A two-phase problem with Robin conditions on the free boundary
Full text
This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ A two-phase problem with Robin conditions on the free boundary © 2021 the Authors Published version Guarino Lo Bianco, Serena; La Manna, Domenico Angelo; Velichkov, Bozhidar Guarino Lo Bianco, S., La Manna, D. A., & Velichkov, B. (2021). A two-phase problem with Robin conditions on the free boundary. Journal de l'École polytechnique : Mathématiques, 8, 1-25. https://doi.org/10.5802/jep.139 2021
Serena Guarino Lo Bianco, Domenico Angelo La Manna, &Bozhidar Velichkov A two-phase problem with Robin conditions on the free boundary Tome 8 (2021), p. 1-25. <http://jep.centre-mersenne.org/item/JEP_2021__8__1_0> © Les auteurs, 2021. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D’ATTRIBUTION CREATIVE COMMONS BY 4.0. https://creativecommons.org/licenses/by/4.0/ L’accès aux articles de la revue « Journal de l’École polytechnique — Mathématiques » (http://jep.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jep.centre-mersenne.org/legal/). Publié avec le soutien du Centre National de la Recherche Scientifique Publication membre du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org
Tome 8, 2021, p.1–25 DOI: 10.5802/jep.139 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 2020 Mathematics Subject Classification. — 35R35, 49Q10. Keywords. — Free boundary problems, two-phase, Robin boundary conditions, regularity. The first author was partially supported by PRIN 2017 Nonlinear Differential Problems via Variational, 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). e-ISSN: 2270-518X http://jep.centre-mersenne.org/
2 S. Guarino Lo Bianco, D. A. La Manna & B. Velichkov 1. Introduction For a fixed a constant β > 0and a smooth bounded open set D⊂Rd,d⩾2, we consider the functional Jβ(u, Ω) = ZD |∇u|2dx +βZ∂∗Ω u2dHd−1, defined on the pairs (u, Ω), where u∈H1(D),Ω⊂Rdis a set of finite perimeter in the sense of De Giorgi (see Section 2) and ∂∗Ωdenotes the reduced boundary of Ω (see Section 2); when Ωis smooth, ∂∗Ωis the topological boundary of Ω. In this paper we study the existence and the regularity of minimizers of the functional Jβamong all pairs (u, Ω), which are fixed outside the domain D. Precisely, throughout the paper, we fix a set E⊂Rdof finite perimeter, a constants m > 0and a function v∈H1 loc(Rd)such that v⩾min Rdand Z∂∗E v2dHd−1<+∞; we define the admissible sets V=u∈H1 loc(Rd) : u−v∈H1 0(D), E=Ω⊂Rd: Per(Ω) <+∞and Ω = Ein RdrD, and we consider the variational minimization problem (1.1) min Jβ(u, Ω) : u∈ V,Ω∈ E. Our main result is the following. Theorem 1.1 (Existence and regularity of minimizers). — Let β >0,D⊂Rd,v,E,V and Ebe 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 in D. (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 part Sing(∂Ω), where: –Reg(∂Ω) is a C∞hypersurface and a relatively open subset of ∂Ω, and the function uis C∞smooth on Reg(∂Ω); –Sing(∂Ω) is a closed set, which is empty if d⩽7, discrete if d= 8, and of Hausdorff dimension d−8, if d > 8. Remark 1.2. — We notice that if (u, Ω) is a solution to (1.1), then uis harmonic in the interior of Ωand DrΩ. Thus, as a consequence of Theorem 1.1(iii), in a neighborhood of a regular point x0∈Reg(∂Ω), the functions u: Ω →Rand u:DrΩ→Rare C∞ up to the free boundary ∂Ω. J.É.P.—M., 2021, tome8
A two-phase problem with Robin conditions on the free boundary 3 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 domains D symmetric with respect to the hyperplane {xd= 0}. 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 (un,Ωn)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 functions unto be bounded from below by a strictly positive constant. Now, notice that we cannot simply replace unby un∨ε, for some ε > 0; this is due to the fact that the second term in Jβis increasing in u: Z∂∗Ωn u2 ndHd−1⩽Z∂∗Ωn (ε∨un)2dHd−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 Jβ(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 un, so we still miss the uniform bound on the perimeter of Ωn. On the other hand, we are able to prove that the sequence unis uniformly Hölder continuous in D(see Section 3, Lemma 3.5). This enables us to extract a subsequence unthat converges locally uniformly in Dto a non-negative Hölder continuous function u∞:D→R(see Section 4). Now, on each of the sets {u∞> t},t > 0, the sequence Ωnhas uniformly bounded perimeter. This enables us to extract a subsequence Ωnthat 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 (un,Ωn)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∞)⩽Jβ(ut,Ωt),where ut=u∞∨tand Ωt= Ω∞∪ {u∞⩽t}, 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 Rdand that the pairs (u∞,Ω∞)is a solution to (1.1). J.É.P.—M., 2021, tome8
4 S. Guarino Lo Bianco, D. A. La Manna & B. Velichkov 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 uit 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 > 0small enough, we have (1.4) Jβ(u, Ω) ⩽Jβ(ut,Ωt)where ut=u∨tand Ωt= Ω ∪ {u⩽t}. In particular, ZD |∇u|2dx +βZ∂∗Ω u2⩽ZD |∇(u∨t)|2dx +βZ∂∗(Ω∪{u<t}) u2 ⩽ZD |∇(u∨t)|2dx +βt2Per({u < t}) + βZ{u>t}∩∂∗Ω u2, which proves that usatisfies the optimality condition (4.1) from Lemma 4.1: (1.5) Z{u<t} |∇u|2dx ⩽β t2Per{u<t}. Now, applying Proposition 4.2, we get that uis bounded from below by a strictly positive constant in D. Finally, Proposition 3.5 gives that uis 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 uto show that a solution Ωis 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 ∂Ωcan be decomposed into a C1,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 x0, the restrictions u+and u−of uon Ωand DrΩare solutions of the following transmission problem: ∆u+= 0 in Ω, ∆u−= 0 in DrΩ, u+=u−=uon ∂Ω, ∂u+ ∂νΩ −∂u− ∂νΩ + 2βu = 0 on ∂Ω, where νΩis 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, J.É.P.—M., 2021, tome8
A two-phase problem with Robin conditions on the free boundary 5 where Fis an explicit (rational) function of ∇u±and u. In particular, this implies that ∂Ωgains one more derivative with respect to u, that is, u∈Ck,α ⇒∂Ω∈Ck+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 (ut,Ωt)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∩ {u⩽t}| ⩽Jβ(ut,Ωt),where ut=u∨tand Ωt= Ω r{u⩽t}, where C > 0. The presence of the constant Cenables 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 uand {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 (1.6) c⩽β1/2Per({u<t})1/2|{u < t}|1/2for every t > 0, where u:= u∞and cis a constant depending on βand d. Then, we notice that this entails c⩽β3/4Per({u<t})1/4|{u < t}|3/4for every t > 0. and we use an iteration procedure to get that c⩽β1−1/2nPer({u < t})1/2n|{u < t}|1−1/2nfor every t > 0. Passing to the limit as n→ ∞, we get that if uis 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 > 0in 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]. J.É.P.—M., 2021, tome8
6 S. Guarino Lo Bianco, D. A. La Manna & B. Velichkov 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 if Ωis a fixed set with smooth boundary and if uminimizes the functional Jβ(·,Ω) in H1(D), then the functions u+:= uon Ωand u−:= uon DrΩ, are harmonic in Ωand DrΩ, and satisfy the following conditions: (1.8) u+=u−and ∂u+ ∂ν+ +β 2u++∂u− ∂ν− +β 2u−= 0 on ∂Ω∩D, 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 (1.9) ∆u= 0 in Ω,∂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 in Ω, the temperature is set to be zero on the surface ∂Ω, which is separated from the interior of Ωby 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 Ωand 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 in Ωto zero in DrΩ. 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= 0 almost-everywhere in D) which satisfy the following constraints: the sum u1+u2should be a Sobolev function (this corresponds to the continuity condition in (1.8)); u2 1and u2 2are 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 that u1and u2are of the form u1=u 1 Ωand u2=u 1 DrΩfor a set of finite perimeter Ω⊂Rd,ubeing 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 J.É.P.—M., 2021, tome8
A two-phase problem with Robin conditions on the free boundary 7 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 uis non-degenerate and Hölder continuous, the regularity of the free boundary ∂Ωfollows immediately since the set Ωbecomes an almost-minimizer of the perimeter. 2. Preliminaries 2.1. Sets of finite perimeter. — Let A⊂Rdbe a an open set in Rd. We recall that the set E⊂Rdis said to have a finite perimeter in Aif (2.1) Per(E, A) = sup nZA div ξ(x)dx :ξ∈C1 c(A;Rd),sup x∈Rd |ξ(x)|⩽1o is finite. We say that Ehas a locally finite perimeter in A, if for every open set B⊂Rd such that B⊂A, we have that Per(E, B)<∞. We say that Eis of finite perimeter if Per(E) := Per(E, Rd)<+∞. By the De Giorgi structure theorem (see for instance [7, Th. II.4.9]), if the set E⊂Rd has locally finite perimeter in A, then there is a set ∂∗E⊂A∩∂E called reduced boundary such that Per(E, B) = Hd−1(B∩∂∗E)for every set BbA, where Hd−1is the (d−1)-dimensional Hausdorff measure in Rd. Moreover, there is aHd−1-measurable function νE:∂∗E→Rd, called generalized normal such that |νE|= 1 and ZE div ξ(x)dx =Z∂∗E νE·ξ dHd−1for every ξ∈C1 c(A;Rd). 2.2. Capacity and traces of Sobolev functions. — We define the capacity (or the 2-capacity) of a set E⊂Rdas cap(E) = inf kuk2 H1(Rd):u∈H1(Rd), u ⩾1in a neighborhood of E. Suppose now that d⩾3. It is well-known that the sets of zero capacity have zero d−1dimensional Hausdorff measure (see for instance [6, §4.7.2, Th. 4]): If cap(E)=0,then Hd−1(E)=0. The Sobolev functions are defined up to a set of zero capacity (i.e., quasi-everywhere), that is, if A⊂Rdis an open set and u∈H1(A), then there is a set Nu⊂Rdsuch that cap (Nu)=0and u(x0) = lim r→0 1 |Br|ZBr(x0) u(x)dx for every x0∈ArNu. Moreover, for every function u∈H1(A)there is a sequence un∈C∞(A)∩H1(A) and a set N ⊂ Aof zero capacity such that: –unconverges to ustrongly in H1(A); –u(x) = limn→∞ un(x)for every x∈Ar(N ∪ Nu). J.É.P.—M., 2021, tome8
14 S. Guarino Lo Bianco, D. A. La Manna & B. Velichkov The construction of Ω0is more delicate. First, we fix t > 0and δ > 0and we notice that the perimeter of Ωεnis bounded on the open set {u0> t} ∩ Dδ. Indeed, the uniform convergence of uεnto u0implies that, for nlarge enough (n⩾Nt,δ, for some fixed Nt,δ ∈N), uεn⩾t 2on Dδ∩ {u0> t}. Thus, we have Jβ(v, E)⩾βZDδ∩{u0>t}∩∂∗Ωεn u2 εndHd−1⩾βt2 2PerΩεn;Dδ∩ {u0> t}. Now, if we choose tsuch that Per({u0> t})<∞(which, by the co-area formula, is true for almost-every t > 0), then we have that PerΩεn∩ {u0> t} ∩ Dδ⩽Ct,δ for every n⩾Nt,δ, for some constant Ct,δ >0. Now, since all the sets Ωεn∩ {u0> t} ∩ Dδare contained in Dand have uniformly bounded perimeter, we can find a set Ω0and a subsequence for which 1 Ωεn∩{u0>t}∩Dδ(x)−→ 1 Ω0∩{u0>t}∩Dδ(x)for almost-every x∈D. Thus, by a diagonal sequence argument, we can extract a subsequence of εn(still denoted by εn) and we can define the set Ω0⊂Rdas the pointwise limit 1 Ω0(x) = lim n→∞ 1 Ωεn∩{u0>0}(x)for almost-every x∈ {u0>0}, and we notice that, by construction, Ω0⊂ {u0>0}. Notice that, we do not know a priori that Ω0has finite perimeter. We only know that Per (Ω0∩ {u0> t} ∩ Dδ)<∞for every δ > 0and almost-every t > 0. which means that Ω0∩ {u0> t}has locally finite perimeter in Dfor 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. Lemma 4.1 (The optimality condition at the limit). — Let u0and Ω0be as in Section 4.1. Then, for almost-every t > 0, we have (4.1) Z{u0<t} |∇u0|2dx ⩽β t2Per{u0< t}. Proof. — Let now t > 0be fixed and such that the set {u0< t}has finite perimeter. Then, for nlarge 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 J.É.P.—M., 2021, tome8
A two-phase problem with Robin conditions on the free boundary 15 a.e. t∈(0, m), as observed in the previous section. For the sake of simplicity, we write uεn=un,Ωεn= Ωn,u0=uand Ω0= Ω. Thus, we have ZD |∇un|2dx+βZ{u>t}∩∂∗Ωn u2 ndHd−1 ⩽ZD |∇un|2dx +βZ∂∗Ωn u2 ndHd−1 ⩽ZD |∇(u∨t)|2dx +βZ∂∗(Ω∪{u<t}) u2dHd−1 (4.2) ⩽ZD |∇(u∨t)|2dx +βt2Per({u < t}) + βZ{u>t}∩∂∗Ω u2dHd−1. Now, by the weak convergence of unto u, we get that ZD |∇u|2dx ⩽lim inf n→∞ ZD |∇un|2dx. On the other hand, setting Ut,δ to be the open set Ut,δ =RdrDδ∩ {u⩽t}, for some fixed δ > 0, and applying Lemma 2.4, we have that ZUt,δ∩∂∗Ω u2dHd−1⩽lim inf n→∞ ZUt,δ∩∂∗Ωn u2 ndHd−1⩽lim inf n→∞ Z{u>t}∩∂∗Ωn u2 ndHd−1. Taking the limit as δ→0, by the monotone convergence theorem, we get that lim δ→0ZUt,δ∩∂∗Ω u2dHd−1=ZRdr(D∩{u⩽t})∩∂∗Ω u2dHd−1 Now, since u(x) = h(x)for quasi-every x∈RdrDand for Hd−1-almost-every x∈RdrD, and since h⩾m > t on ∂D, we have that (4.3) ZRdr(D∩{u⩽t})∩∂∗Ω u2dHd−1=Z{u>t}∩∂∗Ω u2dHd−1. Thus, we get that (4.4) Z{u>t}∩∂∗Ω u2dHd−1⩽lim inf n→∞ ZD∩{u>t}∩∂∗Ωn u2 ndHd−1. Now, using (4.4) and (4.2), we obtain ZD |∇u|2dx+βZ{u>t}∩∂∗Ω u2dHd−1 ⩽lim inf n→∞ ZD |∇un|2dx +βZ{u>t}∩∂∗Ωn u2 ndHd−1 ⩽ZD |∇(u∨t)|2dx +βt2Per({u < t}) + βZ{u>t}∩∂∗Ω u2dHd−1, which gives (4.1). J.É.P.—M., 2021, tome8
16 S. Guarino Lo Bianco, D. A. La Manna & B. Velichkov 4.3. Non-degeneracy. — The crucial observation in this section is that the functions usatisfying the optimality condition (4.1) are non-degenerate in the sense of the following proposition. Proposition 4.2 (Non-degeneracy). — Let β > 0,m > 0,Dbe a bounded open set of Rdand u∈H1(D)be a non-negative function in Dsuch that u⩾mon ∂D. Let Ω⊂Dbe a set of finite perimeter in D. Suppose that uand Ωsatisfy the optimality condition (4.5) ZΩt |∇u|2dx ⩽β t2Per(Ωt)where Ωt={u⩽t}, for almost-every t∈(0, m). Then, |Ωt|= 0 for some t > 0. Proof. — By contradiction, suppose that |Ωt|>0for every t > 0. Let t∈(0, m)be fixed. By the co-area formula, the Cauchy-Schwartz inequality and the optimality condition (4.5), we get (4.6) ZΩt |∇u|=Zt 0 Per(Ωs)ds ⩽ZΩt |∇u|21/2 |Ωt|1/2⩽tβ1/2Per(Ωt)1/2|Ωt|1/2. We now set f(t) := Zt 0 Per(Ωs)ds =ZΩt |∇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 Cdsuch that Zt 0 Per(Ωs)ds ⩽tβ1/2CdPer(Ωt)(2d−1)/(2d−2). Using the definition of f, we can re-write this inequality as f(t)(2d−2)/(2d−1) ⩽t(2d−2)/(2d−1)β1/2Cd(2d−2)/(2d−1)f0(t). After rearranging the terms and integrating from 0to t, we obtain f(t)1/(2d−1) −f(0)1/(2d−1) ⩾t1/(2d−1) β1/2Cd(2d−2)/(2d−1) . Now, since uis non-negative in D, we have that f(0) = 0. Thus f(t)⩾t β1/2Cd2d−2. Setting (4.7) C=βCd1−d, we obtain the lower bound f(t)⩾Ct. J.É.P.—M., 2021, tome8
A two-phase problem with Robin conditions on the free boundary 17 In particular, as a consequence of (4.6), we get that (4.8) C⩽β1/2Per(Ωt)1/2|Ωt|1/2. Step 2. Non-degeneracy of |Ωt|. — Let α∈(0,1) be fixed. Then, we have that Zt 0 Per(Ωs)α|Ωs|1−αds ⩽Zt 0 Per(Ωs)dsαZt 0 |Ωs|ds1−α ⩽tβ1/2Per(Ωt)1/2|Ωt|1/2αt|Ωt|1−α =tβα/2Per(Ωt)α/2|Ωt|1−α/2. Thus, we obtain that for fixed T∈(0, m)and C > 0, the following implication holds: (4.9) (If C⩽Per(Ωt)α|Ωt|1−αfor every t∈(0, T ), then C⩽βα/2Per(Ωt)α/2|Ωt|1−α/2for every t∈(0, T ). We claim that, for every n⩾1and every t∈(0, m), we have the inequality (4.10) C⩽β1−1/2nPer(Ωt)1/2n|Ωt|1−1/2n. 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∈Nimplies 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 C⩽β|Ωt|for every t∈(0, T), where Cis given by (4.7). Thus, there is a dimensional constant Cd>0such that (4.11) β−dCd⩽|Ωt|for every t∈[0, m). Step 3. Conclusion. — We now notice that lim t→0|Ωt|=|Ω0|>0. Thus, for every ε > 0, there is Tεsuch that for all t∈(0, Tε)we have ZΩt |∇u|=Zt 0 Per(Ωs)ds ⩽ZΩt |∇u|21/2 |ΩtrΩ0|1/2 ⩽tε1/2Per(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 with εin place of β. Since ε > 0is arbitrary, this is a contradiction. J.É.P.—M., 2021, tome8
18 S. Guarino Lo Bianco, D. A. La Manna & B. Velichkov 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). Proposition 4.3 (Existence of a solution). — There is a dimensional constant Cd>0 such that if Dis a bounded open set of Rdand β > 0is a given positive constant, then the following holds. For every set E⊂Rdof finite perimeter and every v∈H1(Rd) satisfying v⩾mon Dfor 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 some t > 0. In particular, Ω0has finite perimeter in D. Precisely, for every δ > 0, we have Per(Ω0;Dδ)⩽lim inf n→∞ Per(Ωεn;Dδ)⩽4 t2lim inf n→∞ ZDδ∩∂∗Ωεn u2 εndHd−1 ⩽4 βt2lim inf n→∞ Jβuεn,Ωεn⩽4 βt2Jβ(v, E). Passing to the limit as δ→0, we get Per(Ω0;D)⩽4 βt2Jβ(v, E). In particular, this implies that Ω0is a set of finite perimeter in Rd. Indeed, Per(Ω0)⩽Per(Ω0;D) + 2Per(D) + Per(Ω0;RdrD) ⩽4 βt2Jβ(v, E) + 2Per(D) + Per(E;RdrD). 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 Dand such that u−v∈H1 0(D). Let e Ω⊂Rdbe a set of finite perimeter such that e Ω = Eon RdrD. It is sufficient to prove that Jβ(u0,Ω0)⩽Jβ(eu, e Ω). Let ε > 0be fixed. We now use the pairs (eu∨ε, e Ω) to test the optimality of uεn,Ωεn: Jβuεn,Ωεn⩽Jβ(eu∨ε, e Ω). Passing to the limit as ε→0, we get Jβuεn,Ωεn⩽Jβ(eu, e Ω). Now, Lemma 2.4 and the semicontinuity of the H1norm gives that Jβ(u0,Ω0)⩽ Jβ(eu, e Ω), which concludes the proof. J.É.P.—M., 2021, tome8
A two-phase problem with Robin conditions on the free boundary 19 5. 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, ∂Ωcan be decomposed into a regular and a singular part and that the regular part is C1,α manifold. Then, in Theorem 5.2, we prove that the regular part of the free boundary is C∞smooth. Theorem 5.1. — Let (u, Ω) be a solution to (1.1). there is a constant C > 0such that Ωis an almost-minimizer of the perimeter in the following sense: Per Ω ; Br(x0)⩽1 + Cr1/3Per Ω0;Br(x0), for every ball Br(x0)⊂Dand every set Ω0⊂Rdsuch that Ω=Ω0outside Br(x0). In particular, the free boundary ∂Ω∩Dcan be decomposed as the disjoint union of a regular part Reg(∂Ω) and a singular part Sing(∂Ω), where (i) Reg(∂Ω) is a relatively open subset of ∂Ωand is a C1,α smooth manifold; (ii) Sing(∂Ω) is a closed set, which is empty if d⩽7, discrete if d= 8, and of Hausdorff dimension d−8, if d > 8. Proof. — We first notice that by Lemma 3.4, u∈C0,1/3(D). Let δ > 0,x0∈Dδand r < δ/2. We consider a set Ω0⊂Rdsuch that Ω0∆Ω bBr(x0). Testing the optimality of (u, Ω) against (u, Ω0)we get that ZBr(x0)∩∂∗Ω u2dHn−1⩽ZBr(x0)∩∂∗Ω0 u2dHn−1, which implies that min Br(x0)u2PerΩ ; Br(x0)⩽max Br(x0)u2PerΩ0;Br(x0). By regularity of u, we have that max Br(x0)u2⩽min Br(x0)u2+Cr1/3⩽min Br(x0)u21 + C tr1/3, where in the second inequality, we used that u⩾t > 0. Thus, we obtain PerΩ ; Br(x0)⩽1 + C tr1/3PerΩ0;Br(x0), which proves that Ωis an almost-minimizer of the perimeter in D. We next prove that regular part the free boundary Reg(∂Ω) is C∞. Theorem 5.2. — Let (u, Ω) be a solution to (1.1). Let D∩∂Ω = Reg(∂Ω) ∪Sing(∂Ω) be the decomposition of the free boundary from Theorem 5.1. Then, in a neighborhood of any point x0∈Reg(∂Ω),∂Ωis C∞-regular and the function uis C∞on ∂Ω. Proof. — We fix a point x0∈Reg(∂Ω). Without loss of generality, we assume x0= 0. J.É.P.—M., 2021, tome8
20 S. Guarino Lo Bianco, D. A. La Manna & B. Velichkov Step 1. Notation. — For any x∈Rd, we use the notation x= (x0, xd), where x0∈Rd−1 and xd∈R. By the C1,α regularity of Reg(∂Ω), in B0×(−ε, ε)⊂Rd−1×R,∂Ωis the graph of a C1,α regular function η:B0→R, where B0is a ball in Rd−1; the set Ω coincides with the subgraph of ηin a neighborhood of the origin: B0×(−ε, ε)∩Ω = (x0, xd)∈B0×(−ε;ε) : xd< η(x0). and the exterior normal νΩis given by (5.1) νΩ=(−∇x0η, 1) p1 + |∇x0η|2, where ∇x0ηis the gradient of ηin the first d−1variables. Let u+and u−be the restrictions of uon the sets Ωand DrΩ; since uis continuous across ∂Ω, we have u+=u−on ∂Ω. Moreover, we write the gradients of u+and u−as ∇u±=∇x0u±, ∂xdu±∈Rd−1×R. Step 2. Transmission condition and C1,α regularity of u. — In Lemma 5.3, we keep fixed the free boundary ∂Ωand we use vertical perturbations of the function uto obtain a Robin-type transmission condition on ∂Ω. We notice that the recent results [4, 5] imply the C1,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 C2,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 uis Ck,α, for some k⩾1, then ∂Ωis Ck+1,α. Step 4. Bootstrap. — In Lemma 5.4 we use the recent results of [5] to show that if the boundary ∂Ωis Ck,α for some k⩾2, then the solutions u+and u−are also Ck,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 ∂Ωis C∞. Lemma 5.3 (Robin and continuity conditions on ∂Ω). — Suppose that ∂Ωis C1,α regular in the neighborhood of the origin. Let η:B0→R,u+and u−be as above. Then, for every x0∈B0we have (5.2) (∇x0η· ∇x0u+− ∇x0η· ∇x0u−=−∂xdu+−∂xdu−|∇x0η|2 p1 + |∇x0η|2∂xdu+−∂xdu−+βu = 0, where u+,u−and their partial derivatives are calculated in (x0, η(x0)) ∈∂Ω. J.É.P.—M., 2021, tome8
A two-phase problem with Robin conditions on the free boundary 21 Proof. — Let φ∈C∞ c(D)be a smooth function supported in B0×(−ε, ε). Then, the optimality of ugives that 0 = ∂ ∂tt=0Jβ(u+tφ, Ω) = ZDr∂Ω 2∇u· ∇φ dx +βZ∂Ω 2uφ dHd−1 =Z∂Ω 2νΩ· ∇u+−νΩ· ∇u−+βuφ dHd−1, where in the last inequality we integrated by parts u+in Ωand u−in DrΩ. Since φ is arbitrary we get that usatisfies the Robin-type condition on ∂Ω (5.3) νΩ· ∇u+−νΩ· ∇u−+βu on ∂Ω. Now, using (5.1), we can re-write this as (5.4) − ∇x0η· ∇x0u++∂xdu+−− ∇x0η· ∇x0u−+∂xdu−+βup1 + |∇x0η|2= 0. On the other hand uis continuous across ∂Ω. This means that ∇x0u+(x0, η(x0)) + ∂xdu+(x0, η(x0))∇x0η=∇x0u−(x0, η(x0)) + ∂xdu−(x0, η(x0))∇x0η. Multiplying by ∇x0η, we get (5.5) ∇x0η· ∇x0u++∂xdu+|∇x0η|2=∇x0η· ∇x0u−+∂xdu−|∇x0η|2, where u+,u−and their partial derivatives are calculated in (x0, η(x0)). Putting together (5.4) and (5.5), we get (5.2). Lemma 5.4 (Smooth boundary ⇒smooth function). — Let (u, Ω) be a solution of (1.1). Suppose that, in a neighborhood of zero, ∂Ωis Ck,α-regular for some k⩾1. Then, in a neighborhood of the origin, the functions u+and u−are Ck,α up to the boundary ∂Ω. Proof. — We argue by induction. The case k= 1 follows by [5]. We suppose that k⩾2and that the claim holds for k−1. Suppose that ∂Ωis the graph of η:B0→R, η∈Ck,α(B0), and consider the functions v+(x0, xd) := u+(x0, xd+η(x0)) and v−(x0, xd) := u−(x0, xd+η(x0)), defined on the half-space {xd⩾0}. We set Aη=Nd−1−(∇x0η)t −∇x0η|∇x0η|2, where Nd−1is the null (d−1) ×(d−1) matrix and we notice that Aηhas Ck−1,α regular coefficients. Now, since u+and u−are harmonic in Ωand DrΩ, we have that v+and v−are solutions to the transmission problem −div((Id + Aη)∇v+)=0 in {xd>0} −div((Id + Aη)∇v−)=0 in {xd<0} v+=v−on {xd= 0} ∂xdv+−∂xdv−+β 2p1 + |∇x0η|2(v++v−)=0 on {xd= 0}. J.É.P.—M., 2021, tome8
22 S. Guarino Lo Bianco, D. A. La Manna & B. Velichkov We now fix k−1directions i1, . . . , ik−1,ij6=dfor every j, and we consider the functions w+:= ∂i1∂i2. . . ∂ik−1v+and w−:= ∂i1∂i2. . . ∂ik−1v−. We notice that, in {xd>0}and {xd<0}the functions w+and w−are solutions to −div(Id + Aη)∇w±+X I,J div∂IAη∂J∇u±= 0, where the sum is over all multiindices Iand Jsuch that the sets Iand Jare disjoint subsets of {i1, i2, . . . , ik−1},I∪J={i1, i2, . . . , ik−1}and Iis non-empty. In particular, using that Aη∈Ck−1,α and ∇u∈Ck−2,α (since by hypothesis u±∈Ck−1,α), we get that w±solve −div(Id + Aη∇w±) + div(F±)=0 in {±xd>0}, where F+and F−are C0,α continuous functions (depending on i1, . . . , ik). On the other hand, on the boundary {xd= 0}we have that w+=w−and ∂xdw+−∂xdw−+∂i1∂i2. . . ∂ik−1β(u++u−) 2p1 + |∇x0η|2= 0 on {xd= 0}. Reasoning as above, we notice that this condition can be written as ∂xdw+−∂xdw−=gon {xd= 0}, where gis a C0,α function. Now, applying [5, Th. 1.2], we get that w+and w−are C1,α regular up to the boundary {xd= 0}. Thus, the trace u+=u−is Ck,α smooth on {xd= 0}. Finally, the classical Schauder estimates give that u+and u−are Ck,α on {xd⩾0}and {xd⩽0}, respectively. Lemma 5.5 (Smooth function ⇒smooth boundary). — Let (u, Ω) be a solution of (1.1). Suppose that, in a neighborhood of zero, ∂Ωis C1,α-regular and that the functions u+and u−are Ck,α up to the boundary ∂Ω, for some k⩾1. Then, ∂Ωis Ck+1,α-regular in a neighborhood of zero. Proof. — Let ξ∈C∞ c(D;Rd)be a given vector field with compact support in Dand let Ψtbe the function Ψt(x) = x+tξ(x)for every x∈D. Then, for tsmall enough, Ψt:D→Dis a diffeomorphism and setting Φt:= Ψ−1 t, the function ut:= u◦Φtis well-defined and belongs to H1(D); the function t7−→ ZD |∇ut|2dx is differentiable at t= 0 and ∂ ∂tt=0 ZD |∇ut|2dx =ZD−2∇u Dξ · ∇u+|∇u|2div ξdx. It is immediate to check that −2∇u Dξ · ∇u+|∇u|2div ξ=div|∇u|2ξ−2(ξ· ∇u)∇uin Dr∂Ω. J.É.P.—M., 2021, tome8
A two-phase problem with Robin conditions on the free boundary 23 We now take ξto be smooth outside ∂Ωand such that ξ=φνΩon ∂Ω, where νΩis the exterior normal to ∂Ωand φ:∂Ω→Ris continuous and with compact support. Integrating by parts, we get ∂ ∂tt=0 ZD |∇ut|2dx =Z∂Ω|∇u+|2(ξ·νΩ)−2(ξ· ∇u+)(νΩ· ∇u+)dHd−1 −Z∂Ω|∇u−|2(ξ·νΩ)−2(ξ· ∇u−)(νΩ· ∇u−)dHd−1, where u+:= uon Ω, and u−:= uon DrΩ. Now, if ξ=φedand νΩ=(−∇x0η, 1) p1 + |∇x0η|2, then p1 + |∇x0η|2|∇u+|2(ξ·νΩ)−2(ξ· ∇u+)(νΩ· ∇u+) −p1 + |∇x0η|2|∇u−|2(ξ·νΩ)−2(ξ· ∇u−)(νΩ· ∇u−) =φ|∇u+|2− |∇u−|2 −2φ∂xdu+− ∇x0η· ∇x0u++∂xdu+−∂xdu−− ∇x0η· ∇x0u−+∂xdu−. We now suppose that x0∈Reg(∂Ω) and that ∂Ωis the graph of the (C1,α) function η:B0→R, where B0is a ball in Rd−1. Taking ξ=edφand Ωt= Φt(Ω), we have ∂ ∂tt=0 Z∂Ωt u2 tdHd−1=∂ ∂tt=0 ZB0 u2x0, η(x0)p1 + |∇x0η+t∇x0φ|2dx0 =ZB0 u2x0, η(x0) p1 + |∇x0η|2∇x0η· ∇x0φ dx0 =ZB0 u2x0, η(x0)Hx0, η(x0)φ(x0)dx0 −2ZB0 φ(x0)ux0, η(x0)∇x0u+∂xdu∇x0η· ∇x0η p1 + |∇x0η|2dx0. In particular, combining these two computations and using the optimality of (u, Ω), we get 0 = ∂ ∂tt=0Jβ(ut,Ωt) = ZB0 βu2H(x0)φ(x0)dx0+ZB0|∇u+|2− |∇u−|2φ(x0)dx0 −ZB0 21 + |∇x0η|2(∂xdu+)2−(∂xdu−)2φ(x0)dx0 Since φis arbitrary, we obtain that ηis a solution of the problem −divx0∇x0η p1 + |∇x0η|2=f(x0)in B0, J.É.P.—M., 2021, tome8