A regularity result for minimal configurations of a free interface problem

We prove a regularity result for minimal configurations of variational problems involving both bulk and surface energies in some bounded open region Ω⊆Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega \subseteq {\mathbb {R}}^n$$\end{document}. We will deal with the energy functional F(v,E):=∫Ω[F(∇v)+1EG(∇v)+fE(x,v)]dx+P(E,Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}(v,E):=\int _\varOmega [F(\nabla v)+1_E G(\nabla v)+f_E(x,v)]\,dx+P(E,\varOmega )$$\end{document}. The bulk energy depends on a function v and its gradient ∇v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla v$$\end{document}. It consists in two strongly quasi-convex functions F and G, which have polinomial p-growth and are linked with their p-recession functions by a proximity condition, and a function fE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_E$$\end{document}, whose absolute valuesatisfies a q-growth condition from above. The surface penalization term is proportional to the perimeter of a subset E in Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document}. The term fE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_E$$\end{document} is allowed to be negative, but an additional condition on the growth from below is needed to prove the existence of a minimal configuration of the problem associated with F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}$$\end{document}. The same condition turns out to be crucial in the proof of the regularity result as well. If (u, A) is a minimal configuration, we prove that u is locally Hölder continuous and A is equivalent to an open set A~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{A}}$$\end{document}. We finally get P(A,Ω)=Hn-1(∂A~∩Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(A,\varOmega )={\mathscr {H}}^{n-1}(\partial {\tilde{A}}\cap \varOmega $$\end{document}).


Introduction and statement
The problem of finding the minimal energy configuration of a mixture of two materials in a bounded open set Ω ⊆ R n , penalized by the perimeter of the contact interface between the two materials, has been fully examined in mathematical literature (see for example [2,3,6,8,10,15,[17][18][19][20]).
Let p > 1 and define A (Ω) as the set of all subsets of Ω with finite perimeter. Consider F, G ∈ C 1 (R n ) and define f E := g + 1 E h, where E ∈ A (Ω) and g, h : Ω × R → R are two Borel measurable and lower semicontinuous functions with respect to the real vari-B Lorenzo Lamberti llamberti@unisa.it 1 Dipartimento di Matematica, Universitá degli Studi di Salerno, Fisciano, Italy able. We will deal with the following energy functional: where (v, E) ∈ u 0 +W 1, p 0 (Ω) ×A (Ω), with u 0 ∈ W 1, p (Ω). The regularity of minimizers (u, A) of the functional F was recently investigated in [6,9,10] for the constrained problem where the volume of the region A in Ω is prescribed but the forcing term f A is zero. In the quadratic case p = 2 Ambrosio and Buttazzo [3] proved the regularity for minimizers of F in the case that f A is not zero. We are going to extend this result to functionals with polinomial growth.
We assume that there exist some positive constants l, L, α, β and μ ≥ 0 such that -F and G have p-growth: for all ξ ∈ R n . -F and G are strongly quasi-convex: for all ξ ∈ R n and ϕ ∈ C 1 c (Ω). -there exist two positive constants t 0 , a and 0 < m < p such that for every t > t 0 and ξ ∈ R n with |ξ | = 1, it holds where F p and G p are the p-recession functions of F and G (see Definition 2.1).
We remark that the proximity conditions (F3) and (G3) are trivially satisfied if F and G are positively p-homogeneous. The first of the following assumptions on g and h is essential to prove the existence of a minimal configuration. The same condition turns out to be crucial in the proof of the regularity result as well. We assume that there exist a function γ ∈ L 1 (Ω) and two constants C 0 > 0 and k ∈ R, with k < l 2 p−1 λ , being λ = λ(Ω) the first eigenvalue of the p-Laplacian on Ω with boundary datum u 0 , such that g and h satisfy the following assumptions: for almost all (x, s) ∈ Ω × R. g and h satisfy the following growth conditions: We want to study the following problem: The main result of the paper is the following theorem about the regularity of solutions of problem (P).
The idea of its proof is similar to that of Theorem 2.2 in [3], which in turns relies on the ideas introduced in [7]. The regularity of u is proved in Theorem 4.1 and the regularity of A follows from Proposition 5.1. The proof will be discussed in the final section. The same arguments can be used to treat also the volume-constraint problem for some 0 < d < L n (Ω). The following theorem holds true.
The proof of the previous theorem is a straightforward adaptation of the proof of Theorem 1.4 in [6]. The term concerning the function f E can be treated as a constant, thanks to the boundedness stated in Theorem 4.1. We finally remark that the term λ|L n (E) − d| in the functional F λ can be inglobed in f E , since it is bounded. For this reason, Theorem 1.1 is still valid also for minimal configurations of F λ and, consequently, for solutions of problem (Q).

Notation and preliminary results
Throughout the paper we denote by ·, · and · respectively the Euclidean inner product in R n and the associated norm. We write L n for the Lebesgue measure. Furthermore, we denote by B r (x) the ball centered in x ∈ R n with radius r > 0 (if x = 0, we write simply B r ), by ω n the measure of B 1 , and with Q r (x) the cube centered in x ∈ R n with side r > 0. We write the symbols and → referring to weak and strong convergence, respectively. We often denote by c a general constant that could vary from line to line, even within the same line of estimates. Relevant dependencies on parameters and special constants will be suitably emphasized using brackets. Throughout this section we denote with H a function belonging to C 1 (R n ) and satisfying for some positive constantsl andL the same kind of assumptions imposed on F and G: for all ξ ∈ R n and ϕ ∈ C 1 c (Ω). We collect some definitions and well-known results that will be used later. We start giving the definition of p-recession function of H .

Definition 2.1 The p-recession function of H is defined by
for all ξ ∈ R n .

Remark 2.2 It's clear that H p is positively p-homogeneous, which means that
for all ξ ∈ R n and s > 0. It's also true that the growth condition of H implies the following growth condition of H p : for any ξ ∈ R n .
Next lemma establishes strong quasi-convexity of H p , provided H verifies an appropriate growth condition. Its proof is in [12] (Lemma 2.8).

Lemma 2.3 Let H as above.
If there exist two positive constantst 0 ,d and 0 <m < p such that for every t >t 0 and ξ ∈ R n with |ξ | = 1, it holds for all ξ ∈ R n and ϕ ∈ C 1 c (Ω).
Let's recall some other useful lemmas.

Lemma 2.4 Let H be as above. It holds that
for all ξ ∈ R n . Lemma 2.5 Let H as above. There exists a positive constantc =c( p,l,L, μ) such that The proof of Lemma 2.4 can be found in [14] (Lemma 5.2), while Lemma 2.5 is proved in [6] (Lemma 2.3). We define the auxiliary function for all ξ ∈ R n . Next Lemma has been proved in [13] (Lemma 2.1) for p ≥ 2 and in [1] (Lemma 2.1) for 1 < p < 2.

Lemma 2.6
There exists a constant c = c(n, p) such that for all ξ, η ∈ R n .
The proof of the previous lemma follows from Lemma 2.6. If p ≥ 2 the hypothesis of boundedness of {∇u h } h∈N is superfluous. If 1 < p < 2, by Hölder inequality we gain the stated result.
The following theorem has been proved in [12] (Theorem 2.2).

Theorem 2.8 Let H be as above and let v ∈ W 1, p (Ω) be a local minimizer of the functional
Then v is locally Lipschitz-continuous in Ω and there exists a constant c = c(n, p,l,L) > 0 such that

Theorem 3.1 The minimum problem (P) admits at least a solution.
Proof We initially remark that problem (P) can be written as follows: Since F, G are strongly quasi-convex and g, h are lower semicontinuous in the real variable s, the functional F is lower semicontinuous with respect to the weak convergence of ∇v h in L p and the strong converge of v h in L p (see [5] or [16]). Moreover, the coerciveness of is granted by Lemma 2.5. Therefore the minimum problem (3.2) admits a solution. Let Poincaré inequality we obtain Hence, we can extract a subsequence (not relabelled) such that u h u in W 1, p (Ω). By definition of minimum we infer Applying again Ioffe lower semicontinuity result (see for instance [16] or [4], Theorem 5.8) to the integrand where x ∈ Ω, s 1 ∈ [0, 1], s 2 ∈ R and ξ ∈ R n , we obtain Therefore, by the lower semicontinuity of perimeter we finally gain which proves that A is a minimizer of problem (3.1) and so (u, A) is a minimizing couple of problem (P).

Higher integrability and Hölder continuity of minimizers
The following theorem shows that local minimizers of the functional F (·, E), with E ∈ A (Ω) fixed, are Hölder continuous and a higher integrability property for the gradient holds true. The proof of this result is standard and can be carried on adopting the obvious adaptation in the proof of Theorem 3.1 in [3].

Theorem 4.1 Let (u, A) be a solution of (P). Then the following facts hold:
-u is locally bounded in Ω by a constant depending only on n, p, q, α, β, l, L, μ, C 0 , u L p (Ω) and is locally Hölder continuous in Ω.
Then there exist two constants γ > 0 and r > p depending only on n, p, q, β, l, L, μ, for all y ∈ Ω 0 and Q R (y) ⊆ K .

Regularity of the set
The following proposition is the main result of this section and also the main ingredient to prove Theorem 1.1.

Proposition 5.1 Let (u, A) be a solution of (P). Then for every compact set K
The proof of the previous proposition relies on Proposition 5.5, which is an iteration of the decay estimate in Theorem 5.4. The following definition is crucial in the rescaling argument used in the proof of Theorem 5.4 (see (5.11)).
In the proof of Theorem 5.4 we will show that the sequence of appropriately rescaled minimal configurations of problem (P) is asymptotically minimizing. The following theorem is concerned with the behaviour of asymptotically minimizing sequences.
Proof Let's prove (a). The hypothesis (iv) implies that where {η h } h∈N ⊆ R is the infinitesimal sequence in (5.1). By the convexity of F p and G p and Lemma 2.4, it follows that Using the previous one in (5.3), we obtain The second term in the right hand side is infinitesimal; indeed, using the Hölder inequality, we havê , which tends to 0 as h approaches +∞. So we can inglobe the second term in the right hand side of (5.4) in η h . AddˆB 1 ψ1 A G p (∇u h ) dy to both sides in (5.4) in order to obtain where {η h } h∈N ⊆ R is infinitesimal. Thanks to (5.2), we can pass to the upper limit and obtain lim sup Finally, by lower semicontinuity, we gain By the strong quasi-convexity of F p and G p and Lemma 2.6, we havê Let h → +∞ in the previous inequality. By semicontinuity we infer that P(A, B 1 ) = 0. Thanks to isoperimetric inequality it follows that A = ∅ or A = B. We'll discuss the case A = ∅, being the other one similar. For h large enough, by the isoperimetric inequality we have N and ρ ∈ (0, 1), the coarea formula provides that which means that the sequence of functions λ h´∂ B ρ 1 h (ρ) dH n−1 h∈N converges to 0 in L 1 (0, 1). Thus, it converges to 0 for almost every ρ ∈ (0, 1). Then, for every ρ ∈ (0, 1) fixed, we can find a sequence Thus, thanks to (5.7) the sequence {λ h P(A h , B ρ h )} h∈N is infinitesimal and we can conclude that Let's prove c). and ρ ∈ (0, 1) arbitrary. Thanks to a), we can use the dominated convergence theorem in order to pass to the limit as h approaches +∞, obtaininĝ By the arbitrariety of ρ and ϕ we can conclude the proof.
The following theorem is the main tool for proving Proposition 5.1.  ∈ (0, 1). Letc =c( p, l, L, α, β, μ) and c H = c H (n, p, l, L, α, β) the constants of Lemma 2.5 and Corollary 2.9 for Moreover, let τ ∈ (0, 1) such that τ ε < 1 2(1+ω nc ) . Then there exist two positive constants γ and θ such that for any solution (u, A) of the problem (P) and for any ball B ρ (y) with y ∈ K and ρ ∈ (0, δ 2 ) the two estimateŝ imply thatˆB Proof Let's suppose by contradiction that there exist two sequences {γ h } h∈N and {θ h } h∈N which tend to 0, a sequence of minimizing couples {(w h , D h )} h∈N of (P) and a sequence of balls {B ρ h (x h )} h∈N , with x h ∈ K and ρ h ∈ (0, δ 2 ), for all h ∈ N, such that these estimates hold:ˆB By the usual change of variables x := x h + ρ h y, we have: Rescale the estimates (5.8), (5.9) and (5.10), obtaininĝ We want to apply Theorem 5.3 to the sequence Rescale the functions u h : In the sixth line of the previous inequality we need F p and G p in place of F and G, so by (F3) and (G3) we infer Thus by homogeneity, (F3) and (G3) we get In order to prove that {(u h , A h )} h∈N is λ h -asymptotically minimizing, we need to show that
In order to prove (5.18), we can apply Theorem 4.1: there exist two constants γ > 0 and r > p depending only on n, p, q, β, l, L, μ, C 0 , w h L ∞ (K ) such that for all h ∈ N and y ∈ K , with dist(Q 2ρ h (y), K )≤ δ 2 we have the following local higher summability: It can be also shown that the dependence of γ and r on w h L ∞ (K ) is uniform with respect to h, since {w h } h∈N is locally equibounded in Ω. Fix t ∈ (0, 1). By a covering argument it follows that since ξ < τ σ γ θ. Finally, using that is non-decreasing, we have which concludes the proof.

Proof of the main theorem
In this section we give the proof of Theorem 1.1, which makes use of the results we obtained in the previous sections.
Proof (of Theorem 1.1) The assertion 1. follows from Theorem 4.1. Let's prove the statement 2.
Let's prove that A andÃ are equivalent. One one hand, by the definition ofÃ we have which implies that L n (Ã \ A) = 0; on the other hand, since H n−1 (Ω \ Ω 0 ) = H n−1 (∂ * A) < +∞, we deduce that L n (Ω \ Ω 0 ) = 0 and hence The converse inequality can be obtained from the following one that holds true for any Borel set C ⊆ R n and can be obtained by De Giorgi's structure theorem: Choosing C =Ã, we conclude the proof.
Acknowledgements I wish to thank the reviewer for suggesting me several revisions that improved the paper.
Funding Open access funding provided by Universitá degli Studi di Salerno within the CRUI-CARE Agreement.

Conflict of interest
The corresponding author states that there is no conflict of interest.
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