Pierre Bousquet
email: pierre.bousquet@math.univ-toulouse.fr

ON A DEGENERATE PROBLEM IN THE CALCULUS OF VARIATIONS

Keywords: Mathematics Subject Classification. 35A02, 49N60, 49J45 Degenerate and singular problems; regularity and uniqueness of minimizers

We establish the uniqueness of the solutions for a degenerate scalar problem in the multiple integrals calculus of variations. The proof requires as a preliminary step the study of the regularity properties of the solutions and of their level sets. We exploit the uniqueness and the regularity results to explore some of their qualitative properties. In particular, we emphasize the link between the supports of the solutions and the Cheeger problem.

Introduction

Two problems in Optimal Design. We study a problem in the multiple integrals calculus of variations which is both singular and degenerate. This problem arises as the relaxation of the non convex functional introduced by Kohn and Strang in [START_REF] Kohn | Optimal design and relaxation of variational problems[END_REF][START_REF] Kohn | Optimal design and relaxation of variational problems[END_REF][START_REF] Kohn | Optimal design and relaxation of variational problems[END_REF]:

(1.1)

I 0 : u → Ω Φ(∇u)
where Ω is a bounded open set in R N , N ≥ 2, and Φ(y) := 0 if y = 0, 1 2 (|y| 2 + 1) if |y| > 0. The admissible functions u belong to the Sobolev space H 1 (Ω) and must agree with a given function ψ : R N → R on the boundary ∂Ω of Ω.

The functional I 0 is not lower semicontinuous under weak convergence in H 1 (Ω). Therefore, one cannot rely on the direct method in the calculus of variations to find a minimum as the limit of a minimizing sequence. Actually, it may happen that there is no minimum.

However, the infimum of I 0 is equal to the infimum of the relaxed functional (see [START_REF] Kohn | Optimal design and relaxation of variational problems[END_REF]Theorem 1.1]):

(1.2)

I 0 : u → Ω ϕ(∇u)
where ϕ is the convexification of Φ, namely the largest convex function less than or equal to Φ:

(1.3) ϕ(y) = |y| if |y| < 1, 1
2 (|y| 2 + 1) if |y| ≥ 1. In contrast to the original problem, the relaxed problem is lower semicontinuous under weak H 1 convergence. It follows that it has at least one minimizer, and for every such minimizer u, I 0 (u) = inf I 0 . Moreover, the minimizers of I 0 are exactly the weak limits of minimizing sequences of I 0 .

In order to prove the non existence of a minimizer for I 0 , a possible strategy is to establish the uniqueness of the minimizer for I 0 , see [START_REF] Marcellini | A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations[END_REF]. Indeed, assume that I 0 has a unique minimum u such that |∇u| < 1 on a positive measure set. Then I 0 (u) < I 0 (u) (here, we use the fact that ϕ(y) < Φ(y) for every 0 < |y| < 1) and for every admissible v = u, I 0 (u) < I 0 (v) ≤ I 0 (v). Since inf I 0 = inf I 0 = I 0 (u), this proves that I 0 has no minimum.

For some specific choice of Ω and ψ, this situation may arise, as illustrated in [START_REF] Kohn | Optimal design and relaxation of variational problems[END_REF]Example 7.4] where Ω is the unit square and ψ a polynomial function of degree 2. However, generally speaking, this is a delicate matter to prove the uniqueness of minimizers for I 0 , since I 0 is not strictly convex. This is one of the main contributions of this paper to establish this uniqueness property under a mild condition on Ω, see Theorem 1.1 below.

More recently, a related problem was considered in [START_REF] Alibert | A nonstandard free boundary problem arising in the shape optimization of thin torsion rods[END_REF] for the same functional I 0 , when the boundary condition ψ is equal to 0 everywhere and when Ω is an open bounded subset of R 2 that we assume to be simply connected 1 . More precisely in the context of shape optimization of thin elastic structures, Bouchitté and al. proved in [START_REF] Bouchitté | Optimal thin torsion rods and Cheeger sets[END_REF] that the section of an optimal torsion rod can be obtained from the following parametrized problem (1.4) m(s) := inf I 0 (u) ,

Ω u = s , u ∈ H 1 0 (Ω) .
Here, s is a positive parameter which gives the intensity of the applied torsion load. If the infimum in m(s) is attained at some u ∈ H 1 0 (Ω), then the subset [|∇u| > 1] corresponds to the optimal subregion where the material should be placed. The plateau of u, namely the set [∇u = 0], represents the void subregion.

By [START_REF] Alibert | A nonstandard free boundary problem arising in the shape optimization of thin torsion rods[END_REF]Proposition 3.8], m is differentiable on (0, ∞). The constraint Ω u = s can be included in the cost functional through a Lagrange multiplier. More specifically, for every λ ≥ 0, let us define (1.5)

I λ : u → Ω ϕ(∇u) -λ Ω u.
Then for every s > 0, an admissible u is a solution of m(s) if and only if u minimizes I m (s) on H 1 0 (Ω), see [2, (3.5) and Proposition 3.3].

A major issue, still beyond the scope of this paper, is the existence or the non existence of a special solution for m(s); that is, a solution u with the following property: |∇u| ∈ {0} ∪ (1, +∞ ) a.e. in Ω.

Observe that u is a special solution for I 0 if and only if it is a minimizer of I 0 . For every λ ≥ 0, one can construct non spherical domains Ω for which there exist special solutions, see [START_REF] Alibert | A nonstandard free boundary problem arising in the shape optimization of thin torsion rods[END_REF]Section 6]. When such a solution exists, an optimal design contains no homogenized region corresponding to a fine mixture of material and void. It was proved in [START_REF] Alibert | A nonstandard free boundary problem arising in the shape optimization of thin torsion rods[END_REF]Corollary 3.5] that if there exists a special solution, then this is the unique solution of m(s). As a consequence of Theorem 1.1 below, we obtain a deeper result which does not require the existence of a special solution: for every s > 0, m(s) has a unique solution.

In order to embrace the two situations arising in [START_REF] Kohn | Optimal design and relaxation of variational problems[END_REF][START_REF] Kohn | Optimal design and relaxation of variational problems[END_REF][START_REF] Kohn | Optimal design and relaxation of variational problems[END_REF] and [START_REF] Alibert | A nonstandard free boundary problem arising in the shape optimization of thin torsion rods[END_REF][START_REF] Bouchitté | Optimal thin torsion rods and Cheeger sets[END_REF], we consider henceforth any Lipschitz function ψ : R N → R for the boundary condition and any open bounded subset Ω ⊂ R N , N ≥ 2. Given λ ≥ 0, we study the problem:

(P λ )
To minimize u → I λ (u) on H 1 ψ (Ω),

where H 1 ψ (Ω) := ψ + H 1 0 (Ω). 2 Main results. Our first main result is the uniqueness of solutions for (P λ ) under fairly general assumptions on Ω.

Theorem 1.1. Assume that Ω is Lipschitz and that ∂Ω is connected. Then for every λ ∈ R, the solution of (P λ ) is unique.

The proof of Theorem 1.1 heavily relies on the regularity properties of the solutions of (P λ ). There are serious obstacles to establish them: the function ϕ is singular at the origin and is not twice differentiable on the unit sphere. Moreover, the Hessian ∇ 2 ϕ(ξ) of ϕ at ξ has a non trivial kernel for every ξ in the unit ball. Hence, (P λ ) is both singular and degenerate. In spite of these facts, recent results for this class of integrands can be used to obtain the following regularity properties: Proposition 1.2. Let λ ≥ 0 and u a solution of (P λ ). Then u is bounded on Ω and locally Lipschitz continuous. Moreover, there exists an open set U ⊂ Ω such that u is smooth on U . In fact, |∇u| > 1 on U and |∇u| ≤ 1 a.e. on Ω \ U .

When Ω is assumed to be Lipschitz, then the solutions of (P λ ) are Hölder continuous on Ω, see Section 2. On the complement of U , the regularity of u itself is still open. However, generically, the super-level sets [u ≥ t] = {x ∈ Ω : u(x) ≥ t}, with t ∈ R + , satisfy a variational problem on Ω \ U . It then follows that for a.e. t ∈ R + , the level sets ∂[u ≥ t] are C 1 hypersurfaces, up to a small singular term: Here, ∂ e [u ≥ t] is the essential boundary of the set [u ≥ t], namely the set of those x ∈ R N such that for every ρ > 0,

(1.6) 0 < |[u ≥ t] ∩ B ρ (x)| < |B ρ (x)|.
Both Propositions 1.2 and 1.3 are essential steps in the proof of Theorem 1.1. However, when N = 2, almost every level set of a Lipschitz function is a Lipschitz curve, see e.g. [START_REF] Alberti | Structure of level sets and Sard-type properties of Lipschitz maps[END_REF]Theorem 2.5]. It is then possible to rely on this property instead of Proposition 1.3 in order to prove Theorem 1.1.

In the case when Ω is the ball of radius R > 0 and ψ ≡ 0, the solution has the following explicit expression, see Remark 4.2 below:

(1.7) u(x) = - λ 2N |x| 2 - N 2 λ 2 + + λ 2N R 2 - N 2 λ 2 + .
In particular, the solution is Lipschitz continuous, but not even C 1 (except when it is the trivial solution).

The regularity and the uniqueness of the solutions for (P λ ) have important consequences for the study of their qualitative properties. Given Ω and ψ as in Theorem 1.1, for every λ ≥ 0, we denote by u λ the unique solution of (P λ ) on H 1 ψ (Ω).

2 To be more specific, H 1 0 (Ω) is the set of those u ∈ H 1 (Ω) such that the extension of u by 0 on R N belongs to H 1 (R N ).

Proposition 1.4. The map λ ∈ [0, +∞) → u λ ∈ C 0 (Ω) is continuous and nondecreasing 3 .

Assume now that ψ ≡ 0 and Ω is any bounded open Lipschitz set in R N . When λ = 0, the solution is the constant function equal to 0. It turns out that for small values of λ, 0 is still the unique solution of (P λ ). An interesting fact is that the critical value of λ for which 0 is not the solution any more is exactly the Cheeger constant h Ω of Ω:

(1.8) h Ω = inf D⊂Ω Per D |D| .
Here, Per refers to the perimeter in R N (or equivalently in Ω since all the sets that we consider are contained in Ω). The precise link of h Ω with (P λ ) is given in the following statement, which corresponds to [2, (4.5)]:

Proposition 1.5. Let λ > 0 • If λ > h Ω ,
then 0 is not a solution of (P λ ).

• If λ ≤ h Ω , then 0 is the unique solution of (P λ ).

A Cheeger set for Ω is a subset of Ω for which the infimum in (1.8) is attained. Together with the Cheeger constant h Ω , the Cheeger sets play a natural role in the framework of (P λ ):

Theorem 1.6. The set

Ω 0 := ∩ λ>h Ω [u λ > 0] is a solution of the Cheeger problem for Ω.
On some proofs of uniqueness for some multiple integrals variational problems. There are many uniqueness results for variational problems with a lack of strict convexity. We just quote four of them, which have been important sources of inspiration to us. First, in the seminal paper [START_REF] Marcellini | A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations[END_REF]Theorem 3] (see also [START_REF] Marcellini | Some remarks on uniqueness in the calculus of variations, Collège de France seminar[END_REF]), Marcellini considers the problem Minimize v → Ω g(|∇v(x)|) dx with g : [0, +∞) → [0, +∞) an increasing convex function. The bounded open set Ω is assumed to be convex and C 1 . If there exists a solution u ∈ C 1 (Ω) such that ∇u does not vanish on Ω, then u is proved to be the unique solution on the class of Lipschitz functions agreeing with u on ∂Ω. The proof of this result is based on two observations:

• Step 1 Each level set of a solution u intersects the boundary of the domain ∂Ω.

• Step 2 If v is another solution, then v is constant on the level sets of u. Since u and v agree on ∂Ω, it follows from the two above steps that u agrees with v on each level set of u, which finally proves that u = v on Ω. The first step is based on the fact that g is (strictly) increasing. The second step uses in a crucial way that the integrand only depends on the norm of the gradient of u. Both steps exploit the C 1 regularity of the level sets of u. For a general variational problem which is not stricly convex, such a regularity assumption is difficult to establish (see however [START_REF] Raymond | An anti-plane shear problem[END_REF]). In the two dimensional setting, more precisely when Ω is a bounded open set with connected boundary in R 2 , the uniqueness result remains true without the two assumptions: u is C 1 and ∇u does not vanish on Ω, see [START_REF] Lussardi | A uniqueness result for a class of non-strictly convex variational problems[END_REF].

The above strategy has been exploited in many different contexts. In [START_REF] Sternberg | Existence, uniqueness, and regularity for functions of least gradient[END_REF], the authors consider the case when g(t) = t and prove the uniqueness of the solutions in the class {u ∈ BV (Ω) ∩ C 0 (Ω) : u| ∂Ω = ψ| ∂Ω } under additional geometric conditions on Ω. The fact that the (regular components of the) level sets of a solution intersect the boundary of Ω (which corresponds to Step 1 in Marcellini's proof) arises as an essential argument in the proof of [START_REF] Sternberg | Existence, uniqueness, and regularity for functions of least gradient[END_REF]Lemma 3.4]. This is also a key tool in [START_REF] Jerrard | [END_REF], see [START_REF] Jerrard | [END_REF]Lemma 4.2], where the authors consider the more general problem To minimize v → Ω g(x, ∇v(x)) dx with an integrand g such that → g(x, •) is a norm for every x. In this extended framework, the uniqueness is established under the same assumptions on Ω as in Theorem 1.1, namely that Ω is Lipschitz and has a connected boundary.

The common feature of all the papers quoted above is that the integrand does not depend on the variable u. Adding a lower order term of the form λ Ω u as in our problem (P λ ) involves important consequences for the level sets of the solutions, as illustrated by the example of the explicit solution on the ball (1.7). Indeed, generically, level sets do not intersect the boundary of Ω any more. It follows that one cannot directly use the fact that two solutions agree on ∂Ω to deduce therefrom that they agree on Ω.

Moreover, the regularity of the level sets required by the proof in [START_REF] Marcellini | A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations[END_REF] cannot be established in the framework of (P λ ). Indeed, the explicit example on the ball shows that one cannot expect better than Lipschitz regularity for the solutions. In addition, the assumption that ∇u does not vanish on Ω is far from being satisfied: we can even prove that a solution u is necessarily constant on a positive measure set, see Lemma 2.1 below.

In [START_REF] Sternberg | Existence, uniqueness, and regularity for functions of least gradient[END_REF][START_REF] Jerrard | [END_REF], the general regularity theory for area minimizing sets is exploited in an essential way to get the C 1 regularity of the level sets. In our case, the function y → ϕ(y) behaves differently depending on whether y is small or outside the unit ball. As a consequence, the super-level sets of u do not minimize a simple variational problem on the whole Ω. We thus have to use two different strategies to establish the regularity of the level sets of u, first on [|∇u| > 1] and then on its complement.

Plan of the paper. In the next section, we present the proof of the regularity results Proposition 1.2 and 1.3. For the latter, we need to introduce a minimization problem for the super-level sets of solutions, but only on an open subset of Ω where the gradient is small. In section 3, we establish the uniqueness result Theorem 1.1. The qualitative properties Propositions 1.4 and 1.5 are proved in section 4 as well as a more precise version of Theorem 1.6, see Theorem 4.8. Finally, for the convenience of the reader, we have presented in an appendix the proof of the Lipschitz continuity of the solutions which readily follows from the arguments used in [START_REF] Fonseca | An existence result for a nonconvex variational problem via regularity[END_REF]Theorem 2.7].

Acknowledgements. We warmly thank Lorenzo Brasco, Giuseppe Buttazzo and Cédric Galusinski for very interesting discussions on this subject. Numerical solutions obtained by Cédric Galusinski on different two dimensional domains helped us to get a better insight on the qualitative properties of the minima. Part of this work has been written during a visit of the second author to Toulon. The IMATH institute is kindly acknowledged.

Regularity and Euler equation

2.1. Lipschitz regularity. We first enumerate some continuity properties satisfied by the solutions of (P λ ), λ ≥ 0. These properties readily follow from the classical regularity theory for Lagrangians with quadratic growth: By [18, Theorem 7.5, Theorem 7.6], every solution u is locally Hölder continuous in Ω. If one further assumes that Ω is Lipschitz, then [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Theorem 7.8] implies that u is globally Hölder continuous (remember that the boundary condition is given by the restriction to ∂Ω of a Lipschitz function ψ : R N → R).

Regarding the L ∞ estimate on u, we have the following result which holds true on every bounded open set Ω: Lemma 2.1. There exists C > 0 which depends only on N such that for every x ∈ Ω,

min ∂Ω ψ ≤ u(x) ≤ max ∂Ω ψ + Cλ N (u -max ∂Ω ψ) + L 1 (Ω) . Moreover, if sup Ω u > max ∂Ω ψ, then |{x ∈ Ω : u(x) = sup Ω u}| ≥ 1 Cλ N .
Remark 2.2. That sup Ω u > max ∂Ω ψ holds true is closely related to the value of λ. Actually, when Ω is Lipschitz and has a connected boundary, we can prove that there exists λ * = λ * (ψ, Ω) ∈ [0, +∞) such that for every λ ≤ λ * , sup Ω u = max ∂Ω ψ while for every λ > λ * , sup Ω u > max ∂Ω ψ, see Lemma 4.7 below.

Proof of Lemma 2.1. We denote by a := min ∂Ω ψ. We first prove that u ≥ a on Ω. Let v := max(u, a). Then v is admissible for (P λ ). From the minimality of u,

Ω ϕ(∇u) -λu ≤ Ω ϕ(∇v) -λv which implies [u<a] ϕ(∇u) ≤ λ [u<a] u -a.
Since u -a < 0 on the set [u < a] while ϕ(∇u) ≥ 0, it follows that |[u < a]| = 0; that is, u ≥ a on Ω.

We now prove that u is bounded from above. Let b := max ∂Ω ψ. For every t ≥ b, let w := min(u, t). As above, one has

[u>t] ϕ(∇u) ≤ λ [u>t] u -t.
Since ϕ(y) ≥ |y| for every y ∈ R N , this gives

[u>t] |∇u| ≤ λ [u>t] u -t.
By the Sobolev inequality in the left hand side and the Hölder inequality in the right hand side, this implies

(2.1) (u -t) + L N (Ω) ≤ Cλ (u -t) + L N (Ω) |[u > t]| 1 N ,
where C depends only on

N . If (u -t) + L N (Ω) > 0, then |[u > t]| ≥ 1/(Cλ) N . Integrating the latter inequality on an interval [b, T ] for some T > b yields T -b (Cλ) N ≤ [u≥b] min(u, T ) -b ≤ (u -b) + L 1 (Ω) .
This implies that the inequality

|[u > t]| ≥ 1/(Cλ) N can only hold true for t ≤ T 0 := b+(Cλ) N (u- b) + L 1 (Ω)
. Consequently, (u -t) + L N (Ω) = 0 for every t > T 0 and thus u is bounded from above by T 0 on Ω.

Finally, if sup Ω u > max ∂Ω ψ, there exists a sequence

(t i ) i∈N ⊂ (b, sup Ω u) converging to sup Ω u. Applying (2.1) to t i gives |[u > t i ]| ≥ 1/(Cλ) N so that in the limit |[u = sup Ω u]| ≥ 1/(Cλ) N .
By [START_REF] Fonseca | An existence result for a nonconvex variational problem via regularity[END_REF]Theorem 2.7], u is locally Lipschitz continuous in Ω. Actually, this theorem applies to integrands of the form f (x, ∇u) ; however, the proof can be easily generalized to integrands of the form f (∇u) -λu, see Theorem 5.1 in the appendix for a proof in the specific case that we consider in this paper ; for a more general result in this direction, see also [START_REF] Brasco | Global L ∞ gradient estimates for solutions to a certain degenerate elliptic equation[END_REF]Theorem 2.1].

2.2. The Euler equation. Let λ ≥ 0 and u a solution of (P λ ). One can establish the Euler equation for u exactly as in the proof of [START_REF] Alibert | A nonstandard free boundary problem arising in the shape optimization of thin torsion rods[END_REF]Lemma 3.2] where this is done for the problem stated in (1.4).

Lemma 2.3. There exists

σ ∈ (L 2 ∩ L ∞ loc )(Ω; R N ) such that div σ = -λ and (2.2) σ ∈ ∂ϕ(∇u) a.e.
Proof. We introduce the convex function

K ϕ : g ∈ L 2 (Ω; R N ) → Ω ϕ(g(x) + ∇ψ(x)) dx,
and the two continuous linear maps

J : v ∈ H 1 0 (Ω) → -λ Ω v(x) dx , A : v ∈ H 1 0 (Ω) → ∇v ∈ L 2 (Ω; R N ). Since u ∈ H 1 ψ (Ω) is a minimum of (P λ ), 0 ∈ ∂ (K ϕ • A + J) (u -ψ).
where

∂(• • • ) is the convex subgradient. Now, ∂K ϕ (g) is the set of those ζ ∈ L 2 (Ω; R N ) such that (2.3) ζ(x) ∈ ∂ϕ(g(x) + ∇ψ(x)) a.e. x ∈ Ω.
Since A is linear and continuous,

∂(K ϕ • A)(u -ψ) = A * [∂K ϕ (A(u -ψ))].
Finally, J being linear and continuous, one deduces that there exists σ ∈ L 2 (Ω; R N ) such that σ ∈ ∂ϕ(∇u) a.e. on Ω and (2.4) λ

Ω v = Ω σ, ∇v ∀v ∈ H 1 0 (Ω). Equivalently, div σ = -λ. Finally, since ∇u ∈ L ∞ loc (Ω), it follows that σ ∈ L ∞ loc (Ω; R N ). The proof is complete.
One can use the Euler equation together with a regularity result due to Colombo and Figalli [START_REF] Colombo | Regularity results for very degenerate elliptic equations[END_REF] to prove that any solution is smooth on the set where the norm of its gradient is larger than 1. More precisely, On the open set U , |∇u| > 1 a.e. so that the function σ introduced in Lemma 2.3 satisfies σ = ∇u a.e. on U . Hence, u is a (weak and thus a strong) solution of ∆u = -λ. In particular, u is smooth on U . Since |∇u| = H(∇u) + 1 is uniformly continuous on U ∩ Ω for every Ω Ω, it follows that |∇u| extends as a continuous function on U ∩ Ω which is equal to 1 on ∂U ∩ Ω.

Finally, on Ω \ U , H(∇u) = 0 and thus |∇u| ≤ 1 a.e. there.

The proof of Proposition 1.2 is now complete. In a similar way to [2, Proposition 3.1], we have:

Remark 2.5. On the open set V := Ω \ U , |σ(x)| ≤ 1 a.e. and since ϕ is differentiable on R N \ {0}, σ(x) = ∇ϕ(∇u(x)) a.e. on [∇u = 0]. In particular, (2.5) σ(x), ∇u(x) |∇u(x)| = 1 a.e. x ∈ [|∇u| = 0] ∩ V.
In view of (2.4), one also has

(2.6) λ Ω v ≤ [∇u=0] |∇v| + [∇u =0] ∇ϕ(∇u), ∇v , ∀v ∈ H 1 0 (Ω).

2.3.

Minimizing properties of the super-level sets. Using the Euler equation, one can prove that in V = Ω\U , the super-level sets of a solution u have constant mean curvature, in a generalized sense. We first recall some basic results on BV functions. Let u ∈ BV (Ω). The distributional gradient of u is a vector valued Radon measure Du such that |Du|(Ω) < ∞, where |Du| is the total variation of Du. Given a Borel set E ⊂ Ω, we say that E has finite perimeter in Ω if the characteristic function χ E of E belongs to BV (Ω). The perimeter Per (E, Ω) is then defined as the total variation of χ E on Ω:

Per (E, Ω) = Ω |Dχ E | = sup E div g : g ∈ C 1 c (Ω; R N ), |g(x)| ≤ 1 , ∀x ∈ Ω .
The reduced boundary [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]Chapter 3]. By the Besicovitch derivation theorem (see e.g. [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Theorem 2 In view of (2.7), this equality can be formulated as follows:

∂ * E of E in Ω is the set of those x ∈ (supp |Dχ E |) ∩ Ω such that the limit ν E (x) := lim ρ→0 Bρ(x) Dχ E Bρ(x) |Dχ E | exists in R N and satisfies |ν E (x)| = 1. The reduced boundary ∂ * E is a subset of the essential boudary ∂ e E introduced in (1.6), see
.22]), |Dχ E | is concentrated on ∂ * E and Dχ E = ν E |Dχ E |. Moreover, ∂ * E is a countably (N -1) rectifiable set and for every Borel set B ⊂ Ω, (2.7) B |Dχ E | = H N -1 (B ∩ ∂ * E), see e.
(2.9)

B |Du| = R H N -1 (B ∩ ∂ * E s ) ds.
In terms of the characteristic function of B, this gives:

Ω χ B d|Du| = R ds ∂ * Es χ B dH N -1 .
By linearity and the monotone convergence theorem, this implies that for every nonnegative Borel function f : Ω → R + , (2.10)

Ω f d|Du| = R ds ∂ * Es f dH N -1 .
This identity remains true for every bounded measurable map f compactly supported in Ω. The following lemma asserts that the gradient of a Sobolev function u ∈ W 1,1 (Ω) is orthogonal (in a generalized sense) to the level sets of u.

Lemma 2.6. Let u ∈ W 1,1 (Ω). Then for a.e. s ∈ R and H N -1 a.e. x ∈ ∂ * E s , ∇u(x) = 0. Moreover, one has (2.12)

[a≤u≤b] ∇u, κ = Ω ∇u a,b , κ = - Ω u a,b div κ.
On the set [∇u a,b = 0], we define

f (x) := ∇u a,b (x) |∇u a,b (x)| , κ = ∇u(x) |∇u(x)| , κ .
We then extend f by 0 on Ω. Then ∇u a,b , κ = f |∇u a,b | on Ω. Hence, by (2.10),

[a≤u≤b] ∇u, κ = R ds ∂ * [u a,b ≥s] f dH N -1 . For s ≤ 0, [u a,b ≥ s] = Ω while for s > b -a, [u a,b ≥ s] = ∅. Moreover, for s ∈ (0, b -a), [u a,b ≥ s] = [u ≥ s+a]. Moreover, for a.e. s ∈ (0, b-a), ∇u a,b (x) = 0 for H N -1 a.e. x ∈ ∂ * [u a,b ≥ s] and thus f (x) = ∇u(x) |∇u(x)| , κ . It follows that [a≤u≤b] ∇u, κ = b-a 0 ds ∂ * [u a,b ≥s] ∇u |∇u| , κ dH N -1 = b a ds ∂ * Es ∇u |∇u| , κ dH N -1 .
Inserting this identity in (2.12) and using Fubini theorem in the right hand side, one gets Dividing by b -a and letting b -a → 0, this gives for a.e. s ∈ R,

∂ * Es ∇u |∇u| , κ dH N -1 = - Es div κ.
Using (2.7), one has

∂ * Es κ, Dχ Es |Dχ Es | dH N -1 = Ω κ, Dχ Es |Dχ Es | d|Dχ Es | = - Ω χ Es div κ = - Es div κ.
Hence,

∂ * Es κ, Dχ Es |Dχ Es | dH N -1 = ∂ * Es ∇u |∇u| , κ dH N -1 .
Since κ is arbitrary, this implies (2.11). The proof is complete.

We now present the main result of this section : the super-level sets of a solution u satisfy a minimization problem on the subset of Ω where the gradient of u is lower than 1: Proposition 2.7. Given λ ≥ 0, let u be a solution of (P λ ).

Let V := Ω \ U where U is the open set [|∇u| > 1] introduced in Lemma 2.4. For a.e. s ∈ R, for every set F ⊂ Ω with finite perimeter in Ω such that F ∆E s V , (2.13) Per (E s , V ) -λ|E s ∩ V | ≤ Per (F, V ) -λ|F ∩ V |.
Here,

F ∆E s is the set (F \ E s ) ∪ (E s \ F ).
The argument below is inspired from [3, Proposition 4] which in turn is based on [6, Proposition 2.7]. For the convenience of the reader, we present a self-contained proof.

Proof. We divide the proof into three steps.

Step 1. In the first step, we introduce an approximation of the map σ introduced in the Euler equation, see Lemma 2.3. We extend σ by 0 outside Ω and we define σ j := σ * ρ j , where (ρ j ) j≥1 ⊂ C ∞ c (B 1/j ) is a standard mollifier. Then σ j converges to σ a.e. on Ω. Moreover, for every compact K Ω, for every j ≥ 1/dist (K, ∂Ω), div σ j = (div σ) * ρ j = -λ * ρ j = -λ a.e. on K.

We claim that there exists a subsequence, still denoted by (σ j ) j≥1 , such that for a.e. s > 0, for every K Ω, lim j→+∞ K∩∂ * Es |σ j -σ| = 0. Indeed, let K Ω. By the coarea formula (2.10), for every j ≥ 1,

(2.14) R ds K∩∂ * Es |σ j -σ| dH N -1 = K |∇u||σ j -σ|.
Let j 0 ≥ 1 be such that K + B 1/j 0 Ω. For every j ≥ j 0 , the integrand in the right hand side is bounded from above by

∇u L ∞ (K) |( σ j L ∞ (K) + σ L ∞ (K) ) ≤ 2 ∇u L ∞ (K) σ L ∞ (K+B 1/j 0 ) .
Using that (σ j ) converges a.e. to σ on Ω, the dominated convergence theorem implies that

lim j→+∞ K |∇u||σ j -σ| = 0.
In view of (2.14), there exists a subsequence (we do not relabel) such that for a.e. s ∈ R, (2.15) lim

j→+∞ K∩∂ * Es |σ j -σ| dH N -1 = 0.
Let (K n ) n≥1 be a sequence of compact subsets of Ω such that ∪ n≥1 int K n = Ω. By applying the above reasoning to each K n , one can extract, through a diagonal process, a subsequence, still denoted by (σ j ) j≥1 , such that for a.e. s ∈ R, (2.15) holds true for every K n , and thus for every compact subset K Ω.

Step 2. In the second step, we prove that for every F ⊂ Ω as in the statement of the proposition, for a.e. s ∈ R and for every θ ∈ C ∞ c (V ) such that θ ≡ 1 on F ∆E s and 0 ≤ θ ≤ 1, we have

(2.16) λ R N (χ F -χ Es )θ ≤ Per (F, V ) - ∂ * Es θdH N -1 .
By the coarea formula and (2.5), for a.e. s ∈ R, for H N -1 a.e. x ∈ V ∩ ∂ * E s , ∇u(x) = 0 and σ(x), ∇u(x)/|∇u(x)| = 1. We fix any s for which this property as well as (2.11) and (2.15) hold true. Observe that a.e. s satisfies these conditions.

Let

θ ∈ C ∞ c (V ) such that θ ≡ 1 on F ∆E s and 0 ≤ θ ≤ 1 on V . Since ∇θ = 0 on E s ∆F , (2.17) - R N (χ F -χ Es )θdiv σ j = - R N (χ F -χ Es )div (θσ j ).
Remember that σ j = σ * ρ j and |σ| ≤ 1 a.e. on V . By construction, θσ j ∈ C ∞ c (V ) and for every j >

1 dist (supp θ,∂V ) , |θσ j | ≤ 1. Hence, (2.18) R N χ F div (θσ j ) ≤ |Dχ F |(V ) = Per (F, V ). Using (2.11), R N χ Es div (θσ j ) = - R N θ σ j , Dχ Es |Dχ Es | d|Dχ Es | = - R N θ σ j , ∇u |∇u| d|Dχ Es | = - ∂ * Es θ σ j , ∇u |∇u| dH N -1 . (2.19)
The last line relies on (2.7). Now,

∂ * Es θ σ j , ∇u |∇u| dH N -1 - ∂ * Es θ σ, ∇u |∇u| dH N -1 ≤ supp θ∩∂ * Es |σ j -σ| dH N -1 .
In view of (2.15), the right hand side goes to 0 when j → +∞. Hence,

(2.20) lim j→+∞ ∂ * Es θ σ j , ∇u |∇u| dH N -1 = ∂ * Es θ σ, ∇u |∇u| dH N -1 .
But for

H N -1 a.e. x ∈ V ∩ ∂ * E s , σ(x), ∇u(x)/|∇u(x)| = 1. It
lim j→+∞ R N χ Es div (θσ j ) = - ∂ * Es θdH N -1 .
In the left hand side of (2.17), we use the fact that F ∆E s is compactly contained in Ω, so that for every j sufficiently large, div σ j = -λ on F ∆E s . Our claim (2.16) is now a consequence of (2.17), (2.18) and (2.21).

Step 3. Completion of the proof.

We replace the function θ introduced in the previous step by a sequence (θ k ) k≥ such that each θ k satisfies the same assumptions as θ, and θ k → 1 a.e. on V . By letting k → +∞, we thus obtain

(2.22) λ R N (χ F -χ Es ) ≤ Per (F, V ) -H N -1 (V ∩ ∂ * E s ) = Per (F, V ) -Per (E s , V ). Since F \ V = E s \ V , λ(|F ∩ V | -|E s ∩ V |) = λ(|F | -|E s |) = λ R N (χ F -χ Es ).
Together with (2.22), this implies that

Per (E s , V ) -λ|E s ∩ V | ≤ Per (F, V ) -λ|F ∩ V |.
The proof is complete.

2.4. Regularity of the level sets. This section is devoted to the proof of Proposition 1.3 which states that the level sets of a solution are C 1 up to a small singular set.

Proposition 2.8. Given λ ≥ 0, let u be a solution of (P λ ) and U be the open set in Ω defined by

[|∇u| > 1].
W 0 in V = Ω \ U such that ∂ e E t ∩ W 0 is an N -1 dimensional manifold of class C 1,
α for some 0 < α < 1 and H s (V \ W 0 ) = 0 for every s > N -8. The set W = U ∪ W 0 satisfies the required properties.

Remark 2.9. The proof of [26, Theorems 5.1 and 5.2] shows that for every x ∈ W 0 ∩ ∂ e E t , there exists δ > 0 such that

B δ (x) ∩ ∂ e E t = B δ (x) ∩ ∂ * E t .
In particular, the vector ν Et (y) is defined for every y ∈ B δ (x) ∩ ∂ e E t and it is proved to be (Hölder) continuous on this set. If we choose t as in Lemma 2.6: ∇u(x) = 0 for H N -1 a.e. x ∈ ∂ * E t and (2.11) holds true, we deduce that ∇u |∇u| is equal to the continuous function ν Et H N -1 a.e. on W 0 ∩ ∂ e E t . Hence, the smooth hypersurface W 0 ∩ ∂ e E t has a well defined orientation which is given by (the continuous extension of) ∇u |∇u| | W 0 ∩∂ e Et .

Uniqueness

In this section, we present the proof of Theorem 1.1, namely the uniqueness of the solutions of (P λ ). We first explain the strategy that we follow:

Let λ ≥ 0 and u, v be two solutions of (P λ ).

Step 1 The two open sets [|∇u| > 1] and [|∇v| > 1] coincide : this is an easy consequence of the fact that ϕ is strictly convex outside the unit ball. We denote by U this open set. Still by strict convexity, ∇u = ∇v on U . This step corresponds to Lemma 3.1.

Step 2 Every connected component of U intersects the boundary ∂Ω. This is Lemma 3.3.

It follows from the two above steps that u = v on U .

Step 3 The fact that ϕ only depends on the norm of the gradient of u implies that ∇u and ∇v are colinear a.e. (Lemma 3.1 again). By using the regularity of the level sets of u, one can deduce that v is constant on the level sets of u.

Step 4 For a.e. t ∈ R, every connected component of u -1 (t) with positive measure intersect U ∪ ∂Ω. This step involves a geometrical result, see Lemma 3.8, which is a weak generalization of the fact that a compact hypersurface without boundary is the boundary of a bounded open set.

From Step 3 and Step 4 together with the fact that u = v on U , we finally deduce that u = v on almost every level set of u, which is enough to conclude that u = v on Ω. Proof. Since u is a solution of (P λ ),

I λ (u) ≤ I λ u + v 2 .
By convexity of ϕ,

I λ u + v 2 = Ω ϕ ∇u + ∇v 2 -λ u + v 2 ≤ 1 2 Ω (ϕ(∇u) -λu) + 1 2 Ω (ϕ(∇v) -λv) = 1 2 I λ (u) + 1 2 I λ (v). (3.1)
Since v is another solution,

I λ (u) = 1 2 I λ (u) + 1 2 I λ (v).
This implies that a.e. on Ω,

ϕ ∇u + ∇v 2 -λ u + v 2 = 1 2 (ϕ(∇u) + ϕ(∇v)) - λ 2 (u + v),
or equivalently,

ϕ ∇u + ∇v 2 = 1 2 (ϕ(∇u) + ϕ(∇v)).
Hence for a.e. x ∈ Ω, ϕ is affine on the segment [∇u(x), ∇v(x)]. In view of the definition of ϕ, this implies the desired conclusion.

Lemma 3.2. Assume that the boundary condition ψ satisfies ψ ≡ 0. Given λ ≥ 0, let u be a solution of (P λ ). Then for every µ > 0, the super-level set E µ := [u ≥ µ] has finite perimeter and

Per (E µ , Ω) ≤ λ|E µ |.
Proof. Assume by contradiction that U 0 ⊂ Ω. Since U is open and |∇u| can be extended as a uniformly continuous function on U 0 , |∇u| ≡ 1 on ∂U 0 , see Lemma 2.4. But on the set U 0 , where u is smooth,

(3.2) ∆|∇u| 2 = 2 i,j (∂ 2 ij u) 2 + 2 j ∂ j u∂ j ∆u = 2 i,j (∂ 2 ij u) 2 ≥ 0.
Here, we have used that ∆u = -λ on U so that ∂ j ∆u = 0 for every j. This proves that |∇u| 2 is subharmonic on U 0 , and thus |∇u| 2 attains its maximum on ∂U 0 . But this contradicts the facts that |∇u| ≡ 1 on ∂U 0 and |∇u| > 1 on U 0 . Hence, ∂U 0 ∩ ∂Ω = ∅, as desired.

Remark 3.4. It also follows from (3.2) that the interior of the set [|∇u| = 1] is empty, when λ > 0. Indeed, assume by contradiction that there exists an open connected set W ⊂ Ω where |∇u| = 1 a.e. Then the Euler equation on W is Ω ∇u, ∇θ -λθ = 0, for every θ ∈ C ∞ c (W ). Hence, ∆u = -λ on W (in the sense of distributions and thus in the classical sense). Since |∇u| = 1 on W , ∆|∇u| 2 = 0. From (3.2), we deduce that |∇ 2 u| = 0, which implies that u is affine, and thus ∆u = 0, a contradiction with λ > 0. This proves the remark.

In the proof of Theorem 1.1, we will also need the following variant of Lemma 3.3. Lemma 3.5. Given λ ≥ 0, let u be a solution of (P λ ) and

U := [|∇u| > 1]. For every r > 0, let H r := {x ∈ R N : dist (x, R N \ Ω) ≤ r}. If ∂Ω is connected, then the set U ∪ H r is connected.
Proof. We first prove that the set K r := {x ∈ Ω : dist (x, ∂Ω) ≤ r} is connected. Consider a continuous function χ : K r → R with values into {0, 1}. Since K 0 = ∂Ω is connected, χ is constant on K 0 , e.g. χ ≡ 0 on K 0 . Let r * := inf{s ∈ [0, r] : χ ≡ 0 on K s }. Then by continuity of χ, χ -1 (0) is an open set of K r containing K 0 and thus r * > 0. Moreover, for every 0 < r < r * , χ ≡ 0 on K r so that χ ≡ 0 on K r * . Assume by contradiction that r * < r. Then there exists r i ↓ r * and x i ∈ K r i such that χ(x i ) = 1. Up to a subsequence, (x i ) i converges to some x in K r * , which implies χ(x) = 1, a contradiction. This proves that r * = r; that is, χ ≡ 0 on K r . Hence, K r is connected. We can prove in a similar way that Ks := {x ∈ R N \ Ω : dist (x, ∂Ω) ≤ s} is connected for every s ≥ 0. Since every Ks contains ∂Ω, it follows that R N \ Ω = ∪ s≥0 Ks is connected and so is

H r = K r ∪ (R N \ Ω).
For every connected component U 0 of U , Lemma 3.3 implies that U 0 ∩ H r = ∅. Hence, U 0 ∪ H r is connected. It follows that the set U ∪ H r is connected and the same is true for its closure

U ∪ H r = U ∪ H r = U ∪ H r .
The lemma is proved.

We now establish the main result of this section: two solutions agree on the set where their gradients do not belong to the unit ball. Another important tool in the proof of Theorem 1.1 is the fact that a level set of a solution cannot be contained in Ω \ U except if its interior is not empty. Here is a first result in that direction. Lemma 3.7. Given λ ≥ 0, let u be a solution of (P λ ). Let G be an open subset of Ω \ U , where U is the open set defined by U = [|∇u| > 1]. Assume that u is continuous on G4 and that u is constant on ∂G. Then u is constant on G.

Proof. By assumption, there exists c ∈ R such that u| G -c belongs to H 1 0 (G). Moreover, it minimizes the functional

(3.3) v → G ϕ(∇v) -λv.
By Lemma 2.1 applied to this functional on H 1 0 (G), u ≥ c on G. We now prove that u ≤ c on G. Assume by contradiction that max G u > c. Then

A := {x ∈ G : u(x) = max G u} is a compact subset contained in G. By Lemma 3.2 with µ := max G u, Per A = Per (A, G) ≤ λ|A|.
Since A G, sA G for every s sufficiently close to 1. Fix such an s > 1. Then

(3.4) Per sA = s N -1 Per A ≤ s N -1 λ|A| < λ|sA|.
We claim that there exists v ∈ H 1 0 (G) such that

(3.5) G |∇v| < λ G v. Indeed, let (ρ k ) k≥1 ⊂ C ∞ c (B 1 k
) be a regularization kernel. Then for k > In view of (3.4), there exists k ∈ N such that

G |∇v k | < λ G v k .
This proves the claim (3.5) with v := v k . However, by (2.6) applied to the minimization problem (3.3),

λ G v ≤ [∇u=0]∩G |∇v| + [∇u =0]∩G ∇ϕ(∇u), ∇v .
Using the fact that G ⊂ Ω \ U , this yields

λ G v ≤ [∇u=0]∩G |∇v| + [|∇u|≤1]∩G |∇v| ≤ G |∇v|.
This contradicts (3.5). Hence u ≡ c on G.

A geometrical result.

The following lemma will be crucial to prove that generically, the level sets of u which intersect V = Ω \ U are not contained in V .

Lemma 3.8. Let V be an open bounded subset of R N such that R N \ V is connected. Let R be a C 1 orientable hypersurface compactly contained in V . If H N -1 (R) < ∞ and H N -2 (R \ R) = 0,
then there exists a non empty open set E ⊂ V such that ∂E ⊂ R.

Remark 3.9. If one further assumes that R is connected, then one can prove a deeper result: there exists a set F V with finite perimeter such that ∂F = R, see [START_REF] Jerrard | [END_REF]Lemma 4.2] which inspired us for the first paragraph of the proof below. namely the push-forward by the map h(t, x) = tx of the product of the two currents [0, 1] × R:

Proof. Since R is an orientable C 1 hypersurface of R N with H N -1 (R) < ∞, it
H = h ([0, 1] × R).
We now prove that R N \ R is not connected. Assume by contradiction that this is not the case. Since the support of [R] is contained in R, H has no boundary (in the sense of currents) in the open set R N \ R. It follows from the Constancy theorem, see e.g. [START_REF] Giaquinta | Cartesian currents in the calculus of variations. I, Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]Theorem I.2.3.4], that there exists r ∈ R such that for every smooth N form ω compactly supported in R N \ R,

(3.6) H(ω) = r R N \R ω.
Since H is an N dimensional integer rectifiable current and H N (R) = 0, it follows that (3.6) remains true for every smooth N form with compact support. Hence, H is constant as a current on R N and thus ∂H = [R] = 0, a contradiction. This proves that

R N \ R is not connected. Since R N \ R is open, each connected component of R N \ R is open. By assumption, R ⊂ V and thus R N \V ⊂ R N \R. Since R N \R is not connected, there exists at least one connected component E of R N \ R which does not intersect the open connected component of R N \ R containing the connected set R N \ V . This implies that E ⊂ V . Moreover, ∂E ⊂ R.
This completes the proof.

3.4.

Proof of Theorem 1.1. We can now complete the proof of Theorem 1.1. We assume throughout this section that Ω is Lipschitz and has a connected boundary.

Proof. Let u and v be two solutions of (P λ ). By Lemma 3.1, ∇u(x) and ∇v(x) are colinear for a.e. x ∈ Ω.

Let S be the set of those x ∈ Ω such that (1) either u or v is not differentiable at x, (2) or ∇u(x) = 0, (3) or ∇u(x) and ∇v(x) are not colinear. Since S |∇u| = 0, the coarea formula yields the existence of a negligeable set N 0 ⊂ R such that for every t ∈ R \ N 0 , [u ≥ t] is a set of finite perimeter with H N -1 (u -1 (t) ∩ S) = 0. In view of Proposition 2.8 and Remark 2.9, there exists a negligeable set N 1 ⊂ R with the following property: for every t ∈ R \ N 1 , there exists an open set W t such that ∂ e [u ≥ t] ∩ W t is an orientable C 1 hypersurface and H s (Ω \ (W t ∪ ∂U )) = 0 for every s > N -8 (as usual, U is the open subset of Ω defined by [|∇u| > 1]).

Let N := N 0 ∪ N 1 and t ∈ (0, ∞) \ N . We define

E t := [u ≥ t]. Let R be a connected component of ∂ e E t ∩ W t .
Since u is constant on the C 1 hypersurface R and ∇v(x) is colinear to ∇u(x) H N -1 a.e. x ∈ R, it follows that for such x ∈ R, ∇v(x) is orthogonal to R at x. This implies that the tangential gradient of v on the connected manifold R vanishes. Thus v is constant on R and, by continuity of v, on R as well.

We claim that

(3.7) R ∩ (∂Ω ∪ U ) = ∅.
Indeed, assume by contradiction that R∩(∂Ω∪U ) = ∅. Then there exists r > 0 such that R∩H r = ∅ where

H r := {x ∈ R N : dist (x, R N \ Ω) ≤ r}. We set V r := Ω \ (U ∪ H r ) = R N \ (U ∪ H r ). Then by Lemma 3.5, R N \ V r = U ∪ H r is connected. The essential boundary ∂ e E t is a closed subset of R N , as the complement of the open set E 0 t ∪ E 1 t where E 0 t = {x ∈ R N : ∃ρ > 0 such that |E t ∩ B ρ (x)| = 0}, E 1 t = {x ∈ R N : ∃ρ > 0 such that |E t ∩ B ρ (x)| = |B ρ (x)|}. (The fact that E 0 t and E 1 t are open is proved in [17, Proposition 3.1]). Since R ⊂ ∂ e E t , we also have R ⊂ ∂ e E t . Since the closure of a connected component of the C 1 hypersurface ∂ e E t ∩ W t cannot intersect another connected component of this hypersurface, this implies that (R \ R) ∩ W t = ∅ and thus R \ R ⊂ V r \ W t ⊂ Ω \ (W t ∪ ∂U ). It follows that H N -2 (R \ R) = 0
. By Lemma 3.8 applied to V r and R, there exists a non empty open set E ⊂ V r such that ∂E ⊂ R. Observe that u ≡ t on R ⊃ ∂E. Applying Lemma 3.7 to E yields u ≡ t on E. In particular, ∇u = 0 H N a.e. on E and thus H N (E \ S) = 0. Since E ⊂ u -1 (t) and H N -1 (u -1 (t) ∩ S) = 0, one has H N -1 (E ∩ S) = 0. It follows that H N (E) = 0, a contradiction. Our claim (3.7) is thus proved.

Since Ω is assumed to be Lipschitz, the solutions u and v are continuous on Ω, see section 2.1. Moreover, u = v on ∂Ω∪U and v and u are constant on R. It then follows from (3.7)

that v = u = t on R. Since this is true for every component of ∂ e E t ∩ W t , one gets v = t on ∂ e E t ∩ W t .
For every x ∈ u -1 (t) \ S, u is differentiable at x and ∇u(x) = 0. This implies that for every ρ > 0 sufficiently small, Remark 3.10. When ψ is a constant map, the assumption that Ω is Lipschitz is unnecessary. Indeed, this regularity assumption was made to guarantee the continuity of the solutions up to the boundary. But in the case when ψ is constant, the level sets of the solutions do not intersect the boundary (except for the level set corresponding to the value of ψ) and the continuity of the solutions inside Ω is enough for the above argument to remain true.

{y ∈ B ρ (x) : ∇u(x) |∇u(x)| , y-x ≥ 1 2 |y-x|} ⊂ B ρ (x)∩E t ⊂ {y ∈ B ρ (x) : ∇u(x) |∇u(x)| , y-x ≥ -1 2 |y-x|}, which proves that x ∈ ∂ e E t . It follows that u -1 (t) \ S ⊂ ∂ e E t . Since v = t on ∂ e E t ∩ (W t ∪ U ), one thus gets H N -1 (u -1 (t) ∩ [u = v]) ≤ H N -1 u -1 (t) ∩ S ∪ (Ω \ (W t ∪ ∂U )) = 0.

By the coarea formula, this implies

Some qualitative properties of the solutions

In this section, we assume that Ω is Lipschitz. This implies that every solution is continuous on Ω, see section 2.1.

A comparison principle.

Lemma 4.1. Let λ 2 ≥ λ 1 > 0, u 1 a solution of (P λ 1 ) on u 1 + H 1 0 (Ω) and u 2 a solution of (P λ 2 ) on u 2 + H 1 0 (Ω). We also assume that

u 1 | ∂Ω ≤ u 2 | ∂Ω . If λ 2 > λ 1 or if u 1 is the unique minimum of (P λ 1 ) on u 1 + H 1 0 (Ω) or if u 2 is the unique minimum of (P λ 2 ) on u 2 + H 1 0 (Ω), then u 1 ≤ u 2 on Ω. The assumption u 1 | ∂Ω ≤ u 2 | ∂Ω means that (u 1 -u 2 ) + ∈ H 1 0 (Ω). Proof. Since I λ 1 (u 1 ) ≤ I λ 1 (min(u 1 , u 2 )), we have (4.1)
[u 1 >u 2 ] ϕ(∇u 1 ) -λ 1 u 1 ≤ [u 1 >u 2 ] ϕ(∇u 2 ) -λ 1 u 2 . Since I λ 2 (u 2 ) ≤ I λ 2 (max(u 1 , u 2 )), (4.2)
[u 1 >u 2 ] ϕ(∇u 2 ) -λ 2 u 2 ≤ [u 1 >u 2 ] ϕ(∇u 1 ) -λ 2 u 1 .
The sum of (4.1) and (4.2) gives

0 ≤ (λ 2 -λ 1 ) [u 2 <u 1 ] u 2 -u 1 .
We first assume that λ 2 > λ 1 . Since u 2 -u 1 is nonpositive on the set [u 2 < u 1 ], it follows that u 1 ≤ u 2 on Ω, which completes the proof in that case. Otherwise, λ 1 = λ 2 and assume for instance that u 1 is the unique minimum of (P λ 1 ) on u 1 + H 1 0 (Ω). Then (4.1), (4.2) imply

[u 1 >u 2 ] ϕ(∇u 1 ) -λ 1 u 1 = [u 1 >u 2 ] ϕ(∇u 2 ) -λ 1 u 2 .
This yields

I λ 1 (u 1 ) = I λ 1 (min(u 1 , u 2 )).
Since u 1 is the unique minimum, it follows that u 1 = min(u 1 , u 2 ). Hence, u 1 ≤ u 2 on Ω, which completes the proof in that case as well.

As a consequence of Theorem 1.1, the above comparison principle applies when Ω is Lipschitz and has a connected boundary, since under these assumptions, the solutions of (P λ ) are unique. In the particular case when Ω is a ball, we even know the solution explicitly: Remark 4.2. If Ω is the ball B R (x 0 ) of center x 0 ∈ R N and radius R > 0 and if ψ ≡ 0, then the unique solution of (P λ ) on B R (x 0 ) is the function

(4.3) ξ λ,x 0 ,R = -N 2λ λ 2 N 2 |x -x 0 | 2 -1 + - λ 2 N 2 R 2 -1 + .
The above remark can be seen as a consequence of a more general result due to Cellina, see [10, Theorem 1]:

Theorem 4.3. Let Ω ⊂ R N be a bounded open set in R N and F : R N → R a convex function. We assume that F is superlinear: lim x→+∞ F (x) |x| = +∞. Given c ∈ R, x 0 ∈ R N , let h x 0 ,c (x) = -N λ F * ( -λ N (x -x 0 )) + c , x ∈ Ω,
where F * (y) := sup x∈R N ( x, y -F (x)). Then h x 0 ,c is the unique minimum of the variational problem: By convex duality, this implies that

To Minimize u → Ω F (∇u) -λu on h x 0 ,c + H 1 0 (Ω). Proof. Let ζ be the affine map x → -λ N (x -x 0 ). Then h x 0 ,c = -N λ F * • ζ + c. Since F * is convex, h x 0 ,
ζ(x) ∈ ∂F (∇h x 0 ,c (x)). Let u ∈ h x 0 ,c + H 1 0 (Ω).
Then by definition of a convex subgradient, (

Ω F (∇u) -F (∇h x 0 ,c ) -λ(u -h x 0 ,c ) ≥ Ω ζ, ∇u -∇h x 0 ,c -λ(u -h x 0 ,c ). Since u -h x 0 ,c ∈ H 1 0 (Ω), Stokes formula implies Ω F (∇u) -F (∇h x 0 ,c ) -λ(u -h x 0 ,c ) ≥ Ω λ(u -h x 0 ,c ) -λ(u -h x 0 ,c ) = 0. 4.5)
This proves that h x 0 ,c is a minimum. If u is another minimum, then the left hand side in (4.5) is 0 which implies that

F (∇h x 0 ,c (x)) = F (∇u(x)) + ζ(x), ∇h x 0 ,c (x) -∇u(x) a.e. x ∈ Ω.
Hence, ζ(x) ∈ ∂F (∇u(x)). By convex duality together with (4.4), we get for a.e. x ∈ Ω,

∇u(x) ∈ ∂F * (ζ(x)) = {∇h x 0 ,c (x)}.
Hence ∇u(x) = ∇h x 0 ,c (x) a.e. and thus h x 0 ,c = u. This proves that h x 0 ,c is the unique solution on h x 0 ,c + H 1 0 (Ω).

Observe that Remark 4.2 follows from Theorem 4.3 since the convex conjugate of F = ϕ is F * (y) = 1 2 (|y| 2 -1) + . We can now use the explicit solution on the ball as a barrier which, together with the comparison principle Lemma 4.1, yields certain bounds on the solutions when Ω is any bounded open set. As an illustration, we give an explicit upper bound on the solutions of (P λ ) which does not depend on the L 1 norm of the solutions, in contrast to Lemma 2.1, but only on Ω, λ and the boundary condition ψ: Lemma 4.4. Let λ > 0 and u a solution of (P λ ) on ψ| Ω + H 1 0 (Ω). Then

max Ω u ≤ N 2λ λ 2 N 2 R 2 -1 + + max ∂Ω ψ,
where R = diam Ω.

Proof. The proof is based on the following elementary observation. If v is the unique solution of (P λ ) on v + H 1 0 (Ω 1 ), for some bounded open set Ω 1 ⊂ R N , then for every bounded open subset Ω 2 ⊂ Ω 1 , v| Ω 2 is the unique solution on v| Ω 2 + H 1 0 (Ω 2 ). Indeed, if ṽ were another solution on Ω 2 , then the extension of ṽ by v on Ω 1 would be another solution on Ω 1 , which would contradict the uniqueness of the solution on Ω 1 . Now, let R = diam Ω and x 0 ∈ Ω. Then Ω ⊂ B R (x 0 ). The unique solution of (P λ ) on

H 1 0 (B R (x 0 )) is the map ξ λ,x 0 ,R defined in (4.3). Then v := ξ λ,x 0 ,R | Ω + max ∂Ω ψ is the unique minimum of (P λ ) on v + H 1 0 (Ω). Since ξ λ,x 0 ,R ≥ 0 on B R (x 0 ), it follows that v ≥ max ∂Ω ψ ≥ ψ on ∂Ω. By Lemma 4.1, this implies that on Ω u ≤ v ≤ max B R (x 0 ) ξ λ,x 0 ,R + max ∂Ω ψ = N 2λ λ 2 N 2 R 2 -1 + + max ∂Ω ψ,
and the lemma follows.

The next lemma gives a lower bound on the solutions of (P λ ):

Lemma 4.5. Let B R (x 0 ) ⊂ B R (x 0 ) ⊂ Ω. Let λ ≥ N/R
and u be a solution of (P λ ) on ψ + H 1 0 (Ω). Then

min B R (x 0 ) u ≥ λ 2N (R 2 -R 2 ) + min ∂Ω ψ.
Proof. Let ξ λ,x 0 ,R be the unique solution of (P λ ) on H 1 0 (B R (x 0 )) given by (4.3). Then ξ λ,x 0 ,R + min ∂Ω ψ is the unique solution of (P λ ) on min ∂Ω ψ + H 1 0 (B R (x 0 )). Since u| B R (x 0 ) is a solution of (P λ ) on u| B R (x 0 ) + H 1 0 (B R (x 0 )) and inferring from Lemma 2.1 that u ≥ min ∂Ω ψ on Ω and thus on ∂B R (x 0 ), it follows from Lemma 4.1 that on B R (x 0 ),

u ≥ ξ λ,x 0 ,R + min ∂Ω ψ. In particular, if λ ≥ N/R, then on B R (x 0 ), u ≥ λ 2N (R 2 -R 2 ) + min ∂Ω ψ,
which implies the desired result.

4.2. On the family {u λ } λ>0 . In this section, we assume that Ω is Lipschitz and that ∂Ω is connected. We also fix the boundary condition ψ which is assumed to be Lipschitz continuous. As a consequence of Theorem 1.1, for every λ ≥ 0, there exists a unique solution u λ of (P λ ) on H 1 ψ (Ω) = ψ + H 1 0 (Ω). We then consider the map λ → u λ . A continuity property is established in the next proposition. Proof. Since Ω is Lipschitz, for every λ ≥ 0, u λ is Hölder continuous on Ω. Actually, for every bounded subset Λ ⊂ [0, +∞), there exist C > 0 and α > 0 (which depend on Λ, ψ and Ω) such that for every λ ∈ Λ, for every x, y ∈ Ω, (4.6)

|u λ (x) -u λ (y)| ≤ C|x -y| α .
Indeed, the functional in (P λ ) satisfies the following growth assumptions:

1 2 |∇u| 2 -A|u| ≤ ϕ(∇u) -λu ≤ 1 2 |∇u| 2 + A|u| + 1 2 ,
with A := sup Λ. The proof of [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Theorem 7.8] implies that the Hölder norm and the Hölder exponent of u λ can be estimated 5 only in terms of A, ψ and Ω. In particular, the constants C and α in (4.6) can be estimated independently of λ ∈ Λ. Fix λ ≥ 0. Let λ i → λ and u λ i the solution of (P λ i ). We claim that u λ i uniformly converges to u λ . Indeed, since by (4.6) the sequence (u λ i ) i≥1 is equicontinuous on Ω, Arzela-Ascoli theorem implies that a subsequence (we do not relabel) uniformly converges to a function v ∈ C 0 (Ω). Since (u λ i ) i≥1 is bounded in H 1 ψ (Ω), there exists a subsequence (we do not relabel) which weakly converges in H 1 ψ (Ω), necessarily to v. In particular, v ∈ H 1 ψ (Ω). Now, let w ∈ H 1 ψ (Ω). Then for every i ≥ 1,

Ω ϕ(∇u λ i ) -λ i u λ i ≤ Ω ϕ(∇w) -λ i w.
Letting i → +∞ and using the weak lower semicontinuity in the left hand side, one gets

Ω ϕ(∇v) -λv ≤ Ω ϕ(∇w) -λw.
This proves that v is a solution of (P λ ). By uniqueness for the problem (P λ ), this implies that v = u λ . Hence, by uniqueness of the limit, the whole original sequence (u λ i ) i≥1 uniformly converges to u λ , which completes the proof of the lemma.

In the next lemma, we prove the existence of a critical value of λ for which the supremum of u on Ω becomes larger than the supremum of u on ∂Ω. It follows that lim λ→+∞ (λ) = +∞. Finally, when λ = 0, the constant function v :≡ max ∂Ω ψ is a solution of (P 0 ) and v ≥ u 0 on ∂Ω. It follows from Lemma 4.1 that u 0 ≤ max Ω ψ on Ω and thus (0) = max ∂Ω ψ. Hence ([0, +∞)) = [max ∂Ω ψ, +∞). This proves the existence of the value λ * as in the statement. 4.3. On the supports of the solutions. In this section, we describe some properties of the supports of the solutions and present some connections with the Cheeger subsets of Ω. We assume throughout that Ω is any bounded open Lipschitz set but that the boundary condition ψ is constant and equal to 0. This implies by Lemma 2.1 that every solution is nonnegative on Ω.

As explained in the introduction, the Cheeger constant of Ω is given by

h Ω = inf D⊂Ω Per D |D| .
Alternatively (see for instance [START_REF] Parini | An introduction to the Cheeger problem[END_REF]), h Ω can be defined as the minimum value of a variational problem on H 1 0 (Ω):

h Ω = inf Ω |∇u| : u ∈ H 1 0 (Ω), Ω u = 1 .
This constant arises naturally in the framework of (P λ ), as stated in Proposition 1.5 that we now prove:

Proof of Proposition 1.5. Assume first that 0 is a solution of (P λ ). Then for every v ∈ H 1 0 (Ω),

λ Ω v ≤ Ω ϕ(∇v) ≤ [|∇v|≤1] |∇v| + [|∇v|>1] |∇v| 2 .
By replacing v by sv for some s > 0 and dividing by s, one gets

λ Ω v ≤ [|∇v|≤s -1 ] |∇v| + s [|∇v|>s -1 ] |∇v| 2 .
We now let s → 0 + :

λ Ω v ≤ Ω |∇v|.
This implies that λ ≤ h Ω . Moreover, if u is another solution of (P λ ), then it follows from Lemma 3.1 and the fact that 0 is a solution that |∇u| ≤ 1 a.e. Hence Ω = Ω \ [|∇u| > 1] and u is constant on ∂Ω. From Lemma 3.7, we deduce that u is constant on Ω: u ≡ 0; that is, 0 is the unique solution of (P λ ). Assume now that λ ≤ h Ω . Then for any u ∈ H

1 0 (Ω), Ω ϕ(∇u) -λ Ω u ≥ Ω |∇u| -h Ω Ω u ≥ 0.
This proves that 0 is a solution of (P λ ).

When λ increases and becomes larger than h Ω , a non trivial solution suddenly appears inside the Cheeger sets of Ω. Here is a precise result, which implies Theorem 1.6 stated in the introduction: Theorem 4.8. For λ > h Ω , let u λ be a solution of (P λ ). Then the family ([u λ > 0]) λ>h Ω is nondecreasing. Moreover, the set

Ω 0 := ∩ λ>h Ω [u λ > 0]
is a solution of the Cheeger problem for Ω:

Per Ω 0 |Ω 0 | = h Ω .
Remark 4.9. The Cheeger problem has a unique solution when the domain is convex, see [START_REF] Alter | Uniqueness of the Cheeger set of a convex body[END_REF]. Hence, when Ω is convex, Ω 0 is the unique Cheeger set contained in Ω. When Ω is not convex, we conjecture that Ω 0 is the maximal Cheeger set (for the characterization of the maximal Cheeger set, we refer for instance to [START_REF] Caselles | Some remarks on uniqueness and regularity of Cheeger sets[END_REF]).

Proof of Theorem 4.8.

By Lemma 4.1, if h Ω < λ < λ , then u λ ≤ u λ . Hence [u λ > 0] ⊂ [u λ > 0]
and the first assertion follows. By Lemma 3.2, for every ε > 0,

Per [u λ ≥ ε] ≤ λ|[u λ ≥ ε]|.
Here, we also use the fact that Per

[u λ ≥ ε] = Per ([u λ ≥ ε], Ω) since [u ≥ ε] Ω.
For every λ > h Ω , the of characteristic functions χ [u λ ≥ε] converges to χ [u λ >0] a.e. when ε → 0 and thus, by the dominated convergence theorem, in L 1 (Ω). The lower semicontinuity of the perimeter then yields

(4.7) Per [u λ > 0] ≤ λ|[u λ > 0]|.
On the other hand, by definition of h Ω , for every λ > h Ω ,

) Per [u λ > 0] ≥ h Ω |[u λ > 0]|. When λ → h Ω , |[u λ > 0]| tends to |Ω 0 |. (4.8
> 0]| ≥ (Cλ) -N > 0 and finally letting λ → h Ω , |Ω 0 | ≥ (Ch Ω ) -N > 0.
Together with (4.9), this completes the proof of Theorem 4.8. Remark 4.10. If Ω is convex and a Cheeger set of itself, then Ω = Ω 0 and thus for every λ > h Ω , u λ > 0 on Ω. On the contrary, if Per Ω |Ω| > h Ω , then for λ ∈ (h Ω , Per Ω |Ω| ), we deduce from the inequality (4.7) that [u λ > 0] = Ω; that is, u λ vanishes on a subset of Ω of positive measure. This emphasizes the fact that the strong maximum principle does not apply for this degenerate (not uniformly elliptic) problem.

For a fixed λ > h Ω , the support of a solution u λ is related to the geometry of Ω in the following sense: Proposition 4.11. Given λ > h Ω , let u λ be a solution of (P λ ). Then [u λ > 0] contains the set Ω λ defined as the union of all open convex sets A ⊂ Ω which are Cheeger sets of themselves with h A < λ.

Proof. Indeed, let A ⊂ Ω be an open convex set which is a Cheeger set of itself. Then by [START_REF] Alter | Uniqueness of the Cheeger set of a convex body[END_REF], A is the unique Cheeger set contained in A. Hence A = A 0 , where A 0 is defined in Theorem 4.8, with A instead of Ω. Let λ > h A . By Theorem 1.1, there exists a unique solution u A,λ to the problem (P λ ) on A with a homogeneous Dirichlet boundary condition. It follows from Theorem 4.8 that u A,λ is stricly positive on A. Since u λ | ∂A ≥ 0 = (u A,λ )| ∂A , Lemma 4.1 implies that u λ ≥ u A,λ > 0 on A. Finally, u λ > 0 on Ω λ .

Appendix

In this appendix, we detail the proof of the Lipschitz continuity of the solutions of (P λ ). Our strategy is essentially the same as the one introduced in [13, Theorem 2.7] for Lagrangians which depend only on x and ∇u. In our situation, the dependence on u is linear, so that it plays no role and does not involve any additional difficulty in the proof. Moreover, the specific form of the function ϕ outside the unit ball simplifies the main estimates obtained in [START_REF] Fonseca | An existence result for a nonconvex variational problem via regularity[END_REF].

Theorem 5.1. Let u be a solution of (P λ ). Then the function u is locally Lipschitz on Ω.

We first introduce a sequence of variational problems approximating (P λ ) and for which the solutions v j , j ≥ 1, are known to be smooth. This is the role of Lemma 5.2 below. We then obtain a uniform Lipschitz bound on v j in Lemma 5.3, which finally implies Theorem 5.1. Lemma 5.2. There exists a sequence (ϕ j ) j≥1 of C ∞ uniformly convex 6 functions which converges uniformly to ϕ on bounded sets and such that for every j ≥ 1, (5.1) ∀y ∈ R N \ B 2 (0) , ∇ 2 ϕ j (y) = (1 + 1 j ) Id,

(5.2) ∀y ∈ R N , ϕ(y) ≤ ϕ j (y) ≤ 2(ϕ(y) + 1).

Proof. Let ρ ∈ C ∞ c (B 1 (0)) such that ρ ≥ 0, R N ρ = 1 and ρ(-y) = ρ(y) for every y ∈ R N . We then define ρ j (•) = j N ρ(j•) and

ϕ j (y) = ϕ * ρ j (y) + 1 2j |y| 2 , y ∈ R N .
Then ϕ j is a smooth uniformly convex function which uniformly converges to ϕ on bounded sets. By the change of variables z → -z and the property ρ j (-z) = ρ j (z), one has R N zρ j (z) dz = 0.

6 By this, we mean that there exists mj > 0 such that ∇ 2 ϕj ≥ mj Id.

Lemma 5.3. The function v is locally Lipschitz on B R .

Proof. We only need to prove that there exists a constant C 0 such that for every j ≥ 1,

(5.4) ∇v j L ∞ (B R/2 ) ≤ C 0 .

Since ϕ j is smooth, uniformly convex and has a bounded Hessian, it follows from the standard elliptic regularity theory that v j is smooth on B R and thus satisfies the Euler equation: for every θ ∈ C ∞ c (B R )

B R i,k ∂ ik ϕ j ∂ ks v j ∂ i ζ = 0.
This equality holds true for every ζ ∈ H 1 (B R ) with compact support in B R . Set

V + = 1 + N h=1 (∂ h v j -2) 2 + , V -= 1 + N h=1 (∂ h v j + 2) 2 -.
Let η ∈ C ∞ c (B R ) and take ζ := η 2 (∂ s v j -2) + V β + , where β ≥ 0. This yields

B R η 2 i,k ∂ ik ϕ j ∂ ks v j V β + ∂ i (∂ s v j -2) + + β B R η 2 i,k ∂ ik ϕ j ∂ ks v j V β-1 + (∂ s v j -2) + ∂ i V + = -2 B R η i,k ∂ ik ϕ j ∂ ks v j (∂ s v j -2) + V β + ∂ i η.
Since the integrals above vanish when ∂ s v ≤ 2, one gets

B R η 2 i,k ∂ ik ϕ j ∂ k (∂ s v j -2) + V β + ∂ i (∂ s v j -2) + +β B R η 2 i,k ∂ ik ϕ j ∂ k (∂ s v j -2) + V β-1 + (∂ s v j -2) + ∂ i V + = -2 B R η i,k ∂ ik ϕ j ∂ k (∂ s v j -2) + (∂ s v j -2) + V β + ∂ i η.
Using that ∇ 2 ϕ j (y) = (1 + 1 j ) Id when |y| ≥ 2, this implies

B R η 2 i (∂ i (∂ s v j -2) + ) 2 V β + + β B R η 2 i ∂ i (∂ s v j -2) + V β-1 + (∂ s v j -2) + ∂ i V + = -2 B R η i ∂ i (∂ s v j -2) + (∂ s v j -2) + V β + ∂ i η.
Summing over s and differentiating ∂ i V + in the second term of the left hand side, we get (5.6)

B R η 2 i,s (∂ i (∂ s v j -2) + ) 2 V β + + 2β B R η 2 V β-1 + i s A is 2 = -2 B R η i,s ∂ i (∂ s v j -2) + (∂ s v j -2) + V β + ∂ i η.
where

A is := (∂ s v j -2) + ∂ i (∂ s v j -2) + .
Since N s=1 (∂ s v j -2) 2 + ≤ V + , and using the Cauchy-Schwarz inequality, we obtain

B R η 2 V β-1 + i s A is 2 ≤ B R η 2 i,s (∂ i (∂ s v j -2) + ) 2 V β + .
In view of the above inequality, (5.6) implies

(5.7)

(1 + 2β) B R η 2 V β-1 + i s A is 2 ≤ -2 B R η i,s ∂ i (∂ s v j -2) + (∂ s v j -2) + V β + ∂ i η. Writing A is = 1 2 ∂ i (∂ s v j -2) 2
+ and taking into account the definition of V + , this gives

B R η 2 |∇V + | 2 V β-1 + ≤ C B R ηV β + i ∂ i V + ∂ i η ≤ C B R ηV β + |∇V + ||∇η|,
where C = 4/(1 + 2β). Writing that

V β + = V β-1 2 + V β+1 2 + , this implies B R η 2 |∇V + | 2 V β-1 + ≤ C 2 B R |∇η| 2 V β+1 + .
Inserting now the function ζ = η 2 (∂ s v j + 2) -V β -in (5.5), a similar calculation leads to

B R η 2 |∇V -| 2 V β-1 - ≤ C 2 B R |∇η| 2 V β+1 - .
By summing the two last inequalities, one gets

B R η 2 |∇V | 2 V β-1 ≤ 2C 2 B R |∇η| 2 V β+1 .
where V := max(V + , V -). Equivalently,

B R η 2 |∇V γ | 2 ≤ 2C 2 γ 2 B R |∇η| 2 V 2γ ,
with γ = (β + 1)/2. Hence,

B R |∇(ηV γ )| 2 ≤ (4C 2 γ 2 + 2) B R |∇η| 2 V 2γ .
By the Sobolev inequality and the arbitrariness of β ≥ 0, we get that for every γ ≥ 1/2,

V γ η L 2χ (B R ) ≤ c V γ |∇η| L 2 (B R ) ,
where χ = N/(N -2) if N ≥ 3 or any number > 1 if N = 2, and c is a constant which depends only on N (and on R when N = 2). Considering the sequence of radii r i := (1/2 + 1/2 i )R for i ≥ 0, we apply the above inequality to γ = γ i := χ i /2, and choose η ∈ C 1 c (B r i ) such that η = 1 on B r i+1 , 0 ≤ η ≤ 1, |Dη| ≤ c 0 2 i . This yields

V L 2γ i+1 (B r i +1 ) ≤ (c 1 2 i ) 1 γ i V L 2γ i (Br i ) .
Iterating the above formula and letting i → +∞, one gets

V L ∞ (B R/2 ) ≤ c 2 V L 1 (B R ) ≤ c 3 (1 + ∇v j 2 L 2 (B R ) ).
Since the sequence (v j ) j≥1 is bounded in H 1 (B R ), this implies that V L ∞ (B R/2 ) can be bounded independently of j. In view of the definition of V , (5.4) follows. The proof of Lemma 5.3 is complete.

Proposition 1 . 3 .

 13 Let λ ≥ 0, u a solution of (P λ ) and U as in Proposition 1.2. Then for a.e. t ∈ R, there exists an open set W in Ω such that W ∩ (∂ e [u ≥ t]) is a C 1 hypersurface and H s (Ω \ (W ∪ ∂U )) = 0 for every s > N -8.

Lemma 2 . 4 .

 24 There exists an open subset U ⊂ Ω such that u is smooth on U , |∇u(x)| > 1 for every x ∈ U and |∇u(x)| ≤ 1 for a.e. x ∈ Ω \ U . Moreover, |∇u| is continuous on U ∩ Ω and |∇u| = 1 on ∂U ∩ Ω. Proof. By [11, Theorem 1.1], for every continuous function H : R N → R such that H = 0 on B 1 (0), the function H(∇u) has a continuous representative on Ω. By applying this result to H(y) = (|y| -1) + , one obtains that (|∇u| -1) + has a continuous representative on Ω and the set U := [H(∇u) > 0] is open.

  g.[START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] Theorem 3.59] or[START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF] Theorem 4.4]. For a.e. s ∈ R, the super-level set E s := [u ≥ s] has finite perimeter in Ω and the coarea formula (see e.g.[START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] Theorem 3.40]) asserts that for every Borel set B ⊂ Ω,

  (2.11) ν Es = Dχ Es |Dχ Es | = ∇u |∇u| |Dχ Es | a.e. Proof. Since u ∈ W 1,1 (Ω), the measure Du coincides with the L 1 function ∇u. Let S * be a negligeable Borel set such that every point in R N \ S * is a Lebesgue point of ∇u and let S = S * ∪ [∇u = 0]. By (2.9) with B = S, one gets R H N -1 (∂ * E s ∩ S) ds = S |∇u(x)| dx = 0. Hence, for a.e. s ∈ R, ∇u(x) = 0 for H N -1 a.e. x ∈ ∂ * E s . For every a < b, define u a,b := (min(u, b) -a) + . For every κ ∈ C ∞ c (Ω; R N ),

  Es ∇u |∇u| , κ dH N -1 = -b a ds Es div κ.

3. 1 .

 1 A comparison principle. Lemma 3.1. Let λ ≥ 0 and let u, v be two solutions of (P λ ). Then for a.e. x ∈ Ω, either max(|∇u(x)|, |∇v(x)|) ≤ 1 and ∇u(x), ∇v(x) are colinear or ∇u(x) = ∇v(x).

Lemma 3 . 6 .

 36 Given λ ≥ 0, let u, v ∈ H 1 0 (Ω) be two solutions of (P λ ). Then• the two open sets [|∇u| > 1] and [|∇v| > 1] coincide. We denote by U the corresponding set.• u = v on U .Proof. The first assertion follows from Lemma 3.1. Still by Lemma 3.1, ∇(u -v) = 0 on U and thus u -v is constant on each connected component of U . Since u -v = 0 on ∂Ω, the second assertion is a consequence of Lemma 3.3.

1 dist

 1 (sA,∂G) , the map v k := 1 sA * ρ k belongs to H 1 0 (G) and converges in L 1 (G) to 1 sA . Moreover, by [17, Remark 1.16], lim k→+∞ G |∇v k | = lim k→+∞ R N |∇v k | = Per sA.

  , ∇u = 0 a.e. on the open set [u = v]. Similarly, ∇v = 0 a.e. on [u = v]. Since ∇(u -v) = 0 a.e. on the set [u = v], it follows that ∇(u -v) = 0 a.e. on Ω. But u = v = 0 on ∂Ω. This implies that u = v on Ω. The proof is complete.

  c is concave. Hence, both F * and h x 0 ,c are locally Lipschitz continuous and thus differentiable a.e. Moreover, for a.e. x ∈ Ω, the convex subdifferential ∂F * (ζ(x)) is reduced to the singleton {∇F * (ζ(x))} and (4.4) ∇h x 0 ,c (x) = ∇F * (ζ(x)).

Proposition 4 . 6 .

 46 The function λ → u λ ∈ C 0 (Ω) is continuous.

Lemma 4 . 7 .

 47 There exists λ * = λ * (ψ, Ω) ∈ [0, +∞) such that for every 0 ≤ λ ≤ λ * , sup Ω u = max ∂Ω ψ while for every λ > λ * , sup Ω u > max ∂Ω ψ Proof. By Proposition 4.6, the function λ → u λ ∈ C 0 (Ω) is continuous. This implies that the function : λ → max Ω u λ is continuous as well. By Lemma 4.1, is also nondecreasing. From Lemma 4.5, we deduce that for every B R (x 0 ) Ω, lim λ→+∞ min B R (x 0 ) u λ = +∞.5 In particular, the Lagrangian F (u, ∇u) = 2ϕ(∇u) -2λu satisfies [18, Assumption (7.2)] with γ = p = 2, s = σ = ∞, ε = 2/N , L = 1, b = 2A and a = 2A + 1.

B

  R ∇ϕ j (∇v j ), ∇θ = λ B R θ. Let 1 ≤ s ≤ N and take θ = ∂ s ζ, with ζ ∈ C ∞ c (B R), in the above equality. By integration by parts, this gives(5.5)

  defines an integer multiplicity rectifiable current that we denote by [R]. We first claim that the boundary (in the sense of currents) of [R] is trivial: ∂[R] = 0. Indeed, ∂[R] is a flat chain as the boundary of an integer multiplicity rectifiable current, see e.

g.

[START_REF] Morgan | Geometric measure theory[END_REF] Section 4.3

]. Since ∂[R] is supported in R \ R and H N -2 (R \ R) = 0, this implies that ∂[R] = 0,

see e.g. [27, Theorem 4.7]. It follows that there exists an N dimensional integer multiplicity rectifiable current H with finite mass such that ∂H = [R], see e.g. [14, Sections I.2.3 and I.2.4]. Actually, one can take for H the cone over [R],

In fact, this assumption that we introduce here for simplicity, was not required in[START_REF] Alibert | A nonstandard free boundary problem arising in the shape optimization of thin torsion rods[END_REF].

This is the case when G ⊂ Ω or when Ω is Lipschitz.

Proof. For every 0 < µ < ν, let

Then u µ,ν ∈ H 1 0 (Ω). Hence I λ (u) ≤ I λ (u µ,ν ) and thus Eµ\Eν ϕ(∇u) ≤ λ Ω (u -u µ,ν ).

Since ϕ(∇u) ≥ |∇u|, this implies

By the coarea formula, we get

By definition of u µ,ν , it follows that

Assume first that µ is a Lebesgue point of the map t → Per (E t , Ω). Then, dividing the above inequality by ν -µ and letting ν → µ yield the desired result. Now, for every µ > 0, there exists an increasing sequence of such Lebesgue points µ i converging to µ. Hence, E µ = ∩ i E µ i and thus, lim i→+∞ |E µ i | = |E µ |. Moreover, by semicontinuity of the total variation on BV (Ω),

By the previous case, for every

The proof is complete.

3.2.

Uniqueness on the set U . A crucial step in the proof of Theorem 1.1 is the uniqueness of the solution on the open set where the norms of the gradients are larger than 1.

The proof below is inspired from the one of [2, Proposition 7.3].

By convexity of ϕ, Jensen inequality implies ϕ * ρ j (y) ≥ ϕ(y). Moreover, for every |y| ≥ 2,

where c j := 1 2 R N |z| 2 ρ j (z) dz belongs to (0, 1 2j 2 ). When |y| ≤ 2, using the fact that ϕ(y -z) ≤ (|y -z| 2 + 1)/2 for every z ∈ R N , the same calculation leads to ϕ * ρ j (y) ≤ ϕ(y) + c j . Then (5.1) and (5.2) easily follow. The proof is complete.

Let B R be a ball compactly contained in Ω. For every j ≥ 1, we consider the variational problem

By the direct method in the calculus of variations, for every j ≥ 1, there exists a unique solution v j to the above problem. By minimality of v j , (

Since for every j ≥ 1, y ∈ R N , 1 2 |y| 2 ≤ ϕ(y) ≤ ϕ j (y) ≤ 2(ϕ(y) + 1), the sequence (v j ) j≥1 is bounded in H 1 (B R ). Hence, we can extract (we do not relabel) a subsequence which converges weakly in H 1 (B R ) and strongly in L 2 (B R ) to a certain function v ∈ H 1 (B R ) which agrees with u on ∂B R . Using that ϕ ≤ ϕ j , one has

By weak lower semicontinuity, this gives

Together with (5.3) and the L 2 convergence of v j , this implies

Since ϕ j ≤ 2(ϕ + 1) and ϕ j converges pointwisely to ϕ, one may apply the dominated convergence theorem to get

By minimality of u, the opposite inequality is true. Hence, v is a minimum. By Lemma 3.1, this implies that |∇v -∇u| ≤ 1. Hence, Theorem 5.1 follows from the following lemma.


Le cinéma américain, vecteur des musiques populaires

François Ribac (UBFC)

Aux origines d'une hégémonie

Dès les années 1930 l'industrie culturelle nord-américaine, en particulier sa composante cinématographique, s'impose en Europe occidentale. Le(s) jazz, les comédies musicales produites à Hollywood, ses stars, ses récits et ses mythologies marquent l'imaginaire commun, les USA incarnent la modernité. Au sortir de la deuxième guerre mondiale, le prestige de la victoire contre le nazisme, le délabrement des industries cinématographiques européennes, l'apport des talents qui ont fui le nazisme et le Plan Marshall (qui garantit la diffusion des films nord-américains) permettent à Hollywood de s'imposer encore un peu plus en Europe. On s'intéressera ici aux façons dont, à partir de 1945, le cinéma nord-américain a promu la comédie musicale puis le rock et à quelques uns des effets de la "mise en cinéma" de ces musiques.

Les musicals des fifties

  figures de cette nouvelle musique, apparaissent dans des films -produits par des petites ou des grosses firmes-incarnant des personnages dans des comédies musicales ou se produisant avec leurs groupes dans des numéros musicaux (voir document n° 3). Ce qui est ici fascinant c'est que nombre de films retracent, de façon idéalisée, les débuts de ces nouvelles stars quelques années auparavant. Les films rejouent donc l'histoire du rock 'n' roll en très léger différé.Significativement, des chanteurs anglais commeTommy Steele et Cliff Richard ou comme Johnny Holliday s'inspirent non seulement de la musique et du corps Elvis mais, comme lui, tournent des films musicaux. Il faudra attendre les Beatles (A Hard Day's Night de Richard Lester, 1964) pour que ceux-ci jouent leurs propres personnages au lieu d'incarner des personnages de fiction. Comme des milliers d'autres apprentis rockers britanniques de la fin des années 1950, les Beatles ont d'ailleurs appris dans les films de Bill Halley, Presley, Johnny Cash etc. à travailler en studio, à se tenir en scène, à s'habiller à embrasser les filles etc.. Mais, n'en déplaise aux critiques de la « culture de masse », dont le théâtre des métropoles du 19e siècle est en fait la première incarnation, la génération des Beatles, des Who et des Rolling Stones ne clone pas longtemps Elvis, elle bâtît son propre langage jusqu'à s'imposer aux USA au milieu des années 1960. C'est l'imitation qui a permis l'émancipation des groupes britanniques... Lorsque les contre cultures nord-américaines, dont la composante rock est