers must have compact support, [8]. This information heavily influenced the shape of the proof in [9]. The observations of this section point in the very same direction. For p ∈ (1, 2), the compact support principle gives a very simple intuition of why the optimal function for (3) has compact support, and then it is natural to take the limit in the Euler-Lagrange equation (4). As we shall see in the next section, symmetrization can be avoided and replaced by the moving plane method, which also puts the focus on (4).

PROOF OF THEOREM 1.2
For readability, we divide the proof into a list of simple statements. Some of these statements are classical and are not fully detailed. Our goal here is to prove that the optimal function u p is supported in a ball with finite radius and investigate the limit of u p as p → 1 + .
1. The non-optimal inequality and the optimal constant. It is elementary to prove that (1) holds for some positive constant C Nash and without loss of generality, we can consider the best possible one. See Section 4 for various proofs of such a statement.

Scalings and Gagliardo-Nirenberg inequalities. Inequality (3) is equivalent to
for all u ∈ H 1 (R d ) ∩ L p (R d ). A scaling argument relates K GN (p, λ) and C GN (p). Indeed, take any u ∈ H 1 (R d ) ∩ L p (R d ), then so does u σ = u(σ·) for any σ > 0, and we have where a = d (2 − p), and b = 2 p as in (3). From this, we may deduce two things.
First, choosing σ 2 = λ b a+b and optimising on u yields Second, an optimization on σ > 0 shows that so that optimizing on u yields Throughout this paper we assume that C GN (p) and K GN (p, λ) are the optimal constants respectively in (3) and (9). 3. Comparison of the optimal constants. By taking the limit in Q p [u] as p → 1 + for an arbitrary smooth function u and arguing by density, one gets that p → C GN (p) is lower semi-continuous and On the other hand, since u p p ≤ u 2−p 1 u 2 (p−1) 2 by Hölder's inequality, we find that from which we conclude that 4. Nonnegative optimal functions. We look for optimal functions in (3) by minimizing Q p . The existence of a minimizer is a classical result in the calculus of variations: see for instance [7]. Without loss of generality, we can consider only nonnegative solutions of (4) because Q p [u] = Q p |u| .
5. Support and regularity. According to [10], as a special case of the compact support principle (see [20,21]), nonnegative solutions of (4) have compact support. By convexity of the function t → t 2/p , we can consider solutions which have only one connected component in their support. For each p, we can pick one such solution and denote it by u p . The standard elliptic theory (see for instance [15]) shows that the solution is continuous and smooth in the interior of its support, which is a connected, closed set in R d . 6. Symmetry by moving planes. According to [10], the solution u p is radially symmetric and supported in a ball of radius R p . The proof relies on moving plane techniques and applies to nonlinearities such as u → u − u p−1 with p ∈ (1, 2). Up to a translation we can assume that the support is a centered ball. The solution is unique according to [ 7. Limit as p → 1 + . This is the point of the proof which requires some care. As p → 1 + , u p converges to a nonnegative solution u 1 of Let us give some details. For any p ∈ (1, 2), the solution u p of (4) satisfies the two identities obtained by testing (4) respectively by u p and by x · ∇u p (Pohozaev's method). The first identity is rewritten as ∇u p because of (10). As a consequence we have that lim p→1 + λ p = λ 1 . The two identities and the definition of λ p provide an explicit expression of ∇u p 2 , u p p and u p 2 in terms of λ p . This proves the strong convergence of u p to some u 1 in H 1 ∩ L 1 (R d ) and, as a consequence, this proves that equality in (1) is achieved by u 1 which solves (11). 8. Convergence of the support. By adapting the results of [4,13], it is possible to prove that lim p→1 + R p =: R 1 ∈ (0, ∞). Let us give an elementary argument. The dynamical system has smooth solutions such that, if initial data coincide, then U (r ) = u 1 (r ) and V = u 1 (r ) on the support of u 1 . As a limit of u p , we know that u 1 is nonnegative. Since the sequence (r n ) n∈N of the zeros of V , with the convention r 0 = 0, is such that the sequence (−1) n U (r n ) n∈N is monotone decaying, u 1 has compact support and R 1 = r 1 . This completes the proof of Theorem 1.2.

OTHER PROOFS OF NASH'S INEQUALITIES
This section is devoted to a brief review of various methods that have been used to derive Nash's inequality and estimate the optimal constant C Nash . The method of E. Carlen and M. Loss is the only one which provides the optimal value of C Nash . A comparison of the various results is summarized in Fig. 1.

Interpolation with Sobolev's inequality
In dimension d ≥ 3, it follows from Hölder's inequality that , and from Sobolev's inequality that is the optimal constant in Sobolev's inequality [1,22,24], so that C Nash ≤ S d .

Nash's inequality and the logarithmic Sobolev inequality are equivalent
In the interpolation strategy, we can replace Sobolev's inequality by the logarithmic Sobolev inequality. The equivalence of Nash's inequality and the logarithmic Sobolev inequality is a well known fact, in which optimality of the constants is not preserved: see for instance [2]. In [5], W. Beckner made the following observation: Jensen's inequality applied to the convex function σ(u) := log(1/u) and the probability measure u −2 2 |u| 2 d x shows that and can be combined with the logarithmic Sobolev inequality in scale invariant form of [23,28], that is, According to [5], this estimate is asymptotically sharp as d → ∞, which can be verified by using classical results on the zeroes of Bessel functions [27]. Notice that an information-theoretic proof of Nash's inequality based on Costa's method of entropy powers can be used to directly prove this inequality (see for instance [26], with application to Nash's inequality in [25,Section 4] This inequality becomes an equality when q = p. We may differentiate it with respect to q at q = p and immediately follows. The equivalence of Nash's inequality and the logarithmic Sobolev inequality (although with non-optimal constants) is a well known fact that has been exploited in various related problems. See for instance [6,3].

The original proof by J. Nash
It is a very simple argument based on Fourier analysis, due originally to E.M. Stein according to J. Nash himself (see [17, page 935]). Let us denote byû the Fourier transform of u defined asû so that by Plancherel's formula for any R > 0. By optimizing the right hand side with respect to R, we obtain the estimate

The method of E. Carlen and M. Loss
Without loss of generality, we can assume that the function u is nonnegative. If u denotes the spherically non-increasing rearrangement of a function u, then [16] u q = u q and ∇u 2 ≤ ∇u 2 , so we can consider nonnegative radial non-increasing functions without loss of generality. For any R > 0, let We observe that On the other hand, using we deduce from the Poincaré inequality for all v ∈ H 1 (R d ) such that B R v d x = 0 and from the definition of u R that using ∇u R 2 ≤ ∇u 2 . By definition of u R , we also know that After summing (13) and (14), we arrive at and notice that Altogether, this means that An optimization on R > 0 determines a unique optimal value R = R and provides the expression of C Nash . Equality is achieved by functions meeting equalities in all previous inequalities, that is functions such that u = u R and u R is an optimal corresponding respectively to an interpolation with Sobolev's inequality (only for d > 2) and to an interpolation with the logarithmic Sobolev inequality are shown as dashed curves. The dotted curve is the estimate d → C 2 (d ) of J. Nash in [17]. The optimal value d → C Nash (d ) is the plain curve and it is numerically well approximated from below by d → 1/(2πd ) deduced from (8).
function for the Poincaré inequality above. This is u(x) = 1 − ϕ 1 (|x|/R ) on B R , extended by u ≡ 0 to R d \ B R .