Regularity for the optimal compliance problem with length penalization

We prove some regularity results for a connected set S in the planar domain O, which minimizes the compliance of its complement O\S, plus its length. This problem, interpreted as to find the best location for attaching a membrane subject to a given external force f so as to minimize the compliance, can be seen as an elliptic PDE version of the average distance problem/irrigation problem (in a penalized version rather than a constrained one), which has been extensively studied in the literature. We prove that minimizers consist of a finite number of smooth curves meeting only by three at 120 degree angles, containing no loop, and possibly touching the boundary of the domain only tangentially. Several new technical tools together with the classical ones are developed for this purpose.

This problem, interpreted as to find the best location for attaching a membrane subject to a given external force f so as to minimize the compliance, can be seen as an elliptic PDE version of the average distance problem/irrigation problem (in a penalized version rather than a constrained one), which has been extensively studied in the literature. We prove that minimizers consist of a finite number of smooth curves meeting only by three at 120 degree angles, containing no loop, and possibly touching ∂Ω only tangentially. Several new technical tools together with the classical ones are developed for this purpose.
Letting u Σ ∈ H 1 0 (Ω \ Σ) stand for the unique minimizer of E(·) over H 1 0 (Ω \ Σ), it is classical that in the weak sense in Ω \ Σ. From physical point of view we think of Ω as a membrane, and Σ as the "glue line" attaching it to some fixed base (i.e. preventing the displacement). Then u Σ is the displacement of the membrane with the boundary fixed subject to force field f . The compliance of the membrane can then be defined as the equivalence of the two above expressions for C being due to (1). In this paper we study the following problem. A similar problem, previously introduced and studied in the literature (see e.g. [8,5,34]) is that of minimizing the compliance C over closed connected sets Σ subject to the constraint on the length H 1 (Σ) ≤ instead of the length penalization. In this paper we however concentrate our attention exclusively on the penalized Problem 1.1, whose physical interpretation is as follows: we are trying to find the best location Σ for the glue to put on a membrane in order to minimize the compliance of the latter, subject to the force f , while the penalization by λH 1 takes into account, for instance, the quantity (or cost) of the glue.
The existence of minimizers of Problem 1.1 follows in a more or less standard way from Blaschke,Šverák, and Go lab theorems, see Proposition 2.5 (or [8]).
A problem similar to Problem 1.1 has been studied in [35], where the compliance C(Σ) has been replaced by λ 1 (Ω \ Σ), the first eigenvalue of the Laplace operator with Dirichlet condition on Σ ∪ ∂Ω, in a constrained version H 1 (Σ) ≤ . In [35], the asymptotic behavior of solutions to this problem when goes to +∞ has been identified. On the other hand, the regularity of minimizers is not known for this problem. Since λ 1 (Ω \ Σ) is yet an energy of elliptic type, we believe that the technics introduced in this paper for the compliance could also serve to study its eigenvalue version, though do not pursue this analysis here.
Another version of shape optimization problem similar to Problem 1.1 may be obtained substituting the standard Laplacian by the p-Laplace operator. In this case, letting p → ∞, one obtains [8,Theorem 3] at the limit the so-called average distance minimization problem of purely geometric nature min Σ⊂Ω closed connected Ω dist (x, Σ)dµ + λH 1 (Σ).
This problem is related to a Monge-Kantorovitch problem with a "free Dirichlet region" which was introduced as a model for an optimal urban traffic network [6,9]. The topological and geometric properties of minimizers of this problem (mostly in its constrained version but also in the easier penalized one) were studied by several authors (see [23] for a review on this problem, and [7,27,31,10,22,29,33] for related results on this and similar problems). Problem 1.1 is much different from the average distance minimization problem, and to certain extent is closer to the Mumford-Shah problem. We will indeed prove at the end of this paper that Problem 1.1 is in a sense dual to the Mumford-Shah problem, and a certain amount of tools used in this paper (monotonicity formula, scheme of proof for the C 1 result) is inspired by the arguments developed for the Mumford-Shah functional (we refer to [2,11,20] for reviews on the latter). Here, we show that the minimizers are locally smoooth inside Ω. This is in sharp contrast with the average distance problem: in fact, for the latter Slepčev [30] (see also [25]) shows that minimizers are, in general, not C 1 and using his construction one may find minimizers with infinite and possibly not closed set of points with lack of regularity [24].

Main results
The following theorem sums up the main results of this paper.

Theorem 1.2.
Let Ω ⊂ R 2 be a C 1 domain (i.e. an open bounded connected set with the boundary locally, up to rotation, a graph of C 1 function), and λ ∈ (0, ∞). If f ∈ L p (Ω) with p > 2, then every solution Σ of Problem 1.1 has the following properties.
Part I: qualitative properties.
(iii) Σ is a chord-arc set, i.e. it satisfies for some constant C > 0, where d Σ denotes the geodesic distance in Σ.
(iv) Σ consists of a finite number of embedded curves, possibly intersecting at "triple points" where the curves are meeting by 3 at 120 degrees angles. In particular, it has finite number of endpoints and finite number of branching points, all of which are triple points as described.
We emphasize that (i) and (iv)-(v) are probably the most interesting items of the theorem, while (ii) and (iii) might at first glance look technical. However, besides being interesting on their own, more remarkably, the results of Part I are needed to prove Part II. Let us stress moreover that several of the announced results still hold without assuming C 1 regularity of ∂Ω. For instance, for a generic bounded open Ω ⊂ R 2 we show in fact that every minimizer Σ of Problem 1.1 satisfies (i) and (ii), i.e. has no loops and is Ahlfors regular, and for every compact Ω Ω is a chord-arc set (with the constant in the estimate (2) depending on Ω ), has a finite number of branching points and endpoints in Σ ∩ Ω , all the branching points are triple points where the curves are meeting by 3 at 120 degrees angles, while the curves composing Σ ∩ Ω are locally C 1,α regular.
With the developed technique at hand, it would be also possible to get some finer results for just Lipschitz domains, as, for instance, the fact that the optimal set Σ will never touch a convex corner of ∂Ω but we do not pursue this analysis in details. Theorem 1.2 is the concatenation of several results contained in the paper. Namely, (i) is Theorem 4.1, (ii) is Theorem 4.3, (iii) is Proposition 6.7. Assertions (iv) and (v) are given by Theorem 8.1, in particular, the finite number of curves comes from Theorem 8.6, the characterization of branching by 120 degrees comes from the classification of blow-up limits (Proposition 7.9) together with the uniqueness of the type of the blow-up (Proposition 8.4). The property (v) for a generic (that is, not necessarily convex) Ω, i.e. the local C 1,α regularity of the curves has two ingredients, namely, the classification of blow-up limits (Proposition 7.9) which together with Proposition 8.4 says that any point which is not a triple point neither an endpoint is necessarily a "flat" point, i.e. blows-up as a line), and then Theorem 6.4 which says that Σ is C 1,α around any "flat" points. The fact that Σ touches ∂Ω tangentially is a consequence of Theorem 7.13 and Proposition 7.14. The case of Ω convex in (v) is Remark 8.3. Finally, (vi) is Proposition 8.8.

Basic techniques and background idea
The proof of the local C 1,α regularity result is contained in Sections 5 and 7. The main strategy is inspired by the regularity theory for the Mumford-Shah functional in dimension 2, more precisely by the approach of Bonnet [3] and David [11]. The rough idea that first comes in mind is to show that every minimizer Σ of Problem 1.1 is an almost minimizer for the length, i.e. to prove that for any competitor Σ satisfying Σ Σ ⊂ B r , where B r stands for a ball of radius r > 0, one has H 1 (Σ ∩ B r ) ≤ H 1 (Σ ∩ B r ) + Cr 1+α , and then to apply the regularity theory for almost minimal sets. In our situation, the error term Cr 1+α may only come from the energy (or, equivalently, compliance) part of the functional, namely, we need to prove an estimate of type To obtain the latter we have to overcome several difficulties which are due to essential differences between Problem 1.1 and the classical free boundary or free discontinuity problems. The first one is a substantially nonlocal behavior of the compliance functional, in the sense that small perturbations of Σ affects the potential u Σ in the whole Ω. This can be overcome by a simple cut-off argument (Lemma 3.1). It shows that if Σ is a competitor for Σ in B r , then the defect of minimality is controlled by the inequality In particular, we notice that the right hand side depends on Σ and one may prefer to have a quantity inherent to Σ which does not depend on the considered competitor. This is why we introduce the quantity ω Σ (x, r) := max Σ connected ;Σ ∆Σ⊂Br(x) 1 r Br(x) |∇u Σ | 2 dx , and we arrive to the estimate To obtain C 1,α regularity, one has to prove that ω Σ (x, r) ≤ Cr α for some α > 0. This is done via a classical monotonicity formula, which is a version of the one of Bonnet [3] (similar to the celebrated Alt-Caffarelli-Friedman monotonicity formula [1]), but adapted for Dirichlet boundary conditions. It implies a suitable decay for ω Σ (x, r), provided that Σ is flat enough in the neighborhood of x. On the other hand the flatness decays suitably fast provided that ω Σ is small enough. In other words, we cannot apply directly the theory of almost minimal sets but we have to reproduce some of its arguments, with the additional difficulty that we have to control both the quantity ω Σ and flatness of Σ at the same time. All this leads to an "ε-regularity" result (Corollary 5.18). At the end we bootstrap all the estimates and so follows the C 1,α regularity.
To get the full regularity result (i.e. to show that every minimizer Σ consists of finite number of smooth injective curves), we perform a blow-up analysis. Again, the main difficulty comes from the nonlocal behavior of the compliance functional. To bypass it, we consider the dual formulation of Problem 1.1 (Proposition 7.2), proving that Problem 1.1 is equivalent to the following one The latter problem is now localizable. As a matter of a fact, it turns out that this dual problem is very close to the Mumford-Shah problem, and more remarkably, the dual formulation of the minimizing problem satisfied by the blow-up limits is exactly the same as the characterization of global minimizers for Mumford-Shah problem in [3]. This gives a possibility to characterize completely the possible blow-up limits and conclude the proof of full regularity. This is done in Section 7. Note however, that the blow-up at the boundary of ∂Ω is much different than what usually happens in the Mumford-Shah problem. Here the blow-up limits at the boundary are tangent to the boundary, whereas for the Mumford-Shah functional they are transversal. Let us furthermore mention that we first prove several other qualitative properties on minimizers, like Ahlfors-regularity, absence of loops, chord-arc estimate (Part I of Theorem 1.2). It is worth mentioning that, curiously enough, all of theses properties are needed to characterize the blow-up limits. More precisely, one of the key ingredients that leads to the classification of the global minimizers for the Mumford-Shah functional is the fact that the blow-up limits are connected in the limit, as required by Bonnet [3] to get the classification of blow-up limits. Although here we deal with a priori connected sets, in general, the blow-up of a connected set may not be connected in the limit, and this is why one needs to use the optimality. To this aim, we prove that every minimizer Σ of Problem 1.1 is a chord-arc set (Proposition 6.7). It is not difficult to see then that the blow-up limit of a chord-arc set is connected. A similar chord-arc estimate has been already proven for each connected component of a Mumford-Shah minimizer, but the proof presented here for the compliance minimizers is much different: here we need to preserve connectedness on competitors. Our strategy is as follows. Let us look closer at the rough idea of proving that a minimizer does not contain loops. The argument is by cutting the loop at a flat point by a piece of set of size r, and estimate the loss in the compliance term in terms of r 1+α , provided that we have found a suitable point where to cut. The proof of (2) is a sort of quantitative version of this argument: if x and y are very close to each others and the curve connecting them in Σ is going far away, then one can cut this curve at some place where it is flat, and add the little segment connecting x to y to preserve connectedness. To perform it, one needs the radius of the ball where Σ is flat to be controlled from below, uniformly with respect to the distance from x to y. This is obtained from the uniform rectifiability of Σ, which follows from the Ahlfors-regularity of the latter (Theorem 4.3).

Notation
We introduce the following notation.
• For {x, y} ⊂ R 2 , |x| denotes the Euclidean norm and d(x, y) := |x − y| the Euclidean distance, x · y the usual inner product.
• For D ⊂ R 2 , 1 D is the characteristic function of D, D c := R 2 \D, D and ∂D are the closure and the topological boundary of D respectively, H 1 (D) is the 1-dimensional Hausdorff measure of D, diam D the diameter of D, and dist (x, D) := inf{d(x, y) : y ∈ D} whenever x ∈ R 2 .
• d H (A, B) is the Hausdorff distance between the sets A and B defined by Let now Ω ⊂ R 2 .
• If Ω is measurable, then L p (Ω) stands for the Lebesgue space of p-integrable real functions for p ∈ [1, +∞) and measurable essentially bounded functions for p = +∞, · p starnding for its standard norm and p for the conjugate exponent 1/p + 1/p = 1; L p (Ω, R 2 ) is the respective Lebesgue space of functions with values in R 2 ; L p loc (Ω) is the space of functions u such that u ∈ L p (K) for all compact K ⊂ Ω. The convergence in L p loc (Ω) means the convergence for the norm · p on every compact K ⊂ Ω.
If Ω is open, then • K(Ω) denotes the set of all compact and connected sets Σ ⊂ Ω; • C ∞ 0 (Ω) stands for the class of infinitely differentiable functions with compact support in Ω; • C k (Ω) (resp. C k,α (Ω)) stands for the class of k times differentiable functions (resp. k times differentiable with α-Hölder k-th derivative) in Ω, where k ∈ N, α ∈ (0, 1); • D (Ω) stands for the usual space of distributions in Ω; • Lip(Ω) stands for the class of all Lipschitz maps f : Ω → R; • H 1 (Ω) is the standard Sobolev space of functions u ∈ L 2 (Ω) having (distributional) derivative in L 2 (Ω); H 1 loc (Ω) is its local version akin to L p loc (Ω); • H 1 0 (Ω) stands for the usual Sobolev space defined by the closure of C ∞ 0 (Ω) for the norm u H 1 0 (Ω) := Ω |∇u| 2 dx; if necessary, the functions in H 1 0 (Ω) will be silently assumed to be extended to the whole R n by zero over Ω c . This gives a natural embedding of H 1 0 (Ω) into H 1 (R n ); • Given Ω ⊂ Ω and Σ ∈ K(Ω) we will also denote by An open, bounded, and connected Ω ⊂ R 2 will be called a C 1 domain, if ∂Ω is locally up to rotation the graph a C 1 function.

Estimate of u Σ
Let Ω ⊂ R 2 be an open set. When Σ ⊂ Ω is closed, then Ω \ Σ is open, and it is well-known that if f ∈ L 2 (Ω), then there is a unique function u Σ that minimizes E over H 1 0 (Ω \ Σ). The dependence of u and hence of compliance on Σ is related to the fact that a 1-dimensional set in R 2 has a non-trivial capacity: in fact, if Σ has zero Hausdorff dimension, then C(Σ) = C(∅).
In this paper, we need the following estimate, which is a direct consequence of [17,Theorem 8.16].
Remark 2.2. Under conditions of the above Proposition 2.1 we have further that

Existence
Here we state a particular case ofŠverák's theorem, which will be used several times in this paper.

Theorem 2.3 (Šverák [32]).
Let Ω an open bounded set in R 2 , and f ∈ L 2 (Ω). Let (Σ n ) n be a sequence of connected set in Ω, converging to Σ ⊂ Ω in the Hausdorff distance. Then Remark 2.4. Notice that, a byproduct of Sverák's theorem is the so called Mosco-convergence of H 1 0 (Ω \ Σ n ) that will be used later in the paper. More precisely, for every u ∈ H 1 (Ω \ Σ) there exists a sequence {u n } such that u n ∈ H 1 0 (Ω \ Σ n ) for all n ∈ N and u n → u strongly in H 1 (Ω), and moreover any sequence of {v n } satisfying v n ∈ H 1 0 (Ω \ Σ n ) for all n ∈ N and v n v weakly in H 1 (Ω) satisfies v ∈ H 1 0 (Ω \ Σ) (see [18,Proposition 3.5.4]). The following simple assertion gives existence of solutions to the penalized optimal compliance problem. Proof. Let Σ n be a minimizing sequence for Problem 1.1. From Blaschke's selection principle [2, Theorem 6.1] there exists a subsequence converging for the Hausdorff distance to some closed Σ ⊂ Ω. By Theorem 2.3 u Σn converges strongly in H 1 (Ω) to u Σ and Go lab's theorem [14,Theorem 3.18] gives the lower-semicontinuity of H 1 , which implies that Σ must be a minimizer.
Remark 2.6. In [8] the assumption f ≥ 0 is added in the statement of existence but it is actually unnecessary.
Remark 2.7. If we drop the connectedness assumption on Σ, then the existence of minimizer for F λ fails. Indeed, it is not difficult to construct an example of (highly disconnected) set Σ ⊂ Ω with H 1 (Σ) arbitrary small which spreads into Ω so that u Σ has very small energy, leading to inf Σ⊂Ω,closed F λ (Σ) = 0.

Estimates of the variations of the compliance
Our goal in this section is to obtain an estimate of the variation of the compliance between Σ and Σ when Σ∆Σ is localized, without assuming any regularity for Σ and Σ .

Localization lemma
We first prove the following localization lemma.

Monotonicity formula and decay of energy
The monotonicity of energy will be one of our main tool in all the sequel. We start with a general statement which says that the maximum length for ∂B r \ Σ gives the power of decay for the function u Σ . In what follows we assume that u Σ is extended by 0 outside Ω. Lemma 3.3. Let Σ ⊂ Ω be a closed set, f ∈ L p (Ω), where p > 2, and x 0 ∈ Ω. Let 0 ≤ r 0 < r 1 , We assume and finally we suppose that 1/p > π/γ. Then the function is nondecreasing, where α := 2π/γ and C = C(|Ω|, p, f p , γ) > 0.
Remark 3.4. In particular, setting γ := 2π in the above Lemma, we get that the function Proof. We denote u = u Σ and extend u by zero outside Ω, so that u ∈ H 1 (R 2 ). One has, for a.e. r ∈ (r 0 , r 1 ), for every δ > 0. By Poincaré inequality (Lemma A.1) that can be applied because of (6), we get Notice indeed that the trace of u on ∂B r belongs to H 1 0 (∂B r \ Σ ∪ ∂Ω) for a.e. r. Thus choosing δ := π/(γr), we get The function r → G(r) is indeed absolutely continuous, and its derivative is a.e. equal to r → ∂Br(x 0 ) |∇u| 2 dH 1 .
Recalling that u ∈ L ∞ (Ω) by Proposition 2.1 and using the Hölder inequality together with (3) to estimate with some constant C > 0 depending only on f p , p, |Ω|, we get This gives that In particular, for C 1 domains we obtain the following useful decay at the boundary.
Lemma 3.5. Assume that Ω is a C 1 domain, let Σ ⊂ Ω be a closed set and p > 2. Then there exists an r 0 = r 0 (∂Ω, p) > 0 and a ν = ν(p) > 0 such that for all x ∈ ∂Ω and for all r ∈ (0, r 0 ), with positive constants C 1 and C 2 depending only on |Ω|, p, f p .

Estimate of |C(Σ) − C(Σ )|
We collect now all the necessary estimates on minimizers of E over H 1 0 (Ω \ Σ) in the following statement.
(i) There exists C > 0 depending only on |Ω|, p, f p such that: (ii) There exists C > 0 universal such that: (iii) There exists C > 0 universal such that: Proof. The estimate (10) is directly coming from Hölder inequality together with Proposition 2.1. Next, we can use Poincaré inequality (Lemma A.2), since Σ∩B r (x 0 ) = ∅ and Σ\B 2r (x 0 ) = ∅. We obtain where C P is a universal constant, and in particular does not depend on the geometry of Σ. Using that |∇ϕ| ≤ 1 r .1 B 2r \Br , we directly get (11). It remains to prove (iii). To this aim, we estimate the latter by Hölder inequality. As in (13), from Poincaré inequality in the annulus as desired.

First qualitative properties
In this section, we prove geometrical properties of the minimizers for Problem 1.1.

Absence of loops
The following result holds true. contains no closed curves (homeomorphic images of S 1 ), hence R 2 \ Σ is connected.
The idea of the proof of this result is to see that if Σ has a loop, then we can cut a small piece of this loop and remain connected; this cut decreases the length and increases the energy. If we choose this cut properly, namely where the curve is "flat", then the energy estimate from Section 3.3 shows that these two variations are not of the same order, which leads to a contradiction, see the proof below. Before this proof, we give a geometric lemma asserting that such a cut can be done: Let Σ be a closed connected set in R 2 , containing a simple closed curve Γ (a homeomorphic image of S 1 ) and such that H 1 (Σ) < ∞. Then H 1 -a.e. point x ∈ Γ is such that • "noncut": there is a sequence of (relatively) open sets D n ⊂ Σ satisfying (i) x ∈ D n for all sufficiently large n; (ii) Σ \ D n are connected for all n; (iii) diam D n 0 as n → ∞; (iv) D n are connected for all n.
• "flatness": there exists a "tangent" line P to Σ at x in the sense that x ∈ P and Proof. First, we apply [28,Lemma 5.6], stating that under the hypotheses on Σ, H 1 -a.e. point x ∈ Γ is a noncut point for Σ (i.e. a point such that Σ \ {x} is connected). Then [9, Lemma 6.1] affirms that for every noncut point there are connected neighborhoods that can be cut leaving the set connected, i.e. (i)-(iv) are satisfied for a suitable sequence D n . The second requirement relies on the standard fact that closed connected sets with finite length are rectifiable (see, e.g., [26,Proposition 3.4]

or [4, Proposition 2.2]).
Proof of Theorem 4.1. Assume by contradiction that for some λ > 0 a minimizer Σ of F λ over closed connected subsets of Ω contains a closed curve Γ ⊂ Σ. From Lemma 4.2, we have that there exists a point x ∈ Γ which is a noncut point and such that Σ is differentiable at x. Therefore there exist the sets D n ⊂ Σ and a straight line P as in Lemma 4.2. We denote r n := diam D n so that D n ⊂ Σ ∩ B rn (x). The flatness of Σ at x implies that for some given ε > 0, there exists an r 1 > 0 such that arcsin β Σ,P (x, r) ≤ επ/2 for all r ∈ (0, r 1 ]. Thus every connected S ⊂ ∂B r (x) \ Σ satisfies H 1 (S) ≤ (π + 2 arcsin β Σ,P (x, r))r = (1 + ε)πr, r ∈ (0, r 1 ], and hence using Proposition 3.7 with γ := (1 + ε)π and Σ = Σ \ D n when r n ≤ R, we get But since n . Recalling that 2/p > 1 and choosing ε < 1/2 (of course, this affects the value of C and of r 1 ), then one has as r n → 0 + , which for sufficiently small r n contradicts the minimality of Σ and hence proving that Σ does not contain closed curves. About the last assertion in

Ahlfors regularity
It is not difficult to show that the minimizers of Problem 1.1 are Ahlfors regular under quite nonrestrictive conditions on the data. Recall that a set Σ ⊂ R 2 is called Ahlfors regular, if there exist some constants c > 0, r 0 > 0 and C > 0 such that for every r ∈ (0, r 0 ) and for every x ∈ Σ one has a singleton being considered Ahlfors regular by definition. The constants C and c will be further referred to as upper and lower Ahlfors regularity constants of Σ respectively. It is known that Ahlfors regularity of a closed connected set Σ implies uniform rectifiability (a kind of "quantitative rectifiability" which is somewhat stronger than the classical rectifiability used in geometric measure theory) of Σ, which provides several nice analytical properties of Σ, see for example [13], and will be used later several times. If Σ is closed and connected, then the lower estimate in (22) is trivial: in fact, for all r < diam Σ/2 one has Σ ∩ ∂B r (x) = ∅, and hence Hence, for such Σ the proof of Ahlfors regularity reduces to verifying that for every x ∈ Σ and for all r ∈ (0, r 0 ) with some r 0 > 0 independent of x one has  Proof. We show that for every λ > 0, every minimizer Σ of F λ among closed connected subsets of Ω satisfies (23) with some C > 2π and r 0 ∈ (0, diam Σ/4).

Flatness and small energy implies C 1 regularity
In this section we find sufficient conditions in a ball B r (x) which imply that Σ ∩ B r/2 (x) is a nice C 1 curve. The strategy vaguely follows the approach of Guy David in [11] where a similar work is done for Mumford-Shah minimizers, but adapted with the specificities of our problem which is a Dirichlet problem of min-max type instead of a Neumann problem of min-min type in [11]. The connectedness constraint makes also a big difference with the Mumford-Shah problem. The general tools are a decay of energy provided that Σ stays "flat" thanks to the monotonicity formula, and on the other hand, a control on the flatness when the energy is small. We then conclude by bootstraping both estimates.
In this section and all the next ones, we will always assume λ = 1 for simplicity. Of course this is not restrictive regarding to the regularity theory.

Control of the energy when Σ is flat
For any x ∈ Ω and r > 0 such that B r (x) ⊂ Ω we denote by β Σ (x, r) the flatness of Σ in B r (x) defined through where d H is the Hausdorff distance and where the infimum is taken over all affine lines P passing through x. Notice that the inf above is attained, i.e. is actually a minimum. We shall also need a variant where the proximity to affine lines is checked only in an annulus and not in the whole ball. Precisely, for 0 < s < r we denote where again the inf is taken over all affine lines P passing through x.
Observe that for all 0 < s < r, it directly comes from the definition that and the main point about β Σ (x, r, s) is the following obvious fact Observe also that for all a ∈ (0, 1) we have Our first aim is to use the monotonicity formula to control the energy of u Σ around points where β Σ is small: this is done in Proposition 5.5 below. For this purpose we seek for a lemma similar to Lemma 3.3, but where the assumption (6) is replaced by an assumption on β Σ (x, r) only. This leads to new difficulties, because even if Σ is connected and flat around a point x ∈ Σ, it may happen that Σ ∩ B r (x) is not connected (see Figure 1). We state the lemma when γ = π + π/3 (so that 2π/γ = 3/2); we could have chosen any γ = π + ε with ε small (leading to 2π/γ as close as 2 as required), though this specific choice of γ is sufficient for our purpose, since our main goal is to allow an exponent strictly bigger than 1. Moreover, it is more convenient to work on harmonic functions, and then compare u Σ with its harmonic replacement to obtain a similar statement for u Σ . Therefore the following lemmas are stated first in this more convenient framework of harmonic functions.
We call then Dirichlet minimizer u ∈ g+H 1 0 (D\Σ) any function that minimizes the Dirichlet for some 0 < ε ≤ 1/10. Then for every Dirichlet minimizer To get a proof of this result, we need first the following two lemmas: the first one is a version of Lemma 3.3 for harmonic functions, the second one is a purely geometric statement. (5) for a definition of γ Σ ).Then for every Dirichlet minimizer is nondecreasing. The proof of this lemma is exactly the same as the one of Lemma 3.3 (with f = 0) so we omit it.
As we said before, to replace the assumption (6) by another one relying only on the flatness β Σ (x 0 , r 1 , r 0 ), we face the difficulty that it may happen, even if Σ is connected, that ∂B r (x 0 )∩Σ = ∅ for some r ∈ (r 0 , r 1 ). To handle this difficulty, we establish the following topological fact. Lemma 5.3. Let Σ ⊂ R 2 be a closed and arcwise connected set. Assume that x 0 ∈ Σ and 2r 1 ε < r 0 < r 1 < diam (Σ)/2 are such that Then A is an interval of length less than 2εr 1 . Proof. In this proof, every ball is centered at x 0 . The proof relies on the fact that Σ ∩ B r 1 \ B r 0 is localized in a strip that meets any sphere ∂B r for r ∈ [r 0 , r 1 ] by two sides. Let P 0 be the line that realizes the infimum in the definition of β Σ (x 0 , r 1 , r 0 ). Let us identify P 0 ∩ B r 1 with the segment [−r 1 , r 1 ]. By (28) we know that for each point t such that r 0 ≤ |t| ≤ r 1 , there exists a point z(t) ∈ Σ ∩ B r 1 \ B r 0 which is εr 1 close to t. The point z(t) connects a point outside B r 1 via a curve, that must escape the ball B r 1 either on the right or on the left. Let t + be the minimum of t > 0 for which the curve escape from the right, and t − be the maximum of t > 0 for which the corresponding curve escapes from the left. Then the length of the interval (t − , t + ) must be smaller than 2εr 1 otherwise we would have a hole of so big size that would contradict (28). And every sphere ∂B t for t ∈ [r 0 , r 1 ] \ [t − , t + ] necessarily meet a point of Σ in the εr 1 -strip around P 0 . We deduce that A is a interval contained in [t − , t + ] (see Figure 5.1).
Now we are in position to prove Lemma 5.1: Proof of Lemma 5.1. In this proof, every ball is centered at x 0 . Observe first that r ∈ [2εr 1 , r 1 ] → π + 2 arcsin εr 1 r is decreasing, and is therefore always smaller than π + π 3 , achieved for r = 2εr 1 . Let A := (t 1 , t 2 ) be the set of bad radii given by Lemma 5.3. When r ∈ [t 2 , r 1 ], we apply Lemma 5.2 with γ = π + π 3 thus obtaining directly If r ∈ (t 1 , t 2 ). Then since (29) holds for t 2 we can simply write Combining this estimate with (29) applied with r := t 2 , we get We are now ready to state a useful decay result on u Σ .
Then for every r ∈ [r 0 , r 1 ] one has that for some constant C = C(|Ω|, p, f p ) > 0.
Proof. In this proof, every ball is centered at x 0 . Let w be the harmonic replacement of u Σ in B r 1 , i.e. w is a Dirichlet minimizer in B r 1 : it is harmonic in B r 1 \ Σ with Dirichlet condition w = 0 on Σ ∩ B r 1 and w = u Σ on ∂B r 1 \ Σ. This function is found by minimizing the Dirichlet energy Br 1 |∇w| 2 dx among all w satisfying w − u Σ ∈ H 1 0 (B r 1 \ Σ). Then Lemma 5.1 applies to w and says The last inequality comes from the fact that w minimizes the energy in B r 1 and u Σ is a competitor. Now u Σ minimizes E so that, extending w by u Σ outside B r 1 we can write where C = C(|Ω|, f p , p), where in the last estimate we used Hölder inequality together with the fact that u Σ and hence w are bounded (Proposition 2.1). On the other hand, for any r ∈ [r 0 , r 1 ], we have because ∇w and ∇(u Σ − w) are orthogonal in L 2 (B r 1 \ Σ). Combining (33), (34) and (35) we arrive at (32) thus concluding the proof.
Let us now recall the following notation. For any Σ ∈ K(Ω) we denote by Remark 5.6. Problem (36) has a solution. Indeed, if Σ k be a maximizing sequence, then up to a subsequence Σ k → Σ 0 for the Hausdorff distance and clearly Σ 0 ∆Σ ⊂ B r (x). But theň Sverák Theorem 2.3 says that ∇u Σ k converges strongly to ∇u Σ 0 in L 2 (Ω) and Σ 0 is therefore a maximizer.
We now state a first estimate on defect of minimality.
⊂ Ω be a closed connected set, and assume that x 0 ∈ Σ, and β Σ (x 0 , r 1 ) ≤ ε for some 0 < ε < 1/10 and r 1 < diam (Σ)/2. Then for any r ∈ (2εr 1 , r 1 /2] and for any closed Proof. Every ball in this proof is centered at x 0 . Take an arbitrary ϕ ∈ Lip(R 2 ) as in the statement of Lemma 3.6, i.e. such that ϕ ∞ ≤ 1, ϕ = 1 over B c 2r , ϕ = 0 over B r and ∇ϕ ∞ ≤ 1/r. From Lemma 3.1 we have We obtain then the following chain of estimates (with the constant C changing from line to line) concluding the proof.
It is worth observing (though we will not use it in the sequel) that monotonicity holds also for ω Σ , as the following statement asserts. Corollary 5.9. Let Σ ⊂ Ω be a connected and closed set, f ∈ L p (Ω), where p > 2, and x 0 ∈ Σ.
showing the claim.

Density estimates
We now establish our main estimates about the density of the minimizer Σ at the point x 0 ∈ Σ. The first one is weak in the sense that β appears in the error terms, but will be used to find some good radius s for which #Σ ∩ ∂B s (x 0 ) = 2, as stated in the second item. The third item provides a similar estimate without paying β, provided that #Σ ∩ ∂B s (x 0 ) = 2; this will be our main estimate that leads to regularity.
(iii) If both (40) and (42) hold, and r ∈ [r 1 /2, r 1 ] is such that #Σ ∩ ∂B r (x 0 ) = 2, then (iii-1) the two points of Σ∩∂B r (x 0 ) belong to two different connected components of ∂B r (x 0 )∩ {y : dist (y, P 0 ) ≤ β Σ (x 0 , r)r}, where P 0 is a line that realizes the infimum in the definition of β Σ (x 0 , r), Remark 5.11. It is easy to see from the proof that for an internal point x 0 ∈ Ω one can take r Ω := +∞. Thus if one applies this result only to internal points of Ω, then the requirement that Ω ⊂ R 2 be a C 1 domain is unnecessary.
Remark 5.12. In the sequel, when the situation of item (iii-1) will occur we will say, in short, that the two points lie "on both sides".
Proof. Every ball in this proof is centered at x 0 unless otherwise explicitly stated. We first prove (i). Let us construct a competitor in B r for any fixed r ∈ [r 1 /2, r 1 ]. Let x 1 and x 2 be the two points of ∂B r ∩ P 0 and let W be the little wall defined by We consider two cases.
Case A: B r ⊂ Ω. Then consider the competitor (see Figure 5.2) It is easily seen that Σ is a compact connected set satisfying Σ∆Σ ⊂ B r . Since Σ is a minimizer, we have and thus On the other hand, the latter inequality coming from the assumption β Σ (x 0 , r) ≤ 1/10 minding that | arcsin (z)| ≤ 2 for all z ∈ [0, 1/10]. The estimate (43) combined with (44) implies (i) in the case B r ⊂ Ω.
Case B: ∂Ω ∩ B r = ∅. Notice that, since x 0 may be very close to ∂Ω, the competitor Σ as before might not be contained in Ω thus we need to modify it in this situation. Letting z ∈ ∂Ω ∩ B r we have B r ⊂ B 2r (z). Moreover, since Ω is a C 1 domain, and in particular compact, we can argue as in the proof of Lemma 3.5 to find r Ω > 0 such that ∂Ω is very flat in all balls of radius less than r Ω , uniformly on ∂Ω. In other words such that β ∂Ω (x, s) ≤ 10 −10 /2 for all x ∈ ∂Ω and all s ≤ 2r Ω . This means, since 2r ≤ 2r Ω , that ∂Ω ∩ B 2r (z) is localized in a very thin strip of height δ := 4 · 10 −10 r centered at z. Let us assume that this strip is oriented in the e 1 direction and that Ω is situated below (i.e. touching the region {x 2 < 0}). Our aim is to translate locally Σ a little downwards, to insure that it lies in Ω. For this purpose we construct a bi-Lipschitz mapping Φ, equal to Id outside B r(1+δ) , and equal to −δre 2 in B r 0 which will guarantee that Φ(Σ ) ⊂ Ω (see Figure 5.3).
More precisely, we let ϕ : R + → [0, δr 0 ] be a 1-Lipschitz function, equal to 0 on [r(1+δ), +∞), and to rδ on [0, r], and define Φ : We notice that Φ is 2-Lipschitz and maps B r(1+δ) into itself. Next, we define as before the compact connected set Σ : Now we estimate H 1 (Σ ∩ B r(1+δ) ). To this aim we decompose On B r , the mapping Φ is just a translation so that hence using (44) we get On the other hand, since Φ is 2-Lipschitz, where C 0 is the upper Ahlfors regularity constant of Σ, because Σ ∩ B r(1+δ) \ B r is localized in the union of two balls of radius bounded by 2rβ Σ (x 0 , 2r).
Finally, thanks to (40) we still have β Σ (x 0 , r 0 (1+δ)) ≤ 1/10, so that Proposition 5.8 applies. Since Σ is a minimizer, we have and thus using Proposition 5.8 we infer that where in the last inequality (26) has been used. This concludes the proof of (i) also in case (B).
To prove (ii), we use Lemma 5.3 (with ε := 10 −4 , and r 0 := 10 −3 r 1 ) which implies that, Next, applying (i) with r := r 1 , we get, thanks to our conditions on r 1 and ω Σ (x, r 1 ), Let us define three sets In particular, (45) says that H 1 (E 1 ) ≤ 10 −3 r 1 + 2 · 10 −4 r 1 ≤ 10 −2 r 1 , and hence, using also (46) and (47) we deduce that from which we get and therefore E 2 ∩ [r 1 /2, r 1 ] = ∅. It remains to prove (iii). We argue in a similar way, but now since Σ ∩ ∂B r = {a 1 , a 2 }, i.e. #Σ ∩ ∂B r = 2, the wall set W is no more needed to make a competitor, so we get a better estimate by taking as a competitor a replacement of Σ by just a segment inside B r joining the two points a 1 and a 2 , which leads to the estimate in (iii-3).
Thus we only need to prove (iii-1) and (iii-2). Let us first prove (iii-1). Supposing the contrary, we could take as a competitor the set where W is a little wall on one side of length less than 10rβ Σ (x, r 1 ). This would imply hence a desired contradiction.
To prove (iii-2), we note that by Lemma 5.13 below (with D := B r ∩ Σ), if Σ ∩ B r is not connected, then the set Σ \ B r has to be connected. It follows that Σ \ B r is a competitor, and as before, comparing the energy of Σ and Σ := Σ \ B r leads to which is a desired contradiction. Thus we have proven that (iii-2) holds true, so that the proof of (iii) is concluded. Proof. Let ∂D := {a 1 , a 2 }, and consider an arbitrary couple of points {z 1 , z 2 } ⊂ Σ \ D and an arc Γ ⊂ Σ that connects z 1 and z 2 . Then either Γ ∩ D = ∅, in which case Γ ∩ D = ∅, or #Γ ∩ D = 1, in which case again Γ ∩ D = ∅ (because Γ enters into D through, say, a 1 , but it cannot enter D, since then it must exit through the same point a 1 , which is impossible by injectivity of Γ), or else Γ ∩ D = {a 1 , a 2 }, so that Γ enters D at, say a 1 and exits at, say a 2 . In this last case this means that there exists a curve in D that connects a 1 to a 2 , but then since other point in D is connected to z 1 by some curve, that passes necessarily through either a 1 or a 2 , this means that D is arcwise connected.

Flatness estimates
We begin with a standard flatness estimate on curves coming from Pythagoras inequality.
Lemma 5.14. Let Γ be an arc in B r (x 0 ) satisfying β Γ (x 0 , r) ≤ 1/10, and which connects two points x 1 , x 2 ∈ ∂B r (x 0 ) lying on both sides (as defined in Remark 5.12). Then Proof. Assume that z is the most distant point from the segment [x 1 , x 2 ] in Γ and let z be the point making (x 1 , x 2 , z ) an isosceles triangle with same height that we denote h =: dist (z, [x 1 , x 2 ]). Let a := |x 2 − x 1 |/2 and := |z − x 1 |. We have that On the other hand H 1 (Γ) ≥ 2 so that But now since β Γ (x 0 , r) ≤ 1/10 it is easily seen that ≤ √ 101r/10 ≤ 2r and a ≤ r, so follows the Lemma.
Remark 5.15. Notice that, contrary to the similar statements that sometimes can be found in the literature, Lemma 5.14 would not be true replacing the curve Γ by an arbitrary closed connected set. Indeed, in an arbitrary connected set Σ, the curve from x 1 to z and the curve from z to x 2 may overlap, so that the sum of the length of both curves may not be smaller than the length of Σ. Anyway, in the next proposition, we shall apply this lemma in an arbitrary connected set Σ as follows: we first find an injective curve in Σ from x 1 to x 2 and control the distance of that curve to the segment [x 1 , x 2 ]. Then, this curve will have length at least |x 1 − x 2 | and we will control the distance of the remaining parts of Σ to [x 1 , x 2 ] by a density estimate which will say that the total length of the little forgotten pieces is very small (thus, by connectedness, very close in distance as well).
We now can prove the existence of a threshold for which β stays small at smaller scales as soon as it is small at one scale.
Proposition 5.16. Let Ω ⊂ R 2 , f ∈ L p (Ω) with p > 2, Σ ⊂ Ω be a minimizer, and C 0 be its upper Ahlfors regularity constant. Then there exist the numbers τ 1 , τ 2 with 0 < τ 2 < τ 1 < 10 −5 20+C 0 and r 0 > 0 such that whenever x ∈ Σ and 0 < r < r 0 satisfy B r (x) ⊂ Ω and then (i) (51) also holds with r/16 instead of r. (ii) Proof. We first fix r 0 and τ 2 small enough so that where C appears in Proposition 5.8, and we also assume that Take an arbitrary r ∈ (0, r 0 ) as in the statement. The control on ω Σ will be achieved by use of Proposition 5.7 which says that This proves (ii). The proof of (i) will be accomplished as soon as we prove (iii) because provided that we choose τ 2 and r 0 small enough with respect to τ 1 . Next, recalling (52), we apply Proposition 5.10 (ii) in order to find some s ∈ [r/4, r/2] such that #Σ ∩ ∂B s = 2 (notice indeed that β Σ (x, r) ≤ 10 −5 /(20 + C 0 ) and all the assumptions of Proposition 5.10 are fulfilled with x, r instead of x 0 , r 1 respectively). Then assertion (iii) of Proposition 5.10 says that the two points a 1 and a 2 of Σ ∩ ∂B s (x) must lie on both sides, and B s (x) ∩ Σ is connected. Moreover, Proposition 5.10 (iii-3) says Let Γ ⊂ Σ ∩ B s (x) be an injective curve that connects a 1 and a 2 . Lemma 5.14 (with s instead of r) says that Next, since H 1 (Γ) ≥ |a 2 − a 1 | we also get from (54) that Let us estimate the right hand side, with the constant C > 0 possibly changing from line to line ≤ Cr(ω Σ (x, r)) 1 2 + Cr 1 p + 1 2 the latter inequality being due to ω Σ (x, r) ≤ 1, r ≤ 1 and 1 p > 1 2 . Moreover, from (57) we get √ 2rR ≤ Cr(ω Σ (x, r)) This concludes the proof of (iii), and so follows the proof of the Proposition.
Next, we iterate the last proposition to obtain the following one.
Note that at this point, using the uniform rectifiability of Σ and a standard compactness argument, it would not be very difficult to prove that Σ is C 1,α regular outside a set of Hausdorff dimension d < 1. But the blow-up analysis of the next section will actually prove much more, namely that d = 0.
We finish this section by a last statement saying that Corollary 5.18 still holds at the boundary, if the domain is convex.
Proof. If Ω is convex, then Problem 1.1 is exactly the same if we relax the class of competitors {Σ ⊂ Ω, closed and connected} by {Σ ⊂ R 2 , closed and connected}.
Indeed, the projection P Ω onto Ω is 1-Lipschitz thus minimizing on the second class would lead to the same minimizers since the projection of any competitor has a lower value of F. This means that, in the case of convex domains, we may consider Σ as a subset of R 2 with competitors in R 2 , and consider u Σ ∈ H 1 0 (Ω \ Σ) as a function of H 1 (R 2 ) extending it by zero outside Ω, and then one can follow the whole Section 5 line by line and check that it works at the boundary straight away.
Remark 5.20. A consequence of Proposition 5.19 is that, if Ω is convex and C 1 but nowhere C 1,α , then Σ will never want to stay inside ∂Ω for a while because it would contradict the C 1,α regularity. In this case it can only touch ∂Ω pointwise.
6 Blow-up limits of minimizers and first consequences 6

.1 Convergence of blow-up sequences for interior points
Throughout this section, let Σ denote a minimizer for the Problem 1.1, and u := u Σ . Let {x n } ⊂ Σ be a sequence of points and r n → 0 + as n → ∞. We define the blow-up sequence by Our scaling implies that It follows that, for all n, (u n , Σ n ) is a minimizer in Ω n for the optimal compliance problem associated to the function f n (x) := r 3/2 n f (r n x + x n ). For a given closed set Σ ⊂ R 2 we consider the subspace H 1 0,Σ,loc (R 2 ) of H 1 loc (R 2 ) consisting of functions vanishing on Σ defined by where ϕ R is the Lipschitz cut-off function ϕ R (x) := max(R − |x|, 0).
, and a subsequence r n k → 0 such that (u n k , Σ n k ) converges to (u 0 , Σ 0 ) in the following way: for every ball B ⊂ R 2 , (i) Σ n k → Σ 0 in the Kuratowski sense in R 2 , and, moreover, In addition, u 0 is harmonic in R 2 \ Σ 0 .
Proof. For simplicity we will not relabel subsequences. We first extract a subsequence so that (i) holds. To this aim, let us stress that, in general, if a sequence of sets converge for the Hausdorff distance, then their restrictions to subsets may not converge. This is why the notion of local Hausdorff converging sequence in R 2 is delicate (see [11]) and we shall not try to make Σ n ∩ B converging to Σ 0 ∩B for every ball for the Hausdorff disance. Instead, we start by using Blaschke principle in the Alexandrov one-point compactification of R 2 , to extract a subsequence of Σ n converging to some set Σ 0 in the compactified space. This implies the convergence of Σ n in the Kuratowski sense in the compactified space, hence also in R 2 , and it also implies that, for every ball B ⊂ R 2 , (Σ n ∩ B) ∪ ∂B → (Σ 0 ∩ B) ∪ ∂B for the classical Hausdorff distance (in fact, it clearly converges in Kuratowski sense, and hence in Hausdorff distance because (Σ n ∩ B) ∪ ∂B are all included in the same compact set B).
We now claim that (S) for every ball B = B R (0) ⊂ R 2 one may extract a further subsequence (depending on B) such that for some u 0 ∈ H 1 (B) harmonic in B \ Σ 0 and satisfying ϕu 0 ∈ H 1 0 (B \ Σ 0 ) for all ϕ ∈ C ∞ 0 (B), one has u n → u 0 strongly in L 2 (B) and ∇u n → ∇u 0 strongly in L 2 (B R/2 ; R 2 ).
To show claim (S), use first the change of variables to get B |∇u n | 2 dx = 1 where C > 0 depends only on f p and R, the latter estimate being due to Lemma 3.3 applied with γ = 2π (see Remark 3.4). Together with the Poincaré inequality (Lemma A.1) this implies where still C = C( f p , R) > 0. This means that, up to a subsequence, u n u 0 weakly in H 1 (B) for some u 0 ∈ H 1 (B) (hence strongly in L 2 (B)). This shows that ϕu 0 ∈ H 1 0 (B \ Σ 0 ) for any Lipschitz function ϕ vanishing on ∂B (see Remark 2.4).
Integrating over C s := (1 + s)B R/2 \ B R/2 and recalling that ∇w n = ∇ũ n in B, we can write (71) in the form where R n := 2 By the convexity of | · | 2 we get the inequality so that (72) becomes and in view of ψ ≤ 1 we finally get Notice now that R n → 0, because all the functionsũ n , u n , w n converge strongly in L 2 ((1 + s)B) to the same function u 0 , the sequence {H n } is uniformly bounded in L 2 (C s ) ⊂ L 2 (B), and f n → 0 in L 2 ((1 + s)B) ⊂ L 2 (B). Therefore, passing to the limsup in (73) and using the strong convergence ofũ n to u 0 in H 1 ((1 + s) which gives (70) by taking the limit in s → 0 + , hence completing the proof of strong convergence of ∇u n to ∇u in L 2 (B R/2 ) and therefore concluding the proof of claim (S).
Finally, once the claim (S) is proven, it suffices to choose for each m ∈ N a subsequence {n(m, j)} j ⊂ N such that {n(m + 1, j)} j ⊂ {n(m, j)} j and each sequence {u n(m,j) } j is convergent strongly some function in L 2 (B 2m ) with the sequence of gradients {∇u n(m,j) } j convergent strongly to the gradient of the same function in L 2 (B m (x 0 ); R 2 ), with the limit function as in claim (S) (with B 2m instead of B). Taking the diagonal sequence {u n(m,m) } m we have that there is a u 0 ∈ H 1 loc (R 2 ) harmonic in R 2 \ Σ 0 such that ϕu 0 ∈ H 1 0 (B R \ Σ 0 ) for every R > 0 and ϕ vanishing over ∂B R , with {u n(m,m) } m convergent strongly to u in L 2 (B R ) and the sequence of gradients {∇u n(m,m) } m convergent strongly to u in L 2 (B R ; R 2 ). This completes the proof of (ii).

Two compactness estimates
The goal of this section is to prove that, as soon as β Σ (x, r) with x ∈ Σ is small enough, then all the assumptions of the C 1 regularity result are satisfied. To this aim we need to control the energy ω Σ of u from the flatness β Σ , and this is done in the following proposition via a compactness argument. Proposition 6.2. Assume that f ∈ L p (Ω), p > 2, Σ be a minimizer of Problem 1.1 and let Proof. The balls in this proof are all centered in x. By Remark 3.4 we know that the limit in (74) exists, and is finite. Assume by contradiction that the limit is equal to some C > 0. Then by considering the blow-up sequence u Σ (r n y + x)/ √ r n we know that it converges as r n → 0 in R 2 , to some harmonic function u 0 in the complement of a line, with u 0 = 0 on that line. By the strong convergence in H 1 loc (R 2 ) we infer that u 0 has constant normalized Dirichlet energy, equal to C, in other words But then by using a decomposition of u 0 into spherical harmonics, it is easy to see that u 0 must be equal to zero. Indeed, considering u + 0 the restriction of u 0 on one side of the line, one can extend it as a harmonic functionũ 0 in the whole R 2 using a reflexion. This function still satisfies (75). On the other hand it is a sum of harmonic polynomials whose gradients are orthogonal in L 2 (the so called spherical harmonic decomposition) which contradicts the estimate (75) (for more details see for instance [21,Theorem 15] for a similar argument in conical domains).
Then we control ω Σ (x, r) from Br(x) |∇u Σ | 2 by another compactness argument. Proposition 6.3. Assume that f ∈ L p (Ω), p > 2, and let C > 0 be the constant of Remark 3.4 (depending only on |Ω|, f p , p). Then for every r 0 ∈ (0, diam (Σ)/2) there exists a ρ ∈ (0, 1) such that for any x ∈ Σ we have Proof. If the claim is false, one can find an r 0 ∈ (0, diam (Σ)/2) and two sequences ρ n → 0, {x n } ⊂ Σ such that This implies in particular the existence of a sequence of compact connected maximizers Σ n ⊂ Ω for ω Σ such that Σ n ∆Σ ⊂ B ρnr 0 (x n ) and From Remark 3.4 applied with Σ n and x n instead of Σ and x 0 respectively, we get , which together with (76) gives for an arbitrary ε > 0 the estimate the estimate for all sufficiently large n (depending on ε). Then passing to the limit (up to a subsequence) x n → x 0 , Σ n → Σ 0 (which necessarily equals Σ) and applyingŠverák Theorem 2.3 we get and then passing to a limit in ε → 0 + we arrive at a contradiction.

Flat points are C 1 -points
A first consequence of the above compactness estimates is the following statement that we shall need later.
Theorem 6.4. Assume that f ∈ L p (Ω), p > 2, and let Σ be a minimizer of Problem 1.1. There exist an ε > 0 and an a ∈ (0, 1) such that, if x ∈ Σ is such that B r (x) ⊂ Ω and then Σ ∩ B(x, ar) is a C 1,α curve for some α ∈ (0, 1). As a consequence, for any point x ∈ Σ ∩ Ω which admits a line as blow-up limit, there exists r > 0 such that Σ ∩ B r (x) is a C 1,α curve.
Proof. From Proposition 6.2 and Proposition 6.3 we deduce that, provided ε is small enough, one can find r 0 such that both β Σ (x, r 0 ) and ω Σ (x, r 0 ) are small enough to apply Corollary 5.18.
For some technical reasons we shall also need the following more precise version of the above statement, which is just a rephrasing of some of the previous results, and that will be needed only in the proof of Proposition 6.7. Lemma 6.5. Assume that f ∈ L p (Ω), p > 2 and let Σ be a minimizer of Problem 1.1. There exist the numbers ε 0 ∈ (0, 1/100), η > 0 and r 0 > 0 such that whenever x ∈ Σ and 10r ≤ min(r 0 , d(x, ∂Ω)) are such that β Σ (x, 10r) ≤ ε 0 , then B r (x) ∩ Σ is a C 1,α curve for some α ∈ (0, 1) satisfying in addition Br(x) for all t < r and for some C > 0 depending on |Ω|, f p , p. Moreover, for any t < r there exists an s ∈ [t/2, t] such that #Σ ∩ ∂B s (x) = 2 with the points lying on both sides.
Proof. The first part of the claim, (77) and the conclusion about the existence of s ∈ [t/2, t] such that #Σ ∩ ∂B s (x) = 2, are directly coming from the proof of Theorem 6.4 (and from the proofs of previous propositions used for the proof of the later). But then this allows us to apply the monotonicity Lemma 3.3, which says that for all t < r with α = 2π/γ Σ (x, 0, r). Indeed, we may assume that 2 p − 2π γ Σ (x,0,r) > 0 provided that ε 0 is small enough (with respect to p). Thus (78) follows by setting η := 1 − 2π γ Σ (x,0,r) .
Remark 6.6. Though it will not be used, under the assumptions of Lemma 6.5 we also have by Proposition 5.17 that but in the sequel we really need the slightly different version (78).

Chord-arc estimate and connectedness of blow-up limits
Let d Σ (x, y) be the geodesic distance in Σ. We would like to prove the following.
Proposition 6.7. Assume that f ∈ L p (Ω), p > 2. Let Ω be a C 1 domain and let Σ ⊂ Ω be a minimizer of Problem 1.1. Then there exists C > 0 (depending on Σ) such that Remark 6.8. As it will be clear from the proof, without any condition on the boundary of Ω one would have that (79) holds with C depending on the distance between {x, y} and ∂Ω.
The proof of Proposition 6.7 uses the uniform rectifiability of the minimizer Σ, and will be used later to prove that blow-up limits of minimizers are connected sets. It also needs some well known facts about uniformly rectifiable sets. Here is the exact statement that we shall use. Proposition 6.9. For any C > 0 and ε > 0 there exists c 0 ∈ (0, 1) depending only on C and ε, such that the following holds. Let K be a compact connected set which is Ahlfors-regular with constants C, and let Γ ⊂ K be a curve. Then for any x ∈ Γ and r < min(1, diam (Γ)), there exists a ball B s (y) ⊂ B r (x) such that y ∈ Γ, s ≥ c 0 r and β K (y, s) ≤ ε.
Proof. The statement is classical in uniform rectifiability theory, but here we stress that the concluding ball B s (y) where β K (y, s) is small is centered on y ∈ Γ, a given sub curve of K. The proof follows by the same classical argument but we prefer to give the full details since it may not be as standard as when one wants y ∈ K.
Since K is compact, connected, and Ahlfors-regular, then it is a uniformly rectifiable set, in other words contained in an Ahlfors-regular curve [11,Theorem 31.5]. Consequently, the set K satisfies the so-called BWGL (Bilateral Weak Geometric Lemma [13, Definition 2.2, p. 32]), which means the following. For any ε > 0 let us consider the bad set Then there exists some C ε > 0 such that Now let us consider the set Of course 1 B ε ≤ 1 Bε thus we readily have from (80) Now let r 1 = min(1, diam (Γ)) we argue by contradiction and assume, for some given (x, t) ∈ Γ × (0, r 1 ), that β K (y, s) > ε for all s ∈ (ct, t/2), and for all y ∈ Γ ∩ B t/2 (x), where c > 0 has to be chosen later. Since Γ is a curve, x ∈ Γ and since t/2 < diam (Γ) we have Therefore, which provides a contradiction if we chose c > 0 small enough compared to C ε .
We can now prove Proposition 6.7.
Proof of Proposition 6.7. We argue by contradiction. We already know that Σ is Ahlfors regular with upper constant C 0 . Let r 0 , ε 0 , δ > 0 be the constants of Proposition 6.5 while c 0 is the constant given by Proposition 6.9 with the choice ε := ε 0 . We also denote by r 0 the constant of Lemma 3.5. Assume now that for some {x, y} ⊂ Σ with R := |x − y| small enough, any geodesic curve Γ ⊂ Σ connecting them satisfies where Λ > 0 is a large constant that will be fixed during the proof. Namely, Λ will first be chosen large enough, and then R will be taken sufficiently small, depending in particular on this large Λ. Let us now proceed to the proof.

Now we distinguish two cases.
Case A: B 10s 0 (z ) ⊂ Ω. Then, assuming ΛR < r 0 , we can apply Proposition 6.5 with z and s 0 in place of x and r respectively. This implies with C > 0 independent of t and s 0 . In the sequel we will still denote by C a constant that may change from line to line. Then, we consider t satisfying 2R ≤ t ≤ 4R and such that Σ ∩ ∂B t (z ) = 2 with the points lying on both sides (this is possible by the last conclusion of Proposition 6.5). Then we take the competitor it is easy to verify that Σ is connected. In addition, one has Now letΣ = Σ \ B t (z ), so thatΣ Σ ⊂ B s/2 (z ) andΣ ⊂ Σ . Since H 1 0 (Ω \ Σ ) ⊂ H 1 0 (Ω \Σ) and since uΣ minimizes E over H 1 0 (Ω \Σ), we have E(uΣ) ≤ E(u Σ ) which in turn says that Therefore, using that Σ is a compliance minimizer and that Σ is a competitor, we can write with C > 0 independent of t. Plugging (84) into (85) and using (83) we obtain Bs 0 (z ) Bs 0 (z ) Next we apply Remark 3.4 (monotonicity with γ = 2π) to get the following estimate, denoting also by d 0 = diam (Σ)/2, Returning to (86) and using that s 0 ≤ ΛR/20 we have now obtained By choosing now Λ sufficiently large so that C 1 Λ δ ≤ 1 2 we get Finally, recalling that 2/p > 1 we obtain a contraction for R sufficiently small.
Case B: there exists a z ∈ B 10s 0 (z ) ∩ ∂Ω. Then B s 0 (z ) ⊂ B 20s 0 (z ) and assuming that ΛR ≤ r 0 we can apply Lemma 3.5 which implies where C > 0 is independent of R and r 0 := r(∂Ω, p) in the notation of Lemma 3.5. We can also assume that ε 0 ≤ 10 −5 /(20 + C 0 ) and that R is small enough so that the assumptions (40) and (42) of Proposition 5.10 are satisfied, thanks to (89). This allows us to find s ∈ [s 0 /4, s 0 /2] such that Σ ∩ ∂B s (z ) = 2 with the points lying on both sides. We then continue as in Case A, namely, we take the competitor where Γ x,y is the geodesic curve in Ω that connects x and y. Since Ω is a C 1 domain, we have as R → 0 + . Therefore, if R is small enough, Now arguing exactly as for (85) with s instead of t we get the estimate which yields again a contradiction for R small enough. This means that for some R 0 > 0, once x, y ∈ Σ satisfy |x − y| ≤ R 0 , then the geodesic curve Γ ⊂ Σ connecting them does not satisfy (81), or, in other words, that (79) holds for any such couple of points. But if |x − y| ≥ R 0 we obviously have that which concludes the proof.
One of the main purposes of proving the chord-arc estimate is to obtain the following consequence.
Proposition 6.10. The set Σ 0 given by Proposition 6.1 is an unbounded arcwise connected set, and any connected component of R 2 \ Σ 0 is simply connected.
Proof. Let Σ be a minimizer for Problem 1.1. We first prove that Σ 0 is connected. Let x and y be two arbitrary points in Σ 0 , and set R := |x| + 2|x − y|. Since Σ n → Σ for the Hausdorff distance in B R (0), we can find two sequences x n → x and y n → y satisfying {x n , y n } ⊂ Σ n for all n ∈ N. Let Γ n ⊂ Σ n be a geodesic curve connecting x n to y n in Σ n . By Proposition 6.7 there exists a constant C > 0 (depending only on Σ) such that and hence Γ n ⊂ B CR (0). We can therefore extract a subsequence such that Γ n → Γ in Hausdorff distance for some compact connected Γ ⊂ B CR (0). But then Γ is a connected set contained in Σ 0 which necessarily contains x and y. Since x and y were arbitrary, we have that Σ 0 is a connected set.
It is also quite clear that Σ 0 must be unbounded because diam (Σ n ) → +∞ and Σ n converges to Σ 0 in the sense of Kuratowski. It remains to prove that any connected component of R 2 \ Σ 0 is simply connected. Let U be a connected component of R 2 \ Σ 0 , and let Γ ⊂ U be a simple closed curve. Without loss of generality, we may assume that this curve is polygonal. Since Γ is a Jordan curve, R 2 \ Γ has two connected components, A − and A + . Let A − be the bounded one. Since Σ 0 is unbounded, we must have Σ 0 ∩ A + = ∅. And since Σ 0 is connected we must have Σ 0 ⊂ A + , because otherwise we could write Σ 0 = (Σ 0 ∩ A + ) ∪ (Σ 0 ∩ A − ), a union of two relatively closed disjoint sets, which would give a contradiction. But now it is clear that, Γ being polygonal, it can easily be retracted to a point in A − , proving the claim.

Dual formulation and classification of blow-up limits
Blow-up limits are not easy to handle in the original formulation of the problem because it is of min-max type, so that the functional is not easily localizable. To overcome this difficulty we use a dual formulation of the problem to transform the min-max into a min-min, and then in many aspects we follow the strategy pursued by Bonnet [3] for Mumford-Shah minimizers.
More precisely, we prove, using a duality argument, that the blow-up limits converges to the following notion of global minimizers. In analogy with the Mumford-Shah problem we define the following notion of a compliance global minimizer.
Assuming furthermore that Σ is connected (which actually occurs in our case), our notion of global minimizer (u, Σ) turns out to be exactly the dual one of Mumford-Shah global minimizers, thus Σ must be of the same type of the ones from the Mumford-Shah functional.

Dual formulation
We first find a dual formulation of our problem, for which we give a completely elementary proof without any abstract convex analysis. In addition, for a given compact connected Σ ⊂ Ω fixed, the choice σ := ∇u Σ solves Proof. For any given u ∈ H 1 0 (Ω \ Σ) and σ ∈ L 2 (Ω; R 2 ), we can write and since we have equality for σ = ∇u, we then have just proven the famous Legendre transform identity Therefore, −C(Σ) = min Now we want to exchange min and max in the above formula. It is quite standard that we can do so by use of abstract convex analysis, but let us give here an elementary proof. It is clear that we always have that min where D stands for the space of σ ∈ L 2 (Ω; R 2 ) such that otherwise the infimum in u would be −∞.
To verify the reverse inequality, we consider the minimizer u Σ for the problem min u∈H 1 0 (Ω\Σ) We observe that the optimality conditions on u Σ yields ∇u Σ ∈ D. Thus, recalling that the maximum in (93) is attained at σ := ∇u for u ∈ H 1 0 (Ω \ Σ), we get min In conclusion, we have proven that and σ = ∇u Σ is a maximizer. It follows that and the proposition follows from the uniqueness of u Σ and the minimizer σ.
and it remains to estimate B |∇u| 2 dx. For this we observe that for every ϕ ∈ H 1 0 (B ∩ Ω) one has B∩Ω ∇u · ∇ϕ dx = B∩Ω ϕf dx, so that taking ϕ := u and using the Poincaré inequality we obtain where C > 0 is universal. We conclude then from (95) the estimate which in particular implies Ahlfors regularity of Σ.

Minimization problem for the blow-up limit
The aim of this section is to prove the following assertion.
Proposition 7.4. The limit (u 0 , Σ 0 ) of Proposition 6.1 is a compliance global minimizer in the sense of Definition 7.1. In addition, for any ball B ⊂ R 2 one has that where Σ n k is as in Proposition 6.1.
Proof. Let Σ n k , u n k , Σ 0 and u 0 be as in Proposition 6.1, and we further write n instead of n k for brevity. We will use the dual formulation (92) which says that for all n ∈ N, the pair (∇u n , Σ n ) solves min (σ,Σ)∈Bn Now let B ⊂ R 2 be a ball. We may assume without loss of generality that B = B 1 (0). All the balls in this proof will by default be centered at the origin. Let the pair (σ, Σ ) be a competitor to (u 0 , Σ 0 ) in B in the sense of Definition 7.1, i.e. satisfying Σ \ B = Σ 0 \ B and Σ ∩ B is connected.
Let ε ∈ (0, 1) be a small parameter. We can choose an s ∈ (1, 1 + ε) such that N := #Σ 0 ∩ ∂B s < +∞ (notice that Σ 0 has finite length due to Go lab's theorem and the Ahlfors regularity of Σ n with same constant). Let δ > 0 be another small parameter, and let us define the set Then since Σ n converges to Σ 0 in the Kuratowski sense in R 2 , it follows that for n large enough, Σ n ∩ B 2 stays inside a δ-neighborhood of Σ 0 ∩ B 2 , thus it is easily seen that the set is arcwise connected. Let now σ ∈ L 2 loc (R 2 ; R 2 ) be as above, a competing vector field associated to Σ in the sense of Definition 7.1, i.e. σ = ∇u 0 in R 2 \ B and divσ = 0 in R 2 \ Σ . We want to prove that For this purpose we modify σ to make it admissible as a competitor for the minimization problem (98), i.e. construct a vector fieldσ n such that (σ n , Σ n ) ∈ B n . We start by considering a 2/(s − 1)-Lipschitz cut-off function ϕ equal to 1 on B 1 and to 0 outside B 1+(s−1)/2 , and we let Notice that divσ n = (σ − ∇u n ) · ∇ϕ + (1 − ϕ)f n =: g n in D (Ω n \ Σ n ), and that g n → 0 strongly in L 2 loc (R 2 ), because σ = ∇u 0 a.e. in R 2 \ B, ∇u n → ∇u 0 strongly in L 2 loc (R 2 ; R 2 ) and f n → 0 strongly in L 2 loc (R 2 ) (recall that u 0 is harmonic by Proposition 6.1). We now add a correction because we would like the divergence to be exactly equal to f n . For this purpose we denote, for convenience, r 0 := 1 + ε and let v n be the solution for the problem In other words we are solving the problem −∆v n = f n − g n in B r 0 \ Σ n , v n = 0 on Σ n ∩ B r 0 and ∇v n · ν = 0 on ∂B r 0 \ Σ n . It follows that −div(1 Br 0 ∇v n ) = (f n − g n )1 Br 0 in D (R 2 \ Σ n ).
In addition one has the estimate Since v n vanishes on the substantially large piece of set Σ 0 ∩ B 1+ε \ B 1 , by Poincaré inequality (Lemma A.2) we infer that
Since bothσ n = ∇u n and Σ n = Σ n outside B r 0 , we get where we have estimated H 1 (S δ ) ≤ cN δ, with c > 0 being a universal constant. In other words, Notice that thanks to Go lab's Theorem (one can just use the classical Go lab theorem applied to the sequence of compact connected sets (Σ n ∩ B) ∪ ∂B which converges to (Σ ∩ B) ∪ ∂B for the Hausdorff distance by Proposition 6.1 (i)).
Taking the liminf in n of (102), using thatσ n → σ strongly in L 2 (B r 0 ; R 2 ) we obtain that We can now let δ → 0 + and then ε → 0 + to get which concludes the proof of the fact that (u 0 , Σ 0 ) is a global minimizer. It remains to prove that Inequality (103) gives the first half. To prove the reverse inequality, we pass to the limsup in n in (102) with the special choice Σ = Σ 0 and σ = ∇u 0 . Using ∇u n → ∇u 0 we obtain lim sup n H 1 (Σ n ∩ B) ≤ H 1 (Σ 0 ∩ B) + cN δ and letting δ → 0 + we conclude the proof.

Another formulation and characterization of global minimizers
In this section we give another formulation of global minimizers and prove that it is equivalent to Mumford-Shah minimizers.
Definition 7.5. A pair (u, K) with K ⊂ R 2 closed is called global Mumford-Shah minimizer, if u ∈ L 2 loc (R 2 \ K) and ∇u ∈ L 2 loc (R 2 ) satisfies for every ball B ⊂ R 2 and for all (v, L) satisfying v = u in R 2 \ B and K \ B = L \ B, and such that L ∩ B is connected.
Remark 7.6. The original definition of Bonnet [3] was slightly different, with a less restrictive topological condition of competitors, namely keeping {x, y} ⊂ R 2 \ B separated by L as soon as they were separated by K. We cannot use here exactly the same definition of competitors because we need our competitors to be connected. However, our definition is stronger, in the sense that any of our compliance global minimizers is automatically a global minimizer in the sense of Bonnet, which is enough to get the classification of global minimizers.
If Ω ⊂ R 2 is open and u is harmonic in Ω, we call harmonic conjugate for u in Ω a harmonic function v in Ω such that ∇v = ∇ T u, where the notation ∇ T u is used for the vector field (−∂ y u, ∂ x u). Proof. Since each component of R 2 \ Σ 0 is simply connected, u 0 admits a harmonic conjugate. Namely, for any U ⊂ R 2 \ Σ 0 connected component, since div∇u 0 = 0 in D (U ), by De Rham's theorem we get the existence of u harmonic in U such that ∇ T u 0 = ∇u in R 2 \ Σ 0 . In particular, ∇u 0 L 2 (K) = ∇u L 2 (K) for all K ⊂ R 2 , compact. Now let (v, Σ ) be a competitor in some ball B ⊂ R 2 , i.e. v = u in B \ R 2 , Σ \ B = Σ 0 \ B and Σ ∩ B is connected. Then we define σ := ∇ T v. In particular, divσ = 0 in R 2 \ Σ and σ = ∇ T u = ∇u 0 in R 2 \ B. By the minimality of (u 0 , Σ 0 ) in the sense of Definition 7.1 we have thus (u, Σ 0 ) is a global Mumford-Shah minimizer.
As a direct consequence we get the following. Proposition 7.9. Any blow-up limit (u 0 , Σ 0 ) given by Proposition 6.1 is one of the following list, up to a translation, rotation, dilatation, or adding a constant for u 0 : (i) Σ 0 is a line and u 0 is a constant on each side of it; (ii) Σ 0 is a propeller (three half lines meeting in a single point by number of 3 at 120 degree angles) and u 0 is a constant in each of the sectors formed by these half lines; (iii) Σ 0 is a half-line and u 0 is the "Dirichlet-craktip" function (r, θ) → r/2π cos(θ/2) written in polar coordinates (r, θ) ∈ [0, +∞) × (−π, π).
Proof. Proposition 6.10, Proposition 7.4 and Proposition 7.8 imply that Σ 0 is a global Mumford-Shah minimizer in the sense of Definition 7.5. To obtain that Σ 0 is the singular set of a global Mumford-Shah minimizer in the sense of Bonnet, it is enough to prove that it satisfyies the topological condition of Bonnet, namely, that each pair of points x and y in Σ 0 \ B that are separated by Σ 0 , are still separated by any competitor L. But this is clear due to the fact that L ∩ B is connected. It follows from Bonnet [3] that it must be one of the list described above (notice that the blow-up limit is never empty since we blow-up a connected set).

Blow-up limit at a boundary point
The purpose of this section is to state results analogous to those of the previous sections, in the particular case of a boundary point, namely when x n → x 0 ∈ ∂Ω. Since the proofs are very similar, we shall only highlight the main differences. We shall put together the results in the following unified statement. We first define the type of minimizing problem that arises at the limit.
Remark 7.12. Our class of boundary compliance global minimizers looks similar to the one dimensional sliding minimal sets studied in [15,12]. Notice however that our class is quite different (and, actually, is simpler) because we do not impose any sliding condition: in our situation, the competitors are not obliged to preserve the points of Σ ∩ ∂H on the boundary, they are free to be detached and move everywhere including inside the domain. We indeed will arrive to a different classification than the one of [15,Lemma 4.3].
where Ω is a C 1 domain. Let (u n , Σ n ) be the blow-up sequence in Ω n defined in (66) and (67) with x n → x 0 ∈ ∂Ω and assume that d(x n k , ∂Ω) ≤ r n for a subsequence (otherwise the blow-up analysis is exactly the same as the interior case). Then there exists a further subsequence (still indexed by n k ) such that Ω n k converges to a half-plane and (i) Σ n k → Σ 0 in the Kuratowski sense in R 2 , (ii) u n k → 0 strongly in H 1 loc (R 2 ), (iii) Σ 0 is a boundary compliance global minimizer in the sense of Definition 7.10, Proof. Let us index the sequence by n instead of n k . The proof is the same as in the case of interior points. The only difference is that now Ω n blows-up to a half-plane instead of the whole R 2 . Indeed, the assumption d(x n , ∂Ω) ≤ r n says that for all n, there exists some z n ∈ ∂Ω with d(z n , x n ) ≤ r n and in particular d(z n , x 0 ) → 0. Since ∂Ω is C 1 , it is clear that the blow-up of Ω at z n is the tangent line of ∂Ω at this point, and converges to the tangent line at point x 0 . Comparing the blow-up at z n and the one at x n (for instance by noticing B rn (z n ) ⊂ B 2rn (x n ) ⊂ B 4rn (z n )), we deduce that Ω n blows up to a half-plane. Let us assume without loss of generality that this half-plane is The proofs of Proposition 6.1 and Proposition 6.10 can be followed line by line without any change even at the boundary, leading to the fact that Σ n → Σ 0 in the Kuratowski sense in R 2 , Σ 0 is arcwise connected and R 2 \ Σ 0 is simply connected, and finally u n converges strongly in H 1 loc (R 2 ) to a harmonic function u 0 in H \ Σ 0 .
We prove now that ∇u 0 = 0. Indeed, Lemma 3.5 implies where C 1 and C 2 depend only on |Ω|, p, f p . This estimate, since B rn (z n ) and B rn (x n ) are comparable in the sense that B rn (z n ) ⊂ B 2rn (x n ) ⊂ B 4rn (z n ), implies 1 r n Br n (xn) |∇u Σ | 2 dx −→ n→+∞ 0, and this yields ∇u 0 = 0 as claimed. Now comes the identification of the minimization problem arising at the limit. For this purpose we follow the proof of Proposition 7.4. Everything works the same way except that now the competitor Σ n may not be admissible, since it may not be inside Ω. We then modify the proof as follows: by the C 1 regularity of ∂Ω , we know that there exists some η n → 0 such that ∂Ω ∩ B rn (z n ) ⊂ C ηn , where C η is a cone of aperture η, namely For every η, let Φ η : H → H \ C η be the following (1 + η)-Lisphchitz mapping Φ η (x, y) = (x − η|y|, y).
Let now Σ be a competitor for Σ 0 in some ball B ⊂ R 2 . Then Σ := Φ ηn (Σ 0 ) ⊂ Ω n and The rest of the proof then follows the same way with Σ in place of Σ .
Proposition 7.14. Let Σ be a compliance global minimizer in the sense of Definition 7.10 with Σ ∩ ∂H = ∅. Then Σ = ∂H.
Proof. Let x 0 ∈ Σ∩∂H, S be a connected component of Σ∩B R containing x 0 and N := S ∩∂B R , where R > 0 is arbitrary, the balls here and below being centered at x 0 . Then one has, in the notation of Lemma 7.15, that S ∈ St(N ), since otherwise for an arbitrary K ⊂ St(N ) taking L := (Σ \ S) ∪ K, one gets, recalling that K ⊂ B R by Lemma 7.15 (see below), the estimate contradicting the assumption that Σ be a compliance global minimizer. But then by Lemma 7.15 the set S may not contain In fact, if either of the points A ± R does not belong to N , then x 0 does not belong to the closed convex envelope of N . This shows {A + R , A − R } ⊂ N , and hence, since R > 0 is arbitrary, then ∂H ⊂ Σ. To show that in fact Σ = ∂H, we assume the contrary, and let now R > 0 be such that #(Σ ∩ ∂B R ) is finite (by coarea inequality, a.e. R > 0 would suit for this purpose because Σ has locally finite length) and a connected component S of Σ ∩ B R containing the line segment l := ∂H ∩ B R does not coincide with the latter. Then this component belongs to St(N ∪l), with N ⊂ Σ ∩ ∂B R and hence by [28,Theorem 7.4] is a finite embedded graph consisting of line segments with exactly one endpoint (we denote it by A) overl. But this cannot happen since S ∈ St(N ), where N := S ∩ ∂B R ; in fact, then there is a line segment (AB) ∈ S such that the angle between (AB) andl is less than 120 degrees, which is impossible for Steiner sets connecting a finite number of points. This contradiction shows that the connected component of Σ ∩ B R containing l must be l itself.
Finally it is easy to see that there exist no other connected components of Σ ∩ B R (x 0 ) different from l ∩ B R (x 0 ). Indeed, assuming the contrary, and letting x be a point belonging to the other component, using that Σ is arcwise connected, one can find a curve in Σ connecting x to x 0 . This curve has to branch on ∂H at some point y 0 . Then reasoning as before with a ball centered at y 0 instead of x 0 , we get a contradiction that concludes the proof. Then for every K ∈ St(N ) of finite length one has that K belongs to the closed convex envelope of N .
Proof. If K does not belong to the closed convex envelope of N , then its projection to the latter still connects N and has strictly lower length, contradicting the optimality of K.
Remark 7.16. A standard strategy to prove Proposition 7.14 would be to follow the usual classification of minimal cones. We used here a different and more elementary approach based on Lemma 7.15, but for the sake of completeness we describe the standard one: it is not difficult to see that if Σ is a cone centered at some point x 0 ∈ ∂H, then Σ can only be a half-line or a line (the latter being true only if Σ = ∂H). It is not difficult to exclude the half-line by a competitor which would cut the "corner" near x 0 , showing that the only minimal cone is ∂H. Next, for an arbitrary compliance global minimizer Σ and x 0 ∈ ∂H ∩ Σ, comparing as usual Σ with a cone over its trace on the boundary of a sphere, one shows that the density H 1 (Σ ∩ B r )/r is monotone in r. Therefore the limiting densities at r → 0 + and r → +∞ do exists. Considering the blow-in and blow-up limits, due to the strong convergence of H 1 along those sequences one can see that those limit densities can only be equal to two, because they are densities of a minimal cone. But then by monotonicity, the density of Σ itself is constant and equal to two so that Σ must be a line, leading to the conclusion Σ = ∂H.
Remark 7.17. Without any attempt to make it more precise here, one could also consider global minimizers in angular sectors instead of half-planes leading to some regularity issues in Lipschitz domains instead of C 1 domains. For instance, it is not difficult to see that in any convex angular sector, the only possible global minimizer is the empty set. This means, for instance, if Ω is a convex polygone, a minimizer Σ of Problem 1.1 will never go through a corner of ∂Ω.

Conclusion and full regularity
In this section, we prove our main result on characterization of minimizers. In particular we are interested in the following statement.
Theorem 8.1. Assume that f ∈ L p (Ω), p > 2, where Ω is a C 1 domain. Then every minimizer Σ for Problem 1.1 consists of a finite number of embedded curves whihc are locally C 1,α inside Ω for some α ∈ (0, 1), meeting only by number of three at 120 degree angles. In particular, Σ has finite number of endpoints, finite number of branching points (which are all triple points where smooth curves meet at 120 degree angles), and at all the other points Σ is locally C 1,α -smooth.
Remark 8.2. From the proof of Theorem 8.1 we deduce that for an arbitrary bounded open Ω ⊂ R 2 , without any smoothness condition of the boundary, one has that the result analogous to Theorem 8.1 holds locally inside Ω, namely, for every Ω Ω and every minimizer Σ for Problem 1.1 one has that Σ ∩ Ω has finite number of branching points (which are all triple points where smooth curves meet at 120 degree angles), and at all the other points Σ is locally C 1,α -smooth for some α ∈ (0, 1). Remark 8.3. If under conditions of Theorem 8.1 one has additionally that Ω is convex, then every minimizer Σ for Problem 1.1 consists of a finite number of C 1,α embedded curves in Ω for some α ∈ (0, 1), meeting only by number of three at 120 degree angles. The fact that the curves of Σ are C 1,α in Ω comes from Proposition 5.19 which says that Σ is C 1,α locally around any "flat point" with low energy ω Σ , and this holds true in Ω provided that Ω is convex (Proposition 5.19). The conclusion then follows similarly to that of Theorem 8.1, in particular any point which is not an endpoint nor a triple point is a flat point and Lemma 3.5 (for the boundary case) or Proposition 6.2 (for the interior case) says that any "flat point" has low energy ω Σ so that Proposition 5.19 applies.
The rest of the section is dedicated to the proof of the above Theorem 8.1.

Finite number of curves
To prove the assertion on the finite number of curves, we follow the approach of Bonnet [3] and start with the following observation.
Proposition 8.4. Assume that f ∈ L p (Ω), p > 2, where Ω is a C 1 domain, Σ ⊂ Ω be a minimizer for the Problem 1.1 and let x ∈ Σ. Then all the possible blow-up limits at point x are of same type. The same result holds without any condition on the boundary of ∂Ω, once x ∈ Σ ∩ Ω.
Proof. By Remark 3.4 we know that, for any x ∈ Σ the limit e(x) := lim exists, and is finite (due to (96)). By the strong convergence of the blow-ups of u Σ in H 1 loc (R 2 ) to their limit, and recalling Proposition 7.9, we deduce that e(x) = 0 in the case when there exists a blow-up limit which is a line or a propeller, and e(x) > 0 only if all the blow-up limits are half-lines.
On the other hand, if a blow-up limit at point x ∈ Σ ∩ Ω is a line, by local C 1 regularity Theorem 6.4 we know that all other blow-up limits must also be a line (and if x ∈ ∂Ω we also know that all blow-up sequences converge to the tangent lines to ∂Ω). We then easily conclude that all the blow-up limits have same type: indeed, either e(x) > 0 and all blow-ups must be a half-line, or e(x) = 0. In the latter case, either there exists a blow-up which is a line, and then all other blow-ups must be also lines, or there is no lines and then all blow-ups are propellers. Proposition 8.4 motivates the following terminology.
Definition 8.5. We define the type of x ∈ Σ as follows. The proof of Theorem 8.1 now directly follows from Theorem 6.4 together with the following result.
Theorem 8. 6. Assume that f ∈ L p (Ω), p > 2, where Ω is a C 1 domain, and Σ be a minimizer for the Problem 1.1. Then the set of triple points and endpoints is a finite set.

Further regularity
In this section we derive some regularity of higher order. It relies on the classical elliptic regularity theory and the Euler-Lagrange equation associated to our problem. Proposition 8.8. Let Ω be an open set, λ ∈ (0, ∞) and f ∈ H 1 (Ω) ∩ L p (Ω) with p > 2. Let Σ be a minimizer for Problem 1.1, and x ∈ Σ ∩ Ω, r > 0, α 0 ∈ (0, 1) such that γ := Σ ∩ B r (x) is C 1,α 0 . Then γ is C 2,α for α = 1 − 2 p , and where u + , u − are the restrictions of u on the two connected components of B r (x) \ γ oriented in a suitable way, and H γ denote the mean curvature. If moreover f is C k,β in B r (x), for k ∈ N and β ∈ (0, 1) then γ is C k+3,β .
Proof. Equality (107) is proven in [5], where H γ is understood in a weak sense. From [17,Theorem 8.34 and the remark at the end of section 8.11], u is C 2,α up to γ from each side, where α = 1 − n p . The regularity theory of the mean curvature equation gives the regularity of γ. The case f ∈ C k,α follows from a bootstrap argument.

A Auxiliary results
We introduce the following notation. For a relatively open set S ⊂ ∂B r (0) ⊂ R n λ 1 (S) the first eigenvalue of the Laplace-Beltrami operator over H 1 0 (S). For instance n = 2 and S is an arc of a circumference, then it is easy to compute λ 1 (S) = π 2 H 1 (S) 2 .
This gives a possibility to formulate the following immediate version of the Poincaré inequality.
A corollary is the following Poincaré inequality on the annulus. for every r ∈ (0, R).