A two-phase problem with Robin conditions on the free boundary

We study for the first time a two-phase free boundary problem in which the solution satisfies a Robin boundary condition. We consider the case in which the solution is continuous across the free boundary and we prove an existence and a regularity result for minimizers of the associated variational problem. Finally, in the appendix, we give an example of a class of Steiner symmetric minimizers.


Introduction
For a fixed a constant β > 0 and a smooth bounded open set D ⊂ R d , d 2, we consider the functional defined on the pairs (u, Ω), where u ∈ H 1 (D), Ω ⊂ R d is a set of finite perimeter in the sense of De Giorgi (see Section 2) and ∂ * Ω denotes the reduced boundary of Ω (see Section 2); when Ω is smooth, ∂ * Ω is the topological boundary of Ω.
In this paper we study the existence and the regularity of minimizers of the functional J β among all pairs (u, Ω), which are fixed outside the domain D. Precisely, throughout the paper, we fix a set E ⊂ R d of finite perimeter, a constants m > 0 and a function v ∈ H 1 loc (R d ) such that v m in R d and we define the admissible sets and we consider the variational minimization problem (1.1) min J β (u, Ω) : u ∈ V, Ω ∈ E .
Our main result is the following.   1.1.1. Existence. -The existence of a solution (u, Ω) and the regularity of u (Hölder regularity and non-degeneracy) are treated simultaneously. The reason is that if (u n , Ω n ) is a minimizing sequence for (1.1), then in order to get the compactness of Ω n , we need a uniform bound (from above) on the perimeter Per(Ω n ), for which we need the functions u n to be bounded from below by a strictly positive constant. Now, notice that we cannot simply replace u n by u n ∨ ε, for some ε > 0; this is due to the fact that the second term in J β is increasing in u: Thus, we select a minimizing sequence which is in some sense optimal. Precisely, we take (u n , Ω n ) to be solution of the auxiliary problem (1.2) min J β (u, Ω) : u ∈ V, Ω ∈ E, u 1/n in D , for which the existence of an optimal set is much easier (see Section 3, Proposition 3.1). Still, we do not have a uniform (independent from n) bound from below for the functions u n , so we still miss the uniform bound on the perimeter of Ω n . On the other hand, we are able to prove that the sequence u n is uniformly Hölder continuous in D (see Section 3,Lemma 3.5). This enables us to extract a subsequence u n that converges locally uniformly in D to a non-negative Hölder continuous function u ∞ : D → R (see Section 4). Now, on each of the sets {u ∞ > t}, t > 0, the sequence Ω n has uniformly bounded perimeter. This enables us to extract a subsequence Ω n that converges pointwise almost-everywhere on {u ∞ > 0} to some Ω ∞ . Thus, we have constructed our candidate for a solution: (u ∞ , Ω ∞ ).
In order to prove that (u ∞ , Ω ∞ ) is an admissible competitor in (1.1), we need to show that Ω ∞ has finite perimeter. We do this in Section 4. We first use the optimality of (u n , Ω n ) to prove that (u ∞ , Ω ∞ ) is optimal when compared to a special class of competitors. This optimality condition can be written as (we refer to Lemma 4.1 for the precise statement): for any t > 0. Next, from this special optimality condition we deduce that the function u ∞ is bounded from below by a strictly positive constant (see Proposition 4.2). From this, in Section 4, we deduce that Ω ∞ has finite perimeter in R d and that the pairs (u ∞ , Ω ∞ ) is a solution to (1.1).

1.1.2.
Hölder continuity and non-degeneracy of u. -Let now (u, Ω) be any solution of (1.1). In order to prove the Hölder continuity and the non-degeneracy of u it is sufficient to exploit some of the estimates that we already used to prove the existence. Indeed, we can test the optimality of (u, Ω) with the competitors from (1.3). Thus, for t > 0 small enough, we have In particular, which proves that u satisfies the optimality condition (4.1) from Lemma 4.1: Now, applying Proposition 4.2, we get that u is bounded from below by a strictly positive constant in D. Finally, Proposition 3.5 gives that u is Hölder continuous in D. This proves Theorem 1.1(iii).

1.1.3.
Regularity of the free boundary. -In order to prove the regularity of the free boundary (Theorem 1.1(iii)), we use the Hölder continuity and the non-degeneracy of u to show that a solution Ω is an almost-minimizer of the perimeter. We do this in Theorem 5.1. Now, from the classical regularity theory for almost-minimizers of the perimeter (see [8]), we obtain that (inside D) the free boundary ∂Ω can be decomposed into a C 1,α -regular part Reg(∂Ω) and a (possibly empty) singular part of Hausdorff dimension smaller than d − 8. Finally, in Theorem 5.2, we prove the C ∞ regularity of Reg(∂Ω). In order to do so, we first show (see Lemma 5.3) that in a neighborhood of a regular point x 0 , the restrictions u + and u − of u on Ω and D Ω are solutions of the following transmission problem: where ν Ω is the normal derivative to ∂Ω. Now, using the recent results [4] and [5], we get that u + and u − are as regular as the free boundary ∂Ω (see Lemma 5.4 where F is an explicit (rational) function of ∇u ± and u. In particular, this implies that ∂Ω gains one more derivative with respect to u, that is, u ∈ C k,α ⇒ ∂Ω ∈ C k+1,α . Thus, by a bootstrap argument, the regular part of the free boundary is C ∞ .

1.2.
On the non-degeneracy of the solutions. -We notice that the competitors (u t , Ω t ) in (1.3) are the two-phase analogue of the ones used by Caffarelli and Kriventsov in [3], where the authors study a one-phase version of (1.1). Nevertheless, the functional in [3] involves the measure of Ω, which means that the optimality condition there corresponds to where C > 0. The presence of the constant C enables us to prove the bound from below by using a differential inequality for a suitably chosen function f (t), which is given in terms of u and {u < t} (see Proposition 4.2 and [3, Th. 3.2]). In Proposition 4.2, we exploit the same idea, but since we do not have the constant C, we can only conclude that f (t) εt (which is not in contradiction with the fact that f (t) is defined for every t > 0). So, we continue, and we use this lower bound to obtain a bound of the form where u := u ∞ and c is a constant depending on β and d. Then, we notice that this entails c β 3/4 Per({u < t}) 1/4 |{u < t}| 3/4 for every t > 0.
Passing to the limit as n → ∞, we get that if u is not bounded away from zero, then (1.7) c β|{u < t}| β|D| for every t > 0.
Now, this means that the measure of the zero-set |{u = 0}| is bounded from below. Thus, using again the optimality of u, we get that (1.6) holds with an arbitrary small ε > 0 in place of β, we get that c ε|{u < t}| for every t > 0, which is impossible. A similar non-degeneracy result was proved by Bucur and Giacomini in [1] by a De Giorgi iteration scheme (1) . Precisely, one can prove that any solution to (1.1) satisfies the optimality condition from [1,Rem. 3.7]. Thus, [1,Th. 3.5] also applies to the solutions of (1.1). Conversely, the argument from 4.2 can be applied to the minimizers of [1] to obtain the bound from below of [1,Th. 3.5]. (1) We are grateful to the anonymous referee for bringing to our attention the reference [1].

One-phase and two-phase problems with Robin boundary conditions
The problem (1.1) is the first instance of a two-phase free boundary problem with Robin boundary conditions. Precisely, we notice that if Ω is a fixed set with smooth boundary and if u minimizes the functional J β (·, Ω) in H 1 (D), then the functions u + := u on Ω and u − := u on D Ω, are harmonic in Ω and D Ω, and satisfy the following conditions: where ν + and ν − are the exterior and the interior normals to ∂Ω. Notice that (1.8) is a two-phase counterpart of the one-phase problem which was studied by Bucur-Luckhaus in [2] and Caffarelli-Kriventsov in [3]. As explained in [3], the Robin condition in (1.9) naturally arises in the physical situation in which the heat diffuses freely in Ω, the temperature is set to be zero on the surface ∂Ω, which is separated from the interior of Ω by an infinitesimal insulator. The two-phase problem (1.8) also may be interpreted in this way, in this case the heat diffuses freely both inside Ω and outside, in D Ω; the temperature is set to be zero on the surface ∂Ω, which is insulated from both sides; the continuity of the temperature means that the heat transfer is allowed also across ∂Ω, which happens for instance if the surface ∂Ω is replaced by a very thin (infinitesimal) net.
Even if the problems in [2,3] and in the present paper lead to the free boundary conditions of the same type, the techniques are completely different. For instance, the problem studied in [2,3] is a free discontinuity problem as the function u jumps from positive in Ω to zero in D Ω. Thus, the corresponding variational minimization problem can be naturally stated in the class of SBV functions, which clearly influences both the existence and the regularity techniques; roughly speaking, the existence is obtained through a compactness theorem in the SBV class, while the regularity relies on techniques related to the Mumford-Shah functional.
In our case, the problem can be stated for the functions (u 1 , u 2 ) with disjoint supports (u 1 u 2 = 0 almost-everywhere in D) which satisfy the following constraints: the sum u 1 + u 2 should be a Sobolev function (this corresponds to the continuity condition in (1.8)); u 2 1 and u 2 2 are SBV functions whose jump sets are contained in the boundary of the positivity sets {u 1 > 0} and {u 2 > 0}. Now, it is reasonable to expect that an existence result can be proved also in this class, but then, in order to prove that a solution to (1.1) exists, one should show that u 1 and u 2 are of the form u 1 = u1 Ω and u 2 = u1 D Ω for a set of finite perimeter Ω ⊂ R d , u being the sum u 1 + u 2 . Summarizing, working in the class of SBV functions would allow to state (1.1) in a weaker form, but it doesn't seem to be a shortcut to the existence of a solution (of (1.1)) as it will require the analysis of the jump sets of the optimal pairs in the SBV class. Thus, we prefer not to rely on the advanced compactness results for SBV functions, but to prove the existence of a solution from scratch.
Finally, as explained in Section 1.1, once we know that an optimal pairs (u, Ω) exists, and that u is non-degenerate and Hölder continuous, the regularity of the free boundary ∂Ω follows immediately since the set Ω becomes an almost-minimizer of the perimeter.

Preliminaries
By the De Giorgi structure theorem (see for instance [

2.2.
Capacity and traces of Sobolev functions. -We define the capacity (or the 2-capacity) of a set E ⊂ R d as It is well-known that the sets of zero capacity have zero d − 1 dimensional Hausdorff measure (see for instance [6, §4.7.2, Th. 4]): The Sobolev functions are defined up to a set of zero capacity (i.e., quasi-everywhere), Moreover, for every function u ∈ H 1 (A) there is a sequence u n ∈ C ∞ (A) ∩ H 1 (A) and a set N ⊂ A of zero capacity such that: u n converges to u strongly in As a consequence of the discussion above, we have the following proposition.
In particular, it is sufficient to results from [6], this time in the space W 1,p (D), for p close to 2.
In the next subsection, we will go through the main properties of this functional, which we will need in the proof of Theorem 1.1.

2.3.
Properties of the functional I. -We first notice that we can use an integration by parts to write I as in (2.1).
Proof. -The first claim follows by a classical approximation argument with functions of the form φ n * u, where φ n is a sequence of mollifiers. In order to prove claim (ii), we notice that Thus, it is sufficient to find a sequence ξ n ∈ C 1 c (A; R d ), |ξ n | 1, such that J.É.P. -M., 2021, tome 8 Let A n be a sequence of open sets such that A n A and 1 An → 1 A . Then Setting M n = sup An u 2 , we can find ξ n ∈ C 1 c (A; R d ) such that |ξ n | 1, and In particular, this implies that which concludes the proof.
be a sequence of functions and Ω n ⊂ R d be a sequence of sets of locally finite perimeter in A such that: Then, Proof. -Notice that, for every u ∈ H 1 (A) and every set of finite perimeter Ω, we have where ν Ω denotes the exterior normal to ∂ * Ω. We use the notation ν n := ν Ωn and ν ∞ := ν Ω∞ .
where in order to pass to the limit we used that the sequence u n 1 Ωn converges strongly in L 2 loc (A) to u ∞ 1 Ω∞ , as a consequence of the fact that it converges pointwise a.e. and is bounded by h. Now, taking the supremum over ξ, we get (2.4).

A family of approximating problems
We use the notations D, β, E, v, E, V from Section 1. Moreover, we fix a constant where m is the lower bound of the function v, and we consider the auxiliary problem Proof. -Let (u n , Ω n ) be a minimizing sequence for (3.1). Since Thus, there are subsequences u n and Ω n such that: u n converges strongly in L 2 (D), weakly in H 1 (D) and pointwise almosteverywhere to a function u ∞ ∈ H 1 (D); -1 Ωn converges to 1 Ω∞ strongly in L 1 (D) and pointwise almost-everywhere.
Moreover, we can assume that u n h on D, where h is the harmonic function: Thus, we can assume that 0 u n h, for every n ∈ N, and so the hypotheses of Lemma 2.4 are satisfied, which means that (2.4) holds. Moreover, by the semicontinuity of the H 1 norm we have which finally implies that Remark 3.3.µ ε is the distributional Laplacian of u ε . We will use the notation µ ε = ∆u ε .
Proof. -Let ϕ u ε be a function in H 1 (D) such that ϕ = u ε on ∂D. Then, testing the optimality of (u ε , Ω ε ) with (ϕ ∨ ε, Ω ε ) and using the fact that u ε ϕ ∨ ε, we get which concludes the proof.
We will next show that the family of solutions u ε ε∈(0,m) is uniformly Hölder continuous. We will use the following lemma, which can be proved in several different ways. Here, we give a short proof based on the mean-value formula for subharmonic functions. Similar argument was used to prove the Lipschitz continuity of the solutions to some free boundary problems (see for instance [9] and the references therein).  Then, there is a constant C depending on δ, h, α and K such that Proof.
-We first notice that the following formula is true for every subharmonic function u ∈ H 1 (D) and for every x 0 ∈ D and 0 < s < t < dist(x 0 , ∂D).
In particular, the function is monotone and we can define the function u pointwise everywhere as As a consequence, for every R < dist(x 0 , ∂D), we have Now, applying (3.2), and integrating in r, we get that if x 0 ∈ D δ and R < δ/2, then which, by the subharmonicity of u, implies Let now x 0 , y 0 ∈ D δ be such that where γ ∈ (0, 1) will be chosen later.
where in the last inequality we used that |x 0 − y 0 | 1−γ 1. Now, using (3.5), we get  Proof. -By Lemma 3.2, we have that u ε is subharmonic and, in particular, 0 u ε h in D, where h is the harmonic extension of v in D. Thus, it is sufficient to prove that (3.2) holds. Let x 0 ∈ D δ and R δ/2. Let ϕ ∈ C ∞ c (B 3R/2 (x 0 )) be such that where M ρ := sup h(x) : x ∈ D ρ . Thus, we obtain , which concludes the proof of (3.2) with α = 1/2.

Existence of an optimal set
4.1. Definition of (u 0 , Ω 0 ). -Now, for any ε ∈ (0, m), we consider the solution (u ε , Ω ε ) of (3.1). As a consequence of Proposition (3.5), we can find a sequence ε n → 0 and a function u 0 ∈ H 1 (D) ∩ C 0,1/3 (D) such that: u εn converges to u 0 uniformly on every set D δ , δ > 0, where D δ is defined in (3.3); u εn converges to u 0 strongly in L 2 (D); u εn converges to u 0 weakly in H 1 (D). Our aim in this section is to show that u 0 is a solution to (1.1).
The construction of Ω 0 is more delicate. First, we fix t > 0 and δ > 0 and we notice that the perimeter of Ω εn is bounded on the open set {u 0 > t} ∩ D δ . Indeed, the uniform convergence of u εn to u 0 implies that, for n large enough (n N t,δ , for some fixed N t,δ ∈ N), Thus, we have Now, if we choose t such that Per({u 0 > t}) < ∞ (which, by the co-area formula, is true for almost-every t > 0), then we have that Per Ω εn ∩ {u 0 > t} ∩ D δ C t,δ for every n N t,δ , for some constant C t,δ > 0. Now, since all the sets Ω εn ∩ {u 0 > t} ∩ D δ are contained in D and have uniformly bounded perimeter, we can find a set Ω 0 and a subsequence for which Thus, by a diagonal sequence argument, we can extract a subsequence of ε n (still denoted by ε n ) and we can define the set Ω 0 ⊂ R d as the pointwise limit 1 Ω0 (x) = lim n→∞ 1 Ωε n ∩{u0>0} (x) for almost-every x ∈ {u 0 > 0}, and we notice that, by construction, Ω 0 ⊂ {u 0 > 0}. Notice that, we do not know a priori that Ω 0 has finite perimeter. We only know that Per (Ω 0 ∩ {u 0 > t} ∩ D δ ) < ∞ for every δ > 0 and almost-every t > 0.
which means that Ω 0 ∩ {u 0 > t} has locally finite perimeter in D for a.e. t > 0.

4.2.
An optimality condition. -As pointed out above, we do not know if the pairs (u 0 , Ω 0 ) is even an admissible competitor for (1.1) (we need to show that Ω 0 ∈ E). Nevertheless, we can still prove that it satisfies a suitable optimality condition.  Proof.
-Let now t > 0 be fixed and such that the set {u 0 < t} has finite perimeter. Then, for n large enough, we can use the pairs (u 0 ∨ t, Ω 0 ∪ {u 0 < t}) to test the optimality of (u εn , Ω εn ). Notice that the set Ω 0 ∪ {u 0 < t} has finite perimeter for a.e. t ∈ (0, m), as observed in the previous section. For the sake of simplicity, we write u εn = u n , Ω εn = Ω n , u 0 = u and Ω 0 = Ω. Thus, we have Now, by the weak convergence of u n to u, we get that On the other hand, setting U t,δ to be the open set for some fixed δ > 0, and applying Lemma 2.4, we have that Taking the limit as δ → 0, by the monotone convergence theorem, we get that and since h m > t on ∂D, we have that Thus, we get that

4.3.
Non-degeneracy. -The crucial observation in this section is that the functions u satisfying the optimality condition (4.1) are non-degenerate in the sense of the following proposition.
Let t ∈ (0, m) be fixed. By the co-area formula, the Cauchy-Schwartz inequality and the optimality condition (4.5), we get We now set Per(Ω s ) ds = Ωt |∇u| dx.
Using (4.6), we will estimate f (t) from below.
Step 1. Non-degeneracy of f . -By the isoperimetric inequality and the estimate (4.6), there is a dimensional constant C d such that Using the definition of f , we can re-write this inequality as After rearranging the terms and integrating from 0 to t, we obtain Setting (4.7) we obtain the lower bound f (t) Ct.
In particular, as a consequence of (4.6), we get that Step 2. Non-degeneracy of |Ω t |. -Let α ∈ (0, 1) be fixed. Then, we have that Thus, we obtain that for fixed T ∈ (0, m) and C > 0, the following implication holds: We claim that, for every n 1 and every t ∈ (0, m), we have the inequality (4.10) In order to prove (4.10), we argue by induction on n. When n = 1, (4.10) is precisely (4.8). In order to prove that the claim (4.10) for n ∈ N implies the same claim for n + 1, we apply (4.9) for α = 2 −n , n ∈ N, which gives precisely (4.10) with n + 1. This concludes the proof of (4.10). Next, passing to the limit as n → ∞, we obtain that where C is given by (4.7). Thus, there is a dimensional constant C d > 0 such that Step 3. Conclusion.
-We now notice that Thus, for every ε > 0, there is T ε such that for all t ∈ (0, T ε ) we have  Now, repeating the argument fro Step 1 and Step 2, we get that (4.11) should hold with ε in place of β. Since ε > 0 is arbitrary, this is a contradiction. Proof. -Let (u 0 , Ω 0 ) be as in Section 4.1. Then, by Lemma 4.1, (u 0 , Ω 0 ) satisfies the optimality condition (4.5). Now, by Proposition 4.2 we get that u 0 t in D, for some t > 0. In particular, Ω 0 has finite perimeter in D. Precisely, for every δ > 0, we have Passing to the limit as δ → 0, we get In particular, this implies that Ω 0 is a set of finite perimeter in R d . Indeed,

Regularity of the free boundary
In this section, we prove the regularity of the free boundary. In Theorem 5.1, we prove that the solutions of (1.1) are almost-minimizers for the perimeter in D. As a consequence, ∂Ω can be decomposed into a regular and a singular part and that the regular part is C 1,α manifold. Then, in Theorem 5.2, we prove that the regular part of the free boundary is C ∞ smooth. Proof. -We first notice that by Lemma 3.4, u ∈ C 0,1/3 (D). Let δ > 0, x 0 ∈ D δ and r < δ/2. We consider a set Ω ⊂ R d such that Ω ∆Ω B r (x 0 ). Testing the optimality of (u, Ω) against (u, Ω ) we get that where in the second inequality, we used that u t > 0. Thus, we obtain which proves that Ω is an almost-minimizer of the perimeter in D.
We next prove that regular part the free boundary Reg(∂Ω) is C ∞ . Proof. -We fix a point x 0 ∈ Reg(∂Ω). Without loss of generality, we assume x 0 = 0.
Step 1. Notation. -For any x ∈ R d , we use the notation x = (x , x d ), where x ∈ R d−1 and x d ∈ R. By the C 1,α regularity of Reg(∂Ω), in B × (−ε, ε) ⊂ R d−1 × R, ∂Ω is the graph of a C 1,α regular function η : B → R, where B is a ball in R d−1 ; the set Ω coincides with the subgraph of η in a neighborhood of the origin: and the exterior normal ν Ω is given by where ∇ x η is the gradient of η in the first d − 1 variables. Let u + and u − be the restrictions of u on the sets Ω and D Ω; since u is continuous across ∂Ω, we have u + = u − on ∂Ω. Moreover, we write the gradients of u + and u − as Step 2. Transmission condition and C 1,α regularity of u. -In Lemma 5.3, we keep fixed the free boundary ∂Ω and we use vertical perturbations of the function u to obtain a Robin-type transmission condition on ∂Ω. We notice that the recent results [4,5] imply the C 1,α -regularity of u + and u − , up to the boundary ∂Ω. Thus, the gradient is well-defined and the transmission conditions (5.2) hold in the classical sense.
Step 3. Optimality condition and C 2,α regularity of Reg(∂Ω). -In Lemma 5.5 we perform variations of the optimal set to find the geometric equation solved by ∂Ω. Precisely, we find that the curvature of the optimal set solves an equation of the form "Mean curvature of ∂Ω" = F (∇u + , ∇u − , u ± ) on ∂Ω.
In particular, this implies that if u is C k,α , for some k 1, then ∂Ω is C k+1,α .
Step 4. Bootstrap. -In Lemma 5.4 we use the recent results of [5] to show that if the boundary ∂Ω is C k,α for some k 2, then the solutions u + and u − are also C k,α regular up to the boundary ∂Ω. Finally, applying this result (Lemma 5.4) and the result from the previous step (Lemma 5.5), we get that ∂Ω is C ∞ .
Lemma 5.3 (Robin and continuity conditions on ∂Ω). -Suppose that ∂Ω is C 1,α regular in the neighborhood of the origin. Let η : B → R, u + and u − be as above.
Then, for every x ∈ B we have where u + , u − and their partial derivatives are calculated in (x , η(x )) ∈ ∂Ω.
-Let φ ∈ C ∞ c (D) be a smooth function supported in B × (−ε, ε). Then, the optimality of u gives that where in the last inequality we integrated by parts u + in Ω and u − in D Ω. Since φ is arbitrary we get that u satisfies the Robin-type condition on ∂Ω Now, using (5.1), we can re-write this as On the other hand u is continuous across ∂Ω. This means that Multiplying by ∇ x η, we get where u + , u − and their partial derivatives are calculated in (x , η(x )). Putting together (5.4) and (5.5), we get (5.2). Proof. -We argue by induction. The case k = 1 follows by [5]. We suppose that k 2 and that the claim holds for k − 1. Suppose that ∂Ω is the graph of η : B → R, η ∈ C k,α (B ), and consider the functions defined on the half-space {x d 0}. We set where N d−1 is the null (d − 1) × (d − 1) matrix and we notice that A η has C k−1,α regular coefficients. Now, since u + and u − are harmonic in Ω and D Ω, we have that v + and v − are solutions to the transmission problem We now fix k − 1 directions i 1 , . . . , i k−1 , i j = d for every j, and we consider the functions We notice that, in {x d > 0} and {x d < 0} the functions w + and w − are solutions to where the sum is over all multiindices I and J such that the sets I and J are disjoint subsets of {i 1 , i 2 , . . . , i k−1 }, I ∪J = {i 1 , i 2 , . . . , i k−1 } and I is non-empty. In particular, using that A η ∈ C k−1,α and ∇u ∈ C k−2,α (since by hypothesis u ± ∈ C k−1,α ), we get that w ± solve where F + and F − are C 0,α continuous functions (depending on i 1 , . . . , i k ). On the other hand, on the boundary {x d = 0} we have that w + = w − and Reasoning as above, we notice that this condition can be written as where g is a C 0,α function. Now, applying [5, Th. 1.2], we get that w + and w − are C 1,α regular up to the boundary {x d = 0}. Thus, the trace u + = u − is C k,α smooth on {x d = 0}. Finally, the classical Schauder estimates give that u + and u − are C k,α on {x d 0} and {x d 0}, respectively. -Let (u, Ω) be a solution of (1.1). Suppose that, in a neighborhood of zero, ∂Ω is C 1,α -regular and that the functions u + and u − are C k,α up to the boundary ∂Ω, for some k 1. Then, ∂Ω is C k+1,α -regular in a neighborhood of zero.
Proof. -Let ξ ∈ C ∞ c (D; R d ) be a given vector field with compact support in D and let Ψ t be the function Then, for t small enough, Ψ t : D → D is a diffeomorphism and setting Φ t := Ψ −1 t , the function u t := u • Φ t is well-defined and belongs to H 1 (D); the function It is immediate to check that −2∇u Dξ · ∇u + |∇u| 2 div ξ = div |∇u| 2 ξ − 2(ξ · ∇u)∇u in D ∂Ω.

Appendix. Examples of minimizers
In this section, we use a calibration argument to prove that if E = {x d > 0} and v ≡ 1, then in any Steiner symmetric set D ⊂ R d , the solution (Ω, u) is unique, u is even with respect to the hyperplane {x d = 0} and Ω is precisely the half-space E. Our main result is the following. Proof. -Let u ∈ V and Ω ∈ E be given. We will prove that J β (u, Ω) J β ( u, Ω), with an equality, if and only if, (u, Ω) = ( u, Ω). First, we notice that, since J β (1 ∧ u ∨ 0, Ω) J β ( u, Ω), we can suppose that 0 u 1. We then write u as u = 1 − ϕ for some ϕ ∈ H 1 0 (D) such that 0 ϕ 1 and we define the function u * = 1 − ϕ * , where ϕ * ∈ H 1 0 (D) is the Steiner symmetrization of ϕ. We will show that (A.2) J β ( u * , Ω) J β ( u, Ω).
Indeed, the Steiner symmetrization decreases the Dirichlet energy: In order to estimate also the second term of the energy J β , we use a calibrationtype argument. We first notice that, by construction, along every line orthogonal to {x d = 0}, the symmetrized function achieves its maximum in zero. Precisely Thus, by the definition of u * , we have where in order to get the last equality we used the divergence theorem in Ω∆ Ω. Now, we notice that div( u 2 * (x , 0)e d ) = 0 and that ν Ω = −e d . Thus, we get which concludes the proof of (A.2). Finally, we notice that the problem has a unique solution u, which is Steiner symmetric, nonnegative and solves (A.1).