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

& Bozhidar Velichkov

A two-phase problem with Robin conditions on the free boundary Tome 8 (2021), p. 1-25.

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A TWO-PHASE PROBLEM WITH ROBIN CONDITIONS ON THE FREE BOUNDARY

by Serena Guarino Lo Bianco, Domenico Angelo La Manna

& Bozhidar Velichkov

Abstract. — 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.

Résumé(Un problème à frontière libre à deux phases avec conditions au bord de Robin) Nous étudions pour la première fois un problème à frontière libre à deux phases pour lequel la solution satisfait à une condition de Robin au bord. Nous considérons le cas où la solution est continue au bord et nous montrons un résultat d’existence et de régularité pour les minimiseurs du problème variationnel associé. Enfin, nous donnons dans l’appendice un exemple d’une classe de minimiseurs avec une symétrie de Steiner.

Contents

1. Introduction. . . 2

2. Preliminaries. . . 7

3. A family of approximating problems. . . 10

4. Existence of an optimal set. . . 13

5. Regularity of the free boundary. . . 19

Appendix. Examples of minimizers. . . 24

References. . . 25

2020Mathematics Subject Classification. — 35R35, 49Q10.

Keywords. — Free boundary problems, two-phase, Robin boundary conditions, regularity.

The first author was partially supported by PRIN 2017Nonlinear Differential Problems via Vari- ational, Topological and Set-valued Methods (Grant 2017AYM8XW) and the INdAM-GNAMPA project 2020 “Problemi di ottimizzazione con vincoli via trasporto ottimo e incertezza”. The second author was partially supported by the Academy of Finland grant 314227. The third author has been partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement VAREG, No. 853404).

1. Introduction

For a fixed a constant β > 0 and a smooth bounded open set D ⊂ R^d, d > 2, we consider the functional

Jβ(u,Ω) = Z

D

|∇u|²dx+β Z

∂^∗Ω

u²dH^d−1,

defined on the pairs(u,Ω), whereu∈H¹(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 ofΩ (see Section 2); whenΩis smooth,∂^∗Ωis the topological boundary ofΩ.

In this paper we study the existence and the regularity of minimizers of the func- tional Jβ among all pairs (u,Ω), which are fixed outside the domain D. Precisely, throughout the paper, we fix a setE⊂R^dof finite perimeter, a constantsm >0and a function

v∈H_loc¹ (R^d) such thatv>minR^d and Z

∂^∗E

v²dH^d−1<+∞;

we define the admissible sets V =

u∈H_loc¹ (R^d) : u−v∈H₀¹(D) , E =

Ω⊂R^d: Per(Ω)<+∞andΩ =E inR^drD , and we consider the variational minimization problem

(1.1) min

Jβ(u,Ω) : u∈ V, Ω∈ E .

Our main result is the following.

Theorem1.1(Existence and regularity of minimizers). — Let β >0,D⊂R^d,v,E,V andE be as above. Then the following holds.

(i) There exists a solution (u,Ω)∈ V × E to the variational problem (1.1).

(ii) For every solution (u,Ω) of (1.1), uis Hölder continuous and bounded from below by a strictly positive constant inD.

(iii) If(u,Ω)is a solution to(1.1), then the free boundary∂Ω∩Dcan be decomposed as the disjoint union of a regular part Reg(∂Ω)and a singular partSing(∂Ω), where:

– Reg(∂Ω)is aC^∞ hypersurface and a relatively open subset of∂Ω, and the function uisC^∞ smooth on Reg(∂Ω);

– Sing(∂Ω)is a closed set, which is empty ifd67, discrete if d= 8, and of Hausdorff dimension d−8, if d >8.

Remark1.2. — We notice that if(u,Ω)is a solution to (1.1), thenuis harmonic in the interior ofΩandDrΩ. Thus, as a consequence of Theorem 1.1(iii), in a neighborhood of a regular pointx0∈Reg(∂Ω), the functionsu: Ω→Randu:DrΩ→RareC^∞ up to the free boundary∂Ω.

1.1. Outline of the proof and organization of the paper. — The main difficulty in the proof of Theorem 1.1 is to prove the existence of a minimizing pairs (u,Ω) and to show that the function uis Hölder continuous and bounded from below by a strictly positive constant in D. The almost-minimality of the solutions is proved in Theorem 5.1. Finally, in the Appendix, we give examples of minimizers in domainsD symmetric with respect to the hyperplane{xd= 0}.

1.1.1. Existence. — The existence of a solution(u,Ω)and the regularity ofu(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 Ωn, we need a uniform bound (from above) on the perimeter Per(Ωn), for which we need the functionsun to be bounded from below by a strictly positive constant.

Now, notice that we cannot simply replaceu_n byu_n∨ε, for someε >0; this is due to the fact that the second term inJ_β is increasing in u:

Z

∂^∗Ωn

u²_ndH^d−16 Z

∂^∗Ωn

(ε∨un)²dH^d−1.

Thus, we select a minimizing sequence which is in some sense optimal. Precisely, we take(un,Ωn)to be solution of the auxiliary problem

(1.2) min

J_β(u,Ω) : u∈ V, Ω∈ E, u>1/ninD ,

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 un, so we still miss the uniform bound on the perimeter ofΩn.

On the other hand, we are able to prove that the sequenceun is uniformly Hölder continuous in D (see Section 3, Lemma 3.5). This enables us to extract a subse- quenceu_nthat converges locally uniformly inDto a non-negative Hölder continuous function u_∞ : D → R (see Section 4). Now, on each of the sets {u_∞ > t}, t > 0, the sequenceΩn has uniformly bounded perimeter. This enables us to extract a sub- sequenceΩ_n that converges pointwise almost-everywhere on {u∞>0} to someΩ_∞. 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 thatΩ∞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):

(1.3) Jβ(u_∞,Ω_∞)6Jβ(ut,Ωt), whereut=u_∞∨t andΩt= Ω_∞∪ {u_∞6t}, for any t >0. Next, from this special optimality condition we deduce that the func- tionu_∞is bounded from below by a strictly positive constant (see Proposition 4.2).

From this, in Section 4, we deduce thatΩ_∞ 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 ofu. — 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, fort >0 small enough, we have

(1.4) Jβ(u,Ω)6Jβ(ut,Ωt) whereut=u∨t andΩt= Ω∪ {u6t}.

In particular, Z

D

|∇u|²dx+β Z

∂^∗Ω

u²6 Z

D

|∇(u∨t)|²dx+β Z

∂^∗(Ω∪{u<t})

u²

6 Z

D

|∇(u∨t)|²dx+βt²Per({u < t}) +β Z

{u>t}∩∂^∗Ω

u²,

which proves thatusatisfies the optimality condition (4.1) from Lemma 4.1:

(1.5)

Z

{u<t}

|∇u|²dx6β t²Per {u < t}

.

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 ofuto show that a solutionΩis 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 (insideD) the free boundary∂Ωcan be decomposed into a C^1,α-regular part Reg(∂Ω)and a (possibly empty) singular part of Hausdorff dimension smaller thand−8.

Finally, in Theorem 5.2, we prove theC^∞regularity of Reg(∂Ω). In order to do so, we first show (see Lemma 5.3) that in a neighborhood of a regular point x0, the restrictionsu₊andu₋ofuonΩandDrΩare solutions of the following transmission problem:















∆u+= 0 in Ω,

∆u₋= 0 in DrΩ,

u₊=u₋ =u on∂Ω,

∂u+

∂νΩ

−∂u−

∂νΩ

+ 2βu= 0 on∂Ω,

where νΩ 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).

On the other hand, using variations of ualong smooth vector fields, we obtain that Reg(∂Ω)solves an equation of the form

“Mean curvature of∂Ω"=F(∇u+,∇u−, u±) on∂Ω,

whereFis an explicit (rational) function of∇u_±andu. In particular, this implies that

∂Ωgains 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 isC^∞. 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

Jβ(u,Ω) +C|Ω∩ {u6t}|6Jβ(ut,Ωt), whereut=u∨tandΩt= Ωr{u6t}, 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 ofuand{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 constantC, we can only conclude thatf(t)>εt(which is not in contradiction with the fact thatf(t)is defined for everyt >0). So, we continue, and we use this lower bound to obtain a bound of the form

(1.6) c6β^1/2Per({u < t})^1/2|{u < t}|^1/2 for every t >0,

whereu:=u∞ andc is a constant depending onβ andd. Then, we notice that this entails

c6β^3/4Per({u < t})^1/4|{u < t}|^3/4 for every t >0.

and we use an iteration procedure to get that

c6β^1−1/2nPer({u < t})^1/2n|{u < t}|^1−1/2n for every t >0.

Passing to the limit asn→ ∞, we get that ifuis not bounded away from zero, then (1.7) c6β|{u < t}|6β|D| for everyt >0.

Now, this means that the measure of the zero-set |{u= 0}|is bounded from below.

Thus, using again the optimality ofu, we get that (1.6) holds with an arbitrary small ε >0in place ofβ, we get that

c6ε|{u < t}| for everyt >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].

1.3. 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 ifΩis a fixed set with smooth boundary and ifuminimizes the functionalJβ(·,Ω)inH¹(D), then the functions

u+:=u onΩ and u₋ :=u onDrΩ, are harmonic inΩandDrΩ, and satisfy the following conditions:

(1.8) u₊=u₋ and ∂u₊

∂ν+

+β 2u₊

+∂u₋

∂ν−

+β 2u₋

= 0 on∂Ω∩D, whereν+ andν−are the exterior and the interior normals to∂Ω. Notice that (1.8) is a two-phase counterpart of the one-phase problem

(1.9) ∆u= 0 inΩ, ∂u

∂ν +βu= 0 on∂Ω∩D,

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 inΩ, the temperature is set to be zero on the surface∂Ω, which is separated from the interior ofΩby 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 DrΩ; 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∂Ωis 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 ujumps from positive inΩto zero inDrΩ. 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 (u₁, u₂) with disjoint supports (u1u2= 0almost-everywhere inD) which satisfy the following constraints:

the sum u1+u2 should be a Sobolev function (this corresponds to the continuity condition in (1.8)); u²₁ and u²₂ are SBV functions whose jump sets are contained in the boundary of the positivity sets{u1 >0} and{u2 >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 thatu1 andu2 are of the form u₁ = u1Ω and u₂ = u1DrΩ for a set of finite perimeter Ω ⊂R^d, u being the sum u1+u2. 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 thatuis non-degenerate and Hölder continuous, the regularity of the free boundary∂Ωfollows immediately since the setΩbecomes an almost-minimizer of the perimeter.

2. Preliminaries

2.1. Sets of finite perimeter. — LetA⊂R^d be a an open set inR^d. We recall that the setE⊂R^d is said to have afinite perimeter inAif

(2.1) Per(E, A) = supnZ

A

divξ(x)dx: ξ∈C_c¹(A;R^d), sup

x∈R^d

|ξ(x)|61o

is finite. We say thatEhas alocally finite perimeter inA, if for every open setB⊂R^d such thatB⊂A, we have thatPer(E, B)<∞. We say thatEis of finite perimeterif

Per(E) := Per(E,R^d)<+∞.

By the De Giorgi structure theorem (see for instance [7, Th. II.4.9]), if the setE⊂R^d has locally finite perimeter in A, then there is a set ∂^∗E ⊂ A∩∂E called reduced boundarysuch that

Per(E, B) =H^d−1(B∩∂^∗E) for every setBbA,

where H^d−1 is the(d−1)-dimensional Hausdorff measure in R^d. Moreover, there is a H^d−1-measurable function νE : ∂^∗E → R^d, called generalized normal such that

|ν_E|= 1and Z

E

divξ(x)dx= Z

∂^∗E

νE·ξ dH^d−1 for everyξ∈C_c¹(A;R^d).

2.2. Capacity and traces of Sobolev functions. — We define the capacity (or the 2-capacity) of a setE⊂R^d as

cap(E) = inf

kuk²_H1(R^d): u∈H¹(R^d), u>1 in a neighborhood ofE . Suppose now that d > 3. 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]):

If cap(E) = 0, thenH^d−1(E) = 0.

The Sobolev functions are defined up to a set of zero capacity (i.e.,quasi-everywhere), that is, if A ⊂R^d is an open set andu∈H¹(A), then there is a set N_u ⊂R^d such thatcap (Nu) = 0and

u(x₀) = lim

r→0

1

|Br| Z

B_r(x₀)

u(x)dx for everyx₀∈ArN_u.

Moreover, for every function u∈ H¹(A) there is a sequence un ∈C^∞(A)∩H¹(A) and a setN ⊂Aof zero capacity such that:

– un converges toustrongly in H¹(A);

– u(x) = lim_n→∞u_n(x)for everyx∈Ar(N ∪ Nu).

In particular, ifE⊂R^dis a set of locally finite perimeter in the open setA⊂R^d and ifu∈H¹(A), then the functionu² is definedH^d−1-almost everywhere on∂^∗E and is H^d−1 measurable on∂^∗E. Thus, the integral

I(u, E) :=

Z

A∩∂^∗E

u²dH^d−1 is well-defined.

As a consequence of the discussion above, we have the following proposition.

Proposition2.1. — Let D be a smooth bounded open set inR^d and u∈H¹(D)be a Sobolev function. Then, there is a setN_u⊂D such that H^d−1(N_u) = 0and

u(x0) = lim

r→0

1

|B_r| Z

Br(x0)

u(x)dx for everyx0∈DrNu.

Moreover, if E ⊂ R^d is a set of locally finite perimeter in R^d, then the function u : ∂^∗E∩D → R is defined H^d−1-almost everywhere and is H^d−1-measurable on

∂^∗E∩D. In particular, the integralI(u, D)is well-defined.

Remark2.2. — In the case d= 2, (2.1) still holds. In fact, it is sufficient to notice that ifu∈H¹(D), thenu∈W^1,p(D)for any1< p <2. In particular, it is sufficient to results from [6], this time in the spaceW^1,p(D), forpclose to2.

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 functionalI. — We first notice that we can use an integra- tion by parts to writeI as in (2.1).

Lemma2.3. — Let E⊂R^d be a set of locally finite perimeter in the open setA⊂R^d and letu∈H¹(A)be locally bounded in A. Then, the following holds.

(i) For every ξ∈C_c¹(A;R^d)we have (2.2)

Z

A∩∂^∗E

(ξ·νE)u²dH^d−1= Z

A

div(u²ξ)dx.

(ii) We have the formula (2.3)

Z

A∩∂^∗E

u²dH^d−1= sup Z

A∩∂^∗E

(ξ·ν_E)u²dH^d−1: ξ∈C_c¹(A;R^d), |ξ|61

.

Proof. — The first claim follows by a classical approximation argument with functions of the formφ_n∗u, whereφ_n is a sequence of mollifiers. In order to prove claim (ii), we notice that

Z

A∩∂^∗E

u²dH^d−16sup Z

A∩∂^∗E

(ξ·νE)u²dH^d−1: ξ∈C_c¹(A;R^d), |ξ|61

.

Thus, it is sufficient to find a sequenceξn∈C_c¹(A;R^d),|ξn|61, such that Z

A∩∂^∗E

u²dH^d−1= lim

n→∞

Z

A∩∂^∗E

(ξn·νE)u²dH^d−1.

LetAn be a sequence of open sets such thatAnbAand1A_n→1A. Then Z

A∩∂^∗E

u²dH^d−1= lim

n→∞

Z

An∩∂^∗E

u²dH^d−1.

Setting Mn= sup_Anu², we can find ξn ∈C_c¹(A;R^d)such that |ξn|61, and 06Per(E, An)−

Z

∂^∗E

(ξn·νE)dH^d−16 1 nMn

.

In particular, this implies that 06

Z

An∩∂^∗E

u²dH^d−1− Z

An∩∂^∗E

(ξn·νE)u²dH^d−16 1 n,

which concludes the proof.

Lemma2.4(Main semicontinuity lemma). — Suppose thatA⊂R^d is a bounded open set and that h :A → R is a non-negative function in L¹(A). Let u_n ∈ H¹(A) be a sequence of functions and Ωn ⊂R^d be a sequence of sets of locally finite perimeter inA such that:

(a) 06un 6hinA, for everyn∈N;

(b) there is a functionu∞∈H¹(A)such thatunconverges tou∞weakly inH¹(A) and pointwise almost-everywhere in A;

(c) there is a setΩ_∞⊂R^d of locally finite finite perimeter in A such that the se- quence of characteristic functions1Ω_n converges to1Ω∞ pointwise almost-everywhere inA.

Then,

(2.4)

Z

A∩∂^∗Ω∞

u²_∞dH^d−16lim inf

n→∞

Z

A∩∂^∗Ωn

u²_ndH^d−1.

Proof. — Notice that, for everyu∈H¹(A)and every set of finite perimeterΩ, we have Z

A∩∂^∗Ω

u²dH^d−1= sup Z

A∩∂^∗Ω

(ξ·νΩ)u²dH^d−1: ξ∈C_c¹(A;R^d), |ξ|61

,

whereν_Ωdenotes the exterior normal to∂^∗Ω. We use the notation νn:=νΩn and ν∞:=νΩ∞.

Let nowξ∈C_c¹(A;R^d), |ξ|61 be fixed. By the divergence theorem, we have lim inf

n→∞

Z

A∩∂^∗Ωn

u²_ndH^d−1>lim inf

n→∞

Z

A∩Ωn

div u²_nξ

dx= lim inf

n→∞

Z

A

1Ω_ndiv u²_nξ dx

= lim inf

n→∞

Z

A

2 un1Ωnξ

· ∇un+ un1Ωn

(un divξ) dx

= Z

A

2 u_∞1Ω∞ξ

· ∇u∞+ u_∞1Ω∞

(u_∞ divξ) dx

= Z

A∩Ω∞

div u²_∞ξ dx=

Z

A∩∂^∗Ω∞

(ξ·ν_∞)u²_∞dH^d−1,

where in order to pass to the limit we used that the sequenceun1Ω_nconverges strongly inL²_loc(A)tou∞1Ω∞, as a consequence of the fact that it converges pointwise a.e. and is bounded byh. Now, taking the supremum overξ, we get (2.4).

3. A family of approximating problems

We use the notationsD, β, E, v,E,V from Section 1. Moreover, we fix a constant ε∈[0, m),

wheremis the lower bound of the functionv, and we consider the auxiliary problem

(3.1) min

Jβ(Ω, u) : Ω∈ E, u∈ V, u>εinR^d .

Proposition 3.1 (Existence of a solution). — Let E and V be as above. Then, for every 0< ε < m, there is a solution to the problem (3.1).

Proof. — Let(un,Ωn)be a minimizing sequence for (3.1). Since Z

D

|∇un|²dx+ Z

∂^∗Ωn

u²_ndH^d−1=Jβ(un,Ωn)6Jβ(v, E), for everyn∈N, we have

Z

D

|∇un|²dx6Jβ(v, E) and Per(Ωn)6 1

βε²Jβ(v, E).

Thus, there are subsequencesu_n andΩ_n such that:

– u_n converges strongly in L²(D), weakly in H¹(D) and pointwise almost- everywhere to a functionu_∞∈H¹(D);

– 1Ω_n converges to1Ω∞ strongly inL¹(D)and pointwise almost-everywhere.

Moreover, we can assume thatun6honD, wherehis the harmonic function:

∆h= 0 in D, h−v∈H₀¹(D).

Indeed, we have Z

D

|∇un|²dx= Z

D

|∇(un∧h)|²+ Z

D

|∇(un∨h)|²− Z

D

|∇h|²

>

Z

D

|∇(un∧h)|²dx,

Z

∂^∗Ω

u²_ndH^d−1>

Z

∂^∗Ω

(u_n∧h)²dH^d−1, and

which gives that

J_β(u_n∧h,Ω_n)6J_β(u_n,Ω_n).

On the other hand, we have that

Jβ(un∨0,Ωn)6Jβ(un,Ωn).

Thus, we can assume that 0 6 un 6h, for everyn ∈ N, and so the hypotheses of Lemma 2.4 are satisfied, which means that (2.4) holds. Moreover, by the semiconti- nuity of theH¹ norm we have

Z

D

|∇u∞|²dx6lim inf

n→∞

Z

D

|∇un|²dx,

which finally implies that

Jβ(u_∞,Ω_∞)6lim inf

n→∞ Jβ(un,Ωn).

Lemma3.2(Subharmonicity of the solutions). — Let m >0,β >0 andε∈[0, m)be fixed. Let the functionuε∈H¹(D)and the set of finite perimeterΩεbe such that the pairs (uε,Ωε) is a solution to the problem (3.1). Then uε is subharmonic in D and there is a positive Radon measure µ_ε such that

− Z

D

∇uε· ∇ϕ dx= Z

D

ϕ dµε for everyϕ∈H₀¹(D).

Remark 3.3. — µε is the distributional Laplacian of uε. We will use the notation µε= ∆uε.

Proof. — Letϕ6uεbe a function inH¹(D)such thatϕ=uεon∂D. Then, testing the optimality of(u_ε,Ω_ε)with(ϕ∨ε,Ω_ε)and using the fact thatu_ε>ϕ∨ε, we get

Z

D

|∇ϕ|²dx>

Z

D

|∇(ϕ∨ε)|²dx

>

Z

D

|∇uε|²dx+ Z

∂^∗Ω_ε

u²_εdH^d−1− Z

∂^∗Ω_ε

(ϕ∨ε)²dH^d−1>

Z

D

|∇uε|²dx,

which concludes the proof.

We will next show that the family of solutions

uε 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).

Lemma3.4(A general condition for the Hölder continuity). — Let D be a bounded open set in R^d and leth∈L^∞_loc(D). Suppose that u∈H¹(D)is such that

(a) 06u6hin D;

(b) uis subharmonic inD;

(c) there are constantsK >0 andα∈[0,1)such that (3.2) ∆u Br(x0)

6K r^d−1−α for every x0∈Dδ and every r∈(0, δ/2), whereδ >0 and

(3.3) Dδ :=

x∈D: dist(x, ∂D)> δ .

Then, there is a constantC depending onδ,h,αandK such that

|u(x)−u(y)|6C|x−y|(1−α)/(2−α) for everyx, y∈Dδ.

Proof. — We first notice that the following formula is true for every subharmonic functionu∈H¹(D)and for every x0∈D and0< s < t <dist(x0, ∂D).

− Z

∂B_t(x₀)

u dH^d−1− − Z

∂B_s(x₀)

u dH^d−1= 1 dωd

Z t s

r^1−d∆u B_r(x₀) dr.

In particular, the function

r7−→ − Z

∂B_r(x₀)

u dH^d−1,

is monotone and we can define the functionupointwise everywhereas u(x0) := lim

r→0− Z

∂Br(x0)

u dH^d−1.

As a consequence, for everyR <dist(x₀, ∂D), we have

− Z

∂B_R(x₀)

u dH^d−1−u(x₀) = 1 dωd

Z R 0

r^1−d∆u B_r(x₀) dr.

Now, applying (3.2), and integrating inr, we get that ifx0∈Dδ andR < δ/2, then

(3.4) 06 −

Z

∂B_R(x₀)

u dH^d−1−u(x0)6C R^1−α, whereC:= K dωd(1−α), which, by the subharmonicity ofu, implies

(3.5) 06 −

Z

BR(x0)

u dx−u(x0)6C R^1−α. Let nowx0, y0∈Dδ be such that

|x₀−y₀|61 and R:=|x₀−y₀|^γ 6 δ 4, whereγ∈(0,1)will be chosen later.

Now, sinceB_R(x₀)⊂B_R+|x0−y0|(y₀)⊂B_2R(y₀)⊂D, we can estimate u(x0)−u(y0)6 −

Z

B_R(x₀)

u(x)dx−u(y0)

6 R+|x0−y0|^d

R^d −

Z

B_R+|x0−y0|(y0)

u(x)dx−u(y₀)

= 1 +|x0−y₀|^1−γd

− Z

B_R+|x0−y0|(y₀)

u(x)dx−u(y₀)

6 1 +d2^d−1|x0−y0|^1−γ

− Z

B_R+|x

0−y0|(y₀)

u(x)dx−u(y0),

where in the last inequality we used that|x₀−y₀|^1−γ 61. Now, using (3.5), we get u(x₀)−u(y₀)6

− Z

B_R+|x0−y0|(y0)

u(x)dx−u(y₀)

+d2^d−1|x₀−y₀|^1−γkuk_L∞(B_2R(y₀))

6C R+|x0−y0|1−α

+d2^d−1|x0−y0|^1−γkhk_L^∞_(B2R(y0)) 6 2C+d2^d−1M_δ/2

|x0−y0|^1−γ,

whereM_δ/2is the maximum ofhon the setD_δ/2and where we chooseγ= 1/(2−α), which implies thatγ(1−α) = 1−γ and1−γ= (1−α)/(2−α).

Proposition3.5(Hölder continuity of the solution). — Letm >0,β >0andε∈[0, m) be fixed. Let the functionuε∈H¹(D)and the set of finite perimeterΩε be such that the pairs(u_ε,Ω_ε)is a solution to the problem(3.1)with somev∈H¹(D)andE⊂R^d. Then, for every δ >0, there is a constantC depending on D,δandv (but not on ε) such that

|uε(x)−uε(y)|6C|x−y|^1/3 for everyx, y∈Dδ.

Proof. — By Lemma 3.2, we have thatu_εis subharmonic and, in particular,06u_ε6h in D, where h is the harmonic extension of v in D. Thus, it is sufficient to prove that (3.2) holds. Letx0∈Dδ andR6δ/2. Letϕ∈C_c^∞(B3R/2(x0))be such that

ϕ= 1 onBR(x0), |∇ϕ|6 3

R in B_3R/2(x0).

Now, we test the optimality of(uε,Ωε)with(euε,Ωeε), where eu_ε=u_ε+R^1/2ϕ and Ωe_ε= Ω_ε∪B_3R/2(x₀).

Thus, we get Z

D

|∇uε|²dx+β Z

∂^∗Ωε

u²_εdH^d−16 Z

D

|∇euε|²dx+β Z

∂^∗Ωeε

ue²_εdH^d−1 Z

∂^∗Ωe_εeu²_εdH^d−16 Z

∂^∗Ω_εrB_3R/2(x₀)

u²_εdH^d−1+ Z

∂B_3R/2(x₀)

u²_εdH^d−1 and

6 Z

∂^∗Ω_ε

u²_εdH^d−1+C_dR^d−1M_δ/4² ,

whereMρ:= sup

h(x) : x∈Dρ .Thus, we obtain 2R^1/2

Z

B_3R/2(x₀)

−∇uε· ∇ϕ dx6R Z

B_3R/2(x₀)

|∇ϕ|²dx+βC_dR^d−1M_δ/4²

6C_d 1 +βM_δ/4² R^d−1, which implies that

∆uε BR(x0)

6Cd 1 +βM_δ/4²

R^d−3/2,

which concludes the proof of (3.2) withα= 1/2.

4. Existence of an optimal set

4.1. Definition of (u0,Ω0). — Now, for any ε ∈ (0, m), we consider the solution (uε,Ωε)of (3.1). As a consequence of Proposition (3.5), we can find a sequenceεn→0 and a functionu0∈H¹(D)∩C^0,1/3(D)such that:

– u_εnconverges tou₀uniformly on every setD_δ,δ >0, whereD_δis defined in (3.3);

– uε_n converges tou0 strongly inL²(D);

– uε_n converges tou0 weakly inH¹(D).

Our aim in this section is to show thatu0is a solution to (1.1).

The construction of Ω0 is more delicate. First, we fix t > 0 and δ > 0 and we notice that the perimeter of Ωε_n is bounded on the open set{u0> t} ∩Dδ. Indeed, the uniform convergence of u_εn to u₀ implies that, forn large enough (n>N_t,δ, for some fixedN_t,δ∈N),

u_εn> t

2 onD_δ∩ {u₀> t}.

Thus, we have J_β(v, E)>β

Z

D_δ∩{u0>t}∩∂^∗Ω_εn

u²_ε

ndH^d−1>βt²

2 Per Ω_εn;D_δ∩ {u0> t}

.

Now, if we choose t such that Per({u0 > t}) < ∞ (which, by the co-area formula, is true for almost-everyt >0), then we have that

Per Ωε_n∩ {u0> t} ∩Dδ

6Ct,δ for everyn>Nt,δ,

for some constantCt,δ>0. Now, since all the setsΩε_n∩ {u0> t} ∩Dδ are contained inD and have uniformly bounded perimeter, we can find a setΩ0 and a subsequence for which

1Ω_εn∩{u0>t}∩Dδ(x)−→1Ω₀∩{u0>t}∩Dδ(x) for almost-everyx∈D.

Thus, by a diagonal sequence argument, we can extract a subsequence of ε_n (still denoted byεn) and we can define the setΩ0⊂R^d as the pointwise limit

1Ω₀(x) = lim

n→∞1Ω_εn∩{u0>0}(x) for almost-everyx∈ {u0>0},

and we notice that, by construction,Ω0 ⊂ {u0 >0}. Notice that, we do not know a priori thatΩ₀ has finite perimeter. We only know that

Per (Ω₀∩ {u₀> t} ∩D_δ)<∞ for every δ >0and almost-everyt >0.

which means thatΩ₀∩ {u₀> t} has locally finite perimeter inD for a.e.t >0.

4.2. An optimality condition. — As pointed out above, we do not know if the pairs (u0,Ω0) is even an admissible competitor for (1.1) (we need to show that Ω0 ∈ E).

Nevertheless, we can still prove that it satisfies a suitable optimality condition.

Lemma4.1(The optimality condition at the limit). — Let u0 andΩ0 be as in Sec- tion 4.1. Then, for almost-everyt >0, we have

(4.1)

Z

{u0<t}

|∇u0|²dx6β t²Per {u0< t}

.

Proof. — Let nowt >0be fixed and such that the set{u0< t}has finite perimeter.

Then, for n large enough, we can use the pairs (u0∨t,Ω0∪ {u0 < t}) to test the optimality of (uεn,Ωεn). Notice that the set Ω0∪ {u0 < t} has finite perimeter for

a.e.t∈(0, m), as observed in the previous section. For the sake of simplicity, we write uε_n =un,Ωε_n= Ωn, u0=uandΩ0= Ω. Thus, we have

Z

D

|∇un|²dx+β Z

{u>t}∩∂^∗Ω_n

u²_ndH^d−1

6 Z

D

|∇u_n|²dx+β Z

∂^∗Ω_n

u²_ndH^d−1

6 Z

D

|∇(u∨t)|²dx+β Z

∂^∗(Ω∪{u<t})

u²dH^d−1 (4.2)

6 Z

D

|∇(u∨t)|²dx+βt²Per({u < t}) +β Z

{u>t}∩∂^∗Ω

u²dH^d−1.

Now, by the weak convergence ofun to u, we get that Z

D

|∇u|²dx6lim inf

n→∞

Z

D

|∇un|²dx.

On the other hand, settingUt,δ to be the open set Ut,δ =R^dr Dδ∩ {u6t}

, for some fixed δ >0, and applying Lemma 2.4, we have that

Z

U_t,δ∩∂^∗Ω

u²dH^d−16lim inf

n→∞

Z

U_t,δ∩∂^∗Ω_n

u²_ndH^d−16lim inf

n→∞

Z

{u>t}∩∂^∗Ω_n

u²_ndH^d−1.

Taking the limit asδ→0, by the monotone convergence theorem, we get that

δ→0lim Z

U_t,δ∩∂^∗Ω

u²dH^d−1= Z

R^dr(D∩{u6t})

∩∂^∗Ω

u²dH^d−1

Now, since

u(x) =h(x) for quasi-everyx∈R^drD and forH^d−1-almost-everyx∈R^drD, and sinceh>m > t on∂D, we have that

(4.3)

Z

R^dr(D∩{u6t})

∩∂^∗Ω

u²dH^d−1= Z

{u>t}∩∂^∗Ω

u²dH^d−1.

Thus, we get that (4.4)

Z

{u>t}∩∂^∗Ω

u²dH^d−16lim inf

n→∞

Z

D∩{u>t}∩∂^∗Ω_n

u²_ndH^d−1.

Now, using (4.4) and (4.2), we obtain Z

D

|∇u|²dx+β Z

{u>t}∩∂^∗Ω

u²dH^d−1

6lim inf

n→∞

Z

D

|∇un|²dx+β Z

{u>t}∩∂^∗Ω_n

u²_ndH^d−1

6 Z

D

|∇(u∨t)|²dx+βt²Per({u < t}) +β Z

{u>t}∩∂^∗Ω

u²dH^d−1,

which gives (4.1).

4.3. Non-degeneracy. — The crucial observation in this section is that the func- tions u satisfying the optimality condition (4.1) are non-degenerate in the sense of the following proposition.

Proposition4.2(Non-degeneracy). — Let β > 0,m > 0, D be a bounded open set of R^d and u∈H¹(D)be a non-negative function in D such thatu>m on ∂D. Let Ω⊂D be a set of finite perimeter in D. Suppose that uandΩsatisfy the optimality condition

(4.5)

Z

Ω_t

|∇u|²dx6β t²Per(Ωt) whereΩt={u6t},

for almost-everyt∈(0, m). Then,|Ωt|= 0for somet >0.

Proof. — By contradiction, suppose that

|Ωt|>0 for every t >0.

Lett∈(0, m)be fixed. By the co-area formula, the Cauchy-Schwartz inequality and the optimality condition (4.5), we get

(4.6) Z

Ωt

|∇u|= Z t

0

Per(Ωs)ds6 Z

Ωt

|∇u|² 1/2

|Ωt|^1/26tβ^1/2Per(Ωt)^1/2|Ωt|^1/2. We now set

f(t) :=

Z t 0

Per(Ω_s)ds= Z

Ω_t

|∇u|dx.

Using (4.6), we will estimatef(t)from below.

Step 1. Non-degeneracy off. — By the isoperimetric inequality and the estimate (4.6), there is a dimensional constantCd such that

Z t 0

Per(Ωs)ds6tβ^1/2CdPer(Ωt)(2d−1)/(2d−2). Using the definition off, we can re-write this inequality as

f(t)(2d−2)/(2d−1)

6t(2d−2)/(2d−1) β^1/2Cd(2d−2)/(2d−1)

f⁰(t).

After rearranging the terms and integrating from0tot, we obtain f(t)^1/(2d−1)−f(0)^1/(2d−1)> t^1/(2d−1)

β^1/2Cd(2d−2)/(2d−1). Now, sinceuis non-negative in D, we have thatf(0) = 0. Thus

f(t)> t β^1/2Cd

2d−2. Setting

(4.7) C= βCd^1−d

, we obtain the lower bound

f(t)>Ct.

In particular, as a consequence of (4.6), we get that (4.8) C6β^1/2Per(Ω_t)^1/2|Ωt|^1/2.

Step 2. Non-degeneracy of|Ωt|. — Letα∈(0,1)be fixed. Then, we have that Z t

0

Per(Ωs)^α|Ωs|^1−αds6 Z t

0

Per(Ωs)ds

^αZ t 0

|Ωs|ds ^1−α

6

tβ^1/2Per(Ωt)^1/2|Ωt|^1/2α

t|Ωt|1−α

=tβ^α/2Per(Ωt)^α/2|Ωt|^1−α/2.

Thus, we obtain that for fixedT ∈(0, m)andC >0, the following implication holds:

(4.9)

(IfC6Per(Ωt)^α|Ωt|^1−α for everyt∈(0, T),

thenC6β^α/2Per(Ω_t)^α/2|Ωt|^1−α/2 for everyt∈(0, T).

We claim that, for everyn>1 and everyt∈(0, m), we have the inequality (4.10) C6β^1−1/2nPer(Ω_t)^1/2n|Ωt|^1−1/2n.

In order to prove (4.10), we argue by induction on n. When n = 1, (4.10) is pre- cisely (4.8). In order to prove that the claim (4.10) forn∈Nimplies the same claim forn+ 1, we apply (4.9) forα= 2⁻ⁿ,n∈N, which gives precisely (4.10) withn+ 1.

This concludes the proof of (4.10). Next, passing to the limit as n→ ∞, we obtain that

C6β|Ωt| for every t∈(0, T),

whereC is given by (4.7). Thus, there is a dimensional constantC_d>0such that (4.11) β^−dCd6|Ωt| for everyt∈[0, m).

Step 3. Conclusion. — We now notice that

t→0lim|Ωt|=|Ω0|>0.

Thus, for everyε >0, there isTεsuch that for allt∈(0, Tε)we have Z

Ω_t

|∇u|= Z t

0

Per(Ωs)ds6 Z

Ω_t

|∇u|² 1/2

|ΩtrΩ0|^1/2 6tε^1/2Per(Ωt)^1/2|Ωt|^1/2. (4.12)

Now, repeating the argument fro Step 1 and Step 2, we get that (4.11) should hold withεin place ofβ. Sinceε >0is arbitrary, this is a contradiction.

4.4. Existence of a solution. — We are now in position to prove that the pairs (u0,Ω0), constructed in Section 4.1, is a solution to (1.1).

Proposition4.3(Existence of a solution). — There is a dimensional constantCd >0 such that ifDis a bounded open set ofR^d andβ >0is a given positive constant, then the following holds. For every set E⊂R^d of finite perimeter and every v∈H¹(R^d) satisfying

v>m onD for some constant m >0, there is a solution (u,Ω)of the problem (1.1).

Proof. — Let (u0,Ω0) be as in Section 4.1. Then, by Lemma 4.1, (u0,Ω0) satisfies the optimality condition (4.5). Now, by Proposition 4.2 we get that u0>tin D, for somet >0. In particular,Ω0has finite perimeter inD. Precisely, for everyδ >0, we have

Per(Ω0;Dδ)6lim inf

n→∞ Per(Ωε_n;Dδ)6 4 t²lim inf

n→∞

Z

D_δ∩∂^∗Ω_εn

u²_ε

ndH^d−1

6 4

βt²lim inf

n→∞ J_β u_εn,Ω_εn 6 4

βt²J_β(v, E).

Passing to the limit asδ→0, we get

Per(Ω0;D)6 4

βt²Jβ(v, E).

In particular, this implies thatΩ₀ is a set of finite perimeter inR^d. Indeed, Per(Ω0)6Per(Ω0;D) + 2Per(D) + Per(Ω0;R^drD)

6 4

βt²J_β(v, E) + 2Per(D) + Per(E;R^drD).

Thus, the pairs (u₀,Ω₀) is admissible in (1.1); it now remains to prove that it is optimal. Let eu∈ H¹(D) be non-negative on D and such that u−v ∈ H₀¹(D). Let Ωe ⊂R^d be a set of finite perimeter such that Ω =e E on R^drD. It is sufficient to prove that

Jβ(u0,Ω0)6Jβ(u,e Ω).e

Letε >0be fixed. We now use the pairs(eu∨ε,Ω)e to test the optimality of u_εn,Ω_εn : Jβ uε_n,Ωε_n

6Jβ(ue∨ε,Ω).e

Passing to the limit asε→0, we get J_β u_εn,Ω_εn

6J_β(u,e Ω).e

Now, Lemma 2.4 and the semicontinuity of the H¹ norm gives that J_β(u₀,Ω₀) 6

Jβ(u,e Ω), which concludes the proof.e