Regularity results for a free interface problem with Hölder coefficients

We study a class of variational problems involving both bulk and interface energies. The bulk energy is of Dirichlet type albeit of very general form allowing the dependence from the unknown variable u and the position x. We employ the regularity theory of Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}-minimizers to study the regularity of the free interface. The hallmark of the paper is the mild regularity assumption concerning the dependence of the coefficients with respect to x and u that is of Hölder type.

The regularity of minimizers of these kinds of functionals is a rather subtle issue even in the scalar setting especially regarding the free interface ∂E.
In 1993 in the paper [2] L. Ambrosio and G. Buttazzo proved that if (E, u) is a minimizer of the functional (2), then u is locally Hölder continuous in Ω and E is relatively open in Ω.In the same volume of the same journal, F.H. Lin proved a regularity result for the interface ∂E.To clarify the situation we define the set of regular points of ∂E as follows: Reg(E) := x ∈ ∂E ∩ Ω : ∂E is a C 1,γ hypersurface in some I(x) and for some γ ∈ (0, 1) where I(x) denotes a neighborhood of x.Accordingly, we define the set of singular points of ∂E Σ(E) := (∂E ∩ Ω) \ Reg(E).
In [22] F.H. Lin proved that, for minimal configurations of the functional (2), The aforementioned regularity result has been recently improved by G. De Philippis & A. Figalli, and N. Fusco & V. Julin.Using different approaches and different techniques G. De Philippis & A. Figalli in [7] and N. Fusco & V. Julin in [15] proved that for minimal configurations of the functional (2) it turns out that, for some ε > 0 depending only on α, β.Regarding this dependence, it is worth noticing that in [11] it was proven that u ∈ C 0, 1  2 +ε and the reduced boundary ∂ * E of E is a C 1,ε −hypersurface and H s (∂E \ ∂ * E) = 0 for all s > n − 8, assuming that 1 ≤ α β < γ n , for some γ n > 1 depending only on 1 the dimension.In 1999 F.H. Lin and R.V. Kohn in [23] extended the same result that the first author obtained for the model case (2) to the more general setting of integral energy of the type (1), depending also on x and u.More precisely F.H. Lin and R.V. Kohn proved, for minimal configurations (E, u) of (1) under suitable smothness assumption of F and G, that H n−1 (Σ(E)) = 0.A natural question to ask is whether the same dimension reduction of the singular set Σ(E) proved for the model case (2) by G. De Philippis & A. Figalli and N. Fusco & V. Julin can be extended also to the general case of functionals of the type (1).In a very recent paper we give a positive answer to this question.Inded in [12] we prove that for some ε > 0, for optimal configurations of a wide class of quadratic functionals depending also on x and u.Our path to prove the aforementioned result basically follows the same strategy used in [15].The technique used in [12] relies on the linearity of the Euler-Lagrange equation of the functional (1).For this reason we need a quadratic structure condition for the bulk energy.Conversely, the nonquadratic case is less studied and there are few regularity results available (see [4], [5], [10], [19]).
Throughout the paper we will assume that the density energies F and G in (1) satisfy the following structural quadratic assumptions: G(x, s, z) for any (x, s, z) ∈ Ω × R × R n .In the paper [12] we assumed as in [23] that the coefficients a ij , b ij , , a i , b i , a, b belong to the class C 0,1 (Ω × R) with respect to both variables x and s.This C 0,1 assumption of the coefficients with respect to (x, s) is crucial in several respect in order to prove the desired regularity result for ∂E.
In the first place the C 0,1 assumption is strongly used (see Theorem 2 in [12]) to prove that every minimizer of the constrained problem (that is for |E| = d fixed) is a Λ-minimizer of a penalized functional containing the extraterm Λ||E| − d|.In addition the C 0,1 assumption is primarly used to get an Euler-Lagrange-type equations that is one of the main ingredients to prove the desired regularity result (see Proposition 4.9 in [15]) and Theorem 8 in [12]).
In this paper we examine in depth the question of the minimal regularity assumptions of the coeficients we ought to assume in order to get the regularity result quoted in (3).Concerning the coefficients appearing in (4) and (5) we will assume Hölder continuous dependence of (x, s).We exploited the proof strategy in every possible way in order to push to the limit the assumptions concerning the Hölder exponent of the coefficients.In this regard it is important to point out that no restriction is needed for the Hölder exponent β with respect to the s variable quoted below.Precisely we will assume that R), for every x ∈ Ω.
We will denote by L β the greatest Hölder seminorm of the coefficients with respect to the second variable, that is and the same holds true for b ij , a i , b i , a, b.Similarly we will assume about the dependence on the first variable, We will denote by L α the greatest Hölder seminorm of the coefficients with respect to the first variable, that is [a ij (•, s)] α := sup y,z∈Ω, y =z and the same holds true for b ij , a i , b i , a, b.Moreover, to ensure the existence of minimizers we assume the boundedness of the coefficients and the ellipticity of the matrices a ij and b ij , for any (x, s, z) ∈ Ω × R × R n , where ν, N and L are three positive constants.Some comments about the Hölder exponent α are in order.There are two main points in our proof where the assumption α > n−1 n is used.In both cases we have to handle with a perturbation of the set E. The first point concerns the equivalence between the constrained problem and the penalized problem (see the definitions below).In Theorem 2 we perform a suitable "small" perturbation of a minimal set E around a point x ∈ ∂E using a transformation of the type where X ∈ C 1 0 (B r (x)).If we denote by E := Φ σ (E) the perturbed set and by ũ := u • Φ −1 σ the perturbed function, we prove that F (E, u) − F ( E, ũ) = O(σ α ), where α is the Hölder exponent given in (7).On the other hand, in Theorem 2 we prove by contradiction that (E, u) is a minimizer of a penalized functional obtained adding in (1) a penalization term of the type for some suitable Λ to be choosen sufficiently large.Since we can observe that Λ it is clear that we are forced to choose s = σ (see Definition 2 below).Finally it is evident that this new penalization term cannot exceed the perimeter term when we rescale the functional (see Lemma 6) and so we are forced to choose α > n−1 n .The second point concerns the excess improvement given in Theorem 10, where we use a standard rescaling argument to show that the limit g of the rescaled functions whose graph locally represents ∂E is armonic (see Step 1 in Theorem 10).In this step we use the Taylor expansion of the bulk term given in Theorem 7 and the condition α > n−1 n is again crucial, see (79).
In this paper we study the regularity of minimizers of the following constrained problem.
Definition 1.We shall denote by (P c ) the constrained problem where u 0 ∈ H 1 (Ω), 0 < d < |Ω| are given and A(Ω) is the class of all subsets of Ω with finite perimeter in Ω.
The problem of handling with the constraint |E| = d is overtaken using an argument introduced in [11], ensuring that every minimizer of the constrained problem (P c ) is also a minimizer of a penalized functional of the type for some suitable Λ > 0 (see Theorem 2 below).Therefore, we give in addition the following definition.
Definition 2. We shall denote by (P ) the penalized problem where u 0 ∈ H 1 (Ω) is fixed and A(Ω) is the same class defined in Definition 1.
From the point of view of regularity, the extra term Λ |E| − d α is a higher order negligible perturbation, being α > n−1 n .The main result of the paper is stated in the following theorem.Theorem 1.Let (E, u) be a minimizer of problem (P ), under assumptions (4) − (9).Then a) there exists a relatively open set Γ ⊂ ∂E such that Γ is a C 1,µ hypersurface for all 0 < µ < γ 2 , where γ := 1 + n(α − 1) ∈ (0, 1) , b) there exists ε > 0 depending on n, ν, N, L, such that Let us briefly describe the organization of this paper.Section 2 collects known results, notation and preliminary definitions.Moreover, in this section the equivalence between the constrained problem an the penalized problem is proved.As it always happens when different kind of energies compete with each other, the proof of the regularity is based on the study of the interplay between them.In this case we must compare perimeter and bulk energy (see [3], [22]).We point out that the Hölder exponent 1  2 is critical in this respect for solutions u of either (P ) or (P c ), in the sense that, whenever u ∈ C 0, 12 , under appropriate scaling, the bulk term locally has the same dimension n − 1 as the perimeter term.In section 3 we prove suitable energy decay estimates for the bulk energy.The key point of this approach is contained in Lemma 5, where it is proved that the bulk energy decays faster than ρ n−1 , that is, for any µ ∈ (0, 1), ˆBρ(x0) either in the case that min{|E or in the case that there exists an half-space H such that for some ε 0 > 0. The latter case is the hardest one to handle because it relies on the regularity properties of solutions of a transmission problem which we study in subsection 3.1.Let us notice that, for any given E ⊂ Ω, local minimizers u of the functional are Hölder continuous, u ∈ C 0,σ loc (Ω), but the needed bound σ > 1 2 cannot be expected in the general case without any information on the set E. In subsection 3.1 we prove that minimizers of the functional (11) are in C 0,σ for every σ ∈ (0, 1), in the case E is an half-space.In this context the linearity of the equation strongly comes into play ensuring that the derivatives of the Euler-Lagrange equation are again solutions of the same equation.For the proof in section 3 we readapt a technique depicted in the book [3] in the context of the Mumford-Shah functional and recently used in a paper by E. Mukoseeva and G. Vescovo, [25].In section 4, using the estimates obtained in section 3, we are in position to prove some decay estimates for the whole energy including the perimeter term.More precisely, whenever the perimeter of E is sufficiently small in a ball B ρ (x 0 ), then the total energy ˆBr(x0) |∇u| 2 dx + P (E; B r (x 0 )), 0 < r < ρ, decays as r n (see Lemma 7).In the subsequent sections we collect the preliminary results needed to deduce that ∂E is locally represented by a Lipschitz graph, see Theorem 5.
In section 4, making use of the previous results, we are in position to prove the density upper bound and the density lower bound for the perimeter of E which, in turn, are crucial to prove the Lipschitz approximation theorem.In the subsequent sections the proof strategy follows the path traced from the regularity theory for perimeter minimizers.
In section 5 it is proved the compactness for sequences of minimizers which follows in a quite standard way from the density lower bound.
Section 6 is devoted to the Lipschitz approximation theorem which involves the usual main ingredient of the regularity proof, that is the excess e(x, r) = inf In section 7 we prove the reverse Poincaré inequality which is the counterpart of the well-known Caccioppoli's inequality for weak solutions of elliptic equations.Sections 8 contains a Taylor-like expansion formula for the terms appearing in the energy under a small domain perturbation.
In section 9 we finally prove the excess improvement, which is the main ingredient to achieve the regularity of the interface.More precisely, we prove that, whenever the excess e(x, r) goes to zero, for r → 0, the Dirichlet integral ´Bρ(x0) |∇u| 2 dx decays as in (10).With all these results in hand we can conclude the desired result.
In section 10 we provide the proof of Theorem 1 that is a consequence of the excess improvement proved before.

Preliminary notation and definitions
In the rest of the paper we will write ξ, η for the inner product of vectors ξ, η ∈ R n , and consequently |ξ| := ξ, ξ 1 2 will be the corresponding Euclidean norm.As usual ω n stands for the Lebesgue measure of the unit ball in R n .We will denote by p : R n → R n−1 and q : R n → R the horizontal and vertical projections, so that x = (px, qx) for all x ∈ R n .For simplicity of notation we will often write px = x ′ and qx = x n , so that we will write x = (x ′ , x n ), where x ′ ∈ R n−1 and x n ∈ R. Accordingly, we denote ∇ ′ = (∂ x1 , . . ., ∂ xn−1 ) the gradient with respect to the first n − 1 components.The n-dimensional ball in R n with center x 0 and radius r > 0 will be denoted as If x 0 = 0, we will simply write B R instead of B R (x 0 ).The (n − 1)-dimensional ball in R n−1 with center x ′ 0 and radius r > 0 will be denoted with a different letter, that is For any µ ≥ 0 we define the Morrey space L 2,µ (Ω) as In the sequel we will constantly need to denote the difference between α and n−1 n , so that we define The following definition is standard.
Furthermore we say that v is a local minimizer of the integral functional F E if and only if for all B R (x 0 ) ⊂⊂ Ω.

AND 2
It is worth mentioning that for a quadratic integrand F (x, s, z) of the type given in (4) the following growth condition can be immediately deduced from assumptions (8) and (9): The next lemma is very standard and can be found for example in [3,Lemma 7.54].
Lemma 1.Let f : (0, a] → [0, ∞) be an increasing function such that for some constants A, B ≥ 0, 0 < q < p, s > 0. Then there exist R 0 = R 0 (p, q, s, A) and c = c(p, q, A) such that 2.1.From constrained to penalized problem.The next theorem allows us to overcome the difficulty of handling with the constraint |E| = d.Indeed, we prove that every minimizer of the constrained problem (P c ) is also a minimizer of a suitable unconstrained problem with a volume penalization of the type given in (P ).
Theorem 2. There exists Λ 0 > 0 such that if (E, u) is a minimizer of the functional for some Λ ≥ Λ 0 , among all configurations (A, w) such that w = u 0 on ∂Ω, then |E| = d and (E, u) is a minimizer of problem (P c ). Conversely, if (E, u) is a minimizer of problem (P c ), then it is a minimizer of (14), for all Λ ≥ Λ 0 .
Proof.The proof can be carried out as in [11,Theorem 1].For reader's convenience we give here its sketch, emphasizing main ideas and minor differences with respect to the case treated in [11].
The first part of the theorem can be proved by contradiction.
Without loss of generality we may assume that |E h | < d.Indeed, the case |E h | > d can be treated in the same way considering the complement of E h in Ω.Our aim is to show that, for h sufficiently large, there exists a configuration ( E h , ũh ) such that F λ h ( E h , ũh ) < F λ h (E h , u h ), thus proving the result by contradiction.By condition (15), it follows that the sequence (u h ) h is bounded in H 1 (Ω), the perimeters of the sets E h in Ω are bounded and |E h | → d.Therefore, possibly extracting a not relabelled subsequence, we may assume that there exists a configuration (E, u) such that a.e. in Ω, where the set E is of finite perimeter in Ω and |E| = d.The couple (E, u) will be used as reference configuration for the definition of ( E h , ũh ).
Step 1. Construction of ( E h , ũh ).Proceeding exactly as in [11], we take a point x ∈ ∂ * E ∩ Ω and observe that the sets E r = (E − x)/r converge locally in measure to the half-space H = { z, ν E (x) < 0}, i.e., ½ Er → ½ H in L 1 loc (R n ), where ν E (x) is the generalized exterior normal to E at x (see [3,Definition 3.54]).Let y ∈ B 1 (0) \ H be the point y = ν E (x)/2.Given ε (that will be chosen in the Step 2), since ½ Er → ½ H in L 1 (B 1 (0)) there exists 0 < r < 1 such that , where ω n denotes the measure of the unit ball of R n .Then, if we define x r := x + ry ∈ Ω, we have that Let us assume, without loss of generality, that x r = 0. From the convergence of E h to E we have that for all h sufficiently large Let us now define the following bi-Lipschitz function used in [11] which maps B r into itself: for some 0 < σ h < 1/2 n sufficiently small to be chosen later in such a way that, setting We are going to evaluate In order to estimate the contribution of the last integrals we need some preliminary estimates for the map Φ that can be obtained by direct computation (see [11] or [12] for the explicit calculation).We just observe that for |x| < r/2, Φ is simply a homothety and all the estimates that we are going to introduce are trivial.Conversely, for r/2 < |x| < r we have It is clear from this expression that, since σ h is going to zero, ∇Φ is a small perturbation of the identity that can be written as ∇Φ = Id + σ h Z.We can also address the reader to section 17.2 "Taylor's expansion of the determinant close to the identity" in [24] for related estimates.Then we have It is not difficult to find out also that Concerning JΦ, the Jacobian of Φ, from (19) we deduce For r/2 < |x| < r, we can estimate (see also section 3 in [4]): Summarizing we gain the following inequalities for the Jacobian of Φ: JΦ(x) ≤ 1 + 2 n nσ h , for all x ∈ B r .
Now, let us start estimating I 3,h thus proving at the same time that the condition | E h | < d is satisfied.
Step 2. Estimate of I 3,h .First we recall ( 16), ( 17), (22), thus getting Moreover, if we denote δ For this reason let us observe that we have, proceding as before and using (22), Then we will choose Let us observe that in the last condition we imposed also that σ h is comparable with δ h , which is crucial in the following estimate.Resuming (23) we can conclude Step 3. Estimate of I 1,h .Now we can perform the change of variables y = Φ(x) and, observing that The two terms J 1,h and J 2,h , involving F and G in B r and B r ∩ E h respectively, can be treated in the same way.Therefore we just perform the calculation for J 1,h .
To make the argument more clear, since we shall use the structure conditions ( 4) and ( 5) we introduce the following notation.A 2 (x, s) denotes the quadratic form and A 1 (x, s) denotes the linear form defined as follows: for any z ∈ R n .Analogously we set A 0 (x, s) = a(x, s).Accordingly, we can write down We proceed estimating the first difference in the previous equality, being the other similar and indeed easier to handle.
The first term H 1,h can be estimated observing that, as a consequence of (8), we have: If we apply the last inequality to the vectors we are led to estimate |ξ − η|.
We start observing that, being n for |x| < r/2, by also using, (22) we deduce Therefore we have 20) and using also (21), we can deduce

Summarizing we finally get
From the previous estimates we deduce that where Θ is defined in (15).The second term H 2,h can be estimated using the Hölder continuity assumption on a ij and observing that |x − Φ(x)| ≤ σ h r2 n .Therefore we deduce that In conclusion, since the other terms in ( 25) can be estimated in the same way, collecting estimates ( 26) and ( 27) we get Since the same estimate holds true for J 2,h , we conclude that for some constant Step 4. Estimate of I 2,h .In order to estimate I 2,h , we can use the area formula for maps between rectifiable sets.If we denote by T h,x the tangential gradient of Φ along the approximate tangent space to ∂ * E h in x and T * h,x is the adjoint of the map T h,x , the (n − 1)-dimensional jacobian of T h,x is given by (29) We address the reader to [11] where explicit calculations are given.In order to estimate I 2,h , we use the area formula for maps between rectifiable sets ([3, Theorem 2.91]), thus getting Notice that the last integral in the above formula is non-negative since Φ is a contraction in B r/2 , hence J n−1 T h,x < 1 in B r/2 , while from (29) we have Finally to conclude the proof we recall ( 18), ( 24), ( 28) and (30) to obtain sufficiently large.This contradicts the minimality of (E h , u h ), thus concluding the proof.
The previous theorem motivates the following definition.

Decay of the bulk energy
We start quoting higher integrability results both for local minimizers of the functional (1) and for comparison functions that we will use later in the paper.We assume that E is fixed and therefore we consider only the dependence on the bulk term through u.It is worth mentioning that the following lemmata can be applied in general to minimizers of integral functionals of the type assuming that the energy density H satisfies only the structure condition (4) and the growth conditions ( 8) and ( 9), without assuming any continuity on the coefficients.It is clear that functionals of the type (1) belong to this class and in addition the involved estimates only depend on the constants appearing in ( 8) and ( 9) but do not depend on E accordingly.Since the argument is very standard we address the reader to [12] where detalied proofs is given.
Lemma 2. Let u ∈ H 1 (Ω) be a local minimizer of the functional H defined in (31), where H satisfies the structure condition (4) and the growth conditions (8) and (9).There exists s = s(n, ν, N, L) > 1 such that, for every B 2R (x 0 ) ⊂⊂ Ω, it holds where In the next subsection we will prove some energy density estimates by using a standard comparison argument.For this purpose we will need a reverse Hölder inequality for the comparison function defined below.

Definition 5 (Comparison function).
Let u ∈ H 1 (Ω) be a local minimizer of the functional H defined in (31) and B 2R (x 0 ) ⊂⊂ Ω.We shall denote by v the solution of the following problem v := argmin where H(x, z) := H(x, u(x), z) satisfies the structure condition (4) and the growth conditions (8) and (9).
Lemma 3. Let u ∈ H 1 (Ω) be a local minimizer of the functional H defined in (31), where H satisfies the structure condition (4) and the growth conditions (8) and (9).Let v ∈ H 1 (B R (x 0 )) be the comparison function defined in (32).Denoting by s = s(n, ν, N, L) > 1 the same exponent given in Lemma 2, it holds where

3.1.
A decay estimate for elastic minima.In this section we prove a decay estimate for elastic minima that will be crucial for the proof strategy.Indeed, we show that if (E, u) is a (Λ, α)-minimizer of the functional F defined in (1) and x 0 is a point in Ω, where either the density of E is close to 0 or 1, or the set E is asymptotically close to a hyperplane, then for ρ sufficiently small we have for any µ ∈ (0, 1).A preliminary result we want to mention, which will be used later, provides an upper bound for F .The proof is rather standard and is related to the threshold Hölder exponent 1 2 of the function u, when (E, u) is either a solution of the constrained problem (P c ) or a solution of the penalized problem (P ) defined in Section 1.For the proof we address the reader to [23, Lemma 2.3] and [15].A detailed proof in the case of costrained problems and for functionals satisfying general p-polinomial growth is contained in [4].
Making F explicit and getting rid of the common terms, we obtain: Now we want to prove that there exist τ ∈ 0, 1 2 and δ ∈ (0, 1) such that for every M > 0 there exists h 0 ∈ N such that, for any B r (x 0 ) ⊂ U , we have Step 1: Arguing by contradiction, for τ ∈ 0, 1 2 and δ ∈ (0, 1), we choose M > τ δ−n and we assume that, for every h ∈ N, there exists a ball B r h (x h ) ⊂ U such that Note that estimates (33) and (34) yield and so for some positive constant c 0 .
Step 2: We will prove our aim by means of a blow-up argument.We set and, for y ∈ B 1 , we introduce the sequence of rescaled functions defined as We have ∇u(x h + r h y) = ς h ∇v h (y) and a change of variable yields Therefore, there exist a (not relabeled) subsequence of . Moreover, the semicontinuity of the norm implies We rewrite the inequalities (34), ( 35) and (37).They become, respectively, Of course, (39) implies that ς h → ∞, as h → ∞.
Step 3: We claim that the L 2 -norm of v h converges to the L 2 -norm of v. Consider the sets Using the change of variable x = x h + r h y, we deduce for every ψ ∈ H 1 0 (B 1 ), ˆB1 ) and exploit ∇v h + ∇ψ h for reader convenience, For simplicity of notation we will denote w h := u(x h + r h y) + r h ς h η(v − v h ) so that the previous inequality can be read as Using the quadratic structure of F and G we can pull out the terms in red in order to use the convexity in the next step. ˆB1 Using the convexity of F and G and rearranging the terms we obtain The last term and the second to last term can be treated in a standard way using (38), Hölder's inequality, the strong convergence of v h to v and the weak convergence of ∇v h to ∇v.The remaining two terms, which differ only in the second argument, can be treated as follows.
We remark that by definition of v h and Hölder continuity of u h immediately follows r h ς h v h → 0. Therefore, being r h ς h → 0 where v = 0, we deduce also Finally, using the equi-integrability of |∇v h | 2 , resulting from the weak convergence of ∇v h , and the boundedness of the coefficients a ij , a i , a we conclude that Combining the previous inequalities, we get Dividing by ς 2 h , the linear terms in F tend to 0, thus getting Since B r h (x h ) ⊂ U ⊂⊂ Ω for all h ∈ N, we may assume that x h → x, as h → ∞.Letting η ↓ 1 in the previous inequality, passing to the lower limit, as h → ∞, by lower semicontinuity, we finally get Since the matrix a ij (x, u(x)) is elliptic and bounded, it induces a norm which is equivalent to the euclidean norm.Thus we get which contradicts (40), provided we choose M > τ δ−n .
Remark 1.Let (E, u) be a Λ-minimizer of the functional F defined in (1).For every open set U ⊂⊂ Ω there exists a constant In order to prove the main lemma of this section we introduce the following preliminary result.For reader's convenience we give here a sketch of the proof, which can be found in [25].Actually we state here a weaker version that is suitable for our aim.In the following we will denote where for some σ ∈ (0, 1] and A is an elliptic matrix satisfying and for some constants ν, N > 0. Let us denote Then ∇v ∈ L 2,n loc (B 1 ) (see (12)).Moreover, there exist two constants Proof.Fix x 0 ∈ B 1 and let r be such that B r (x 0 ) ⊂ B 1 .Let us denote by a + and a − the averages of A in H ∩ B r (x 0 ) and H c ∩ B r (x 0 ) respectively.In an analogous way we define g + and g − the averages of G in H ∩ B r (x 0 ) and H c ∩ B r (x 0 ).For x ∈ B r (x 0 ) we define Notice that by assumption Let w be the solution of The last equation can be rewritten as where w + := w½ Br (x0)∩H , w − := w½ Br (x0)∩H c .Set where A in is the (i, n)-th entry of the matrix A. We notice that D c w has no jumps on the boundary thanks to the transmission condition in (44).This allows us to prove that the distributional gradient of D c w coincides with the point-wise one.
Step 1: Tangential derivatives of w.Let us denote with τ the general direction tangent to the hyperplane ∂H.Since A and G are both constant along the tangential directions, the classical difference quotient method gives that ∇ τ w ∈ W 1,2 loc (B r (x 0 )) and div(A∇(∇ τ w)) = 0 in B r (x 0 ).
Hence, Caccioppoli's inequality holds: for all balls B 2ρ (x) ⊂ B r (x 0 ) and, by De Giorgi's regularity theorem, ∇ τ w is Hölder continuous and there exists for any ρ ∈ 0, s 2 and max Step We can conclude that D c w ∈ W 1,2 loc (B r (x 0 )).Using Poincaré's inequality and (45), we have for any B 2ρ (x) ⊂ B r (x 0 ).By (46) we infer for any x ∈ B r 4 (x 0 ) and ρ ≤ r 4 .Hence by Lemma 4.2 in [25] (see also [3,Lemma 7.51]), D c w is Hölder continuous and by (47) we get: Step 3: Comparison between v and w.Subtracting the equation for w from the equation for v we get by (47), (48), the minimality of w and Young's inequality we gain which leads to our aim if we apply Lemma 1.
The next lemma is inspired by [15,Proposition 2.4] and is the main result of this section.In the sequel we shall consider the worst Hölder exponent introduced in ( 6) and ( 7), defined as δ := min {α, β}.

Now we use the following identity
in order to deduce that By the Euler-Lagrange equation for v we deduce that the sum of the last two integrals in the previous identity is zero, being also u = v on ∂B r/2 .Therefore, using the ellipticity assumption of A 0 we finally achieve that ν 1 AND 2 Now we prove that u is an ω-minimizer of F 0 .We start writing Estimate of F 0 (u) − F (E, u).We use ( 6), ( 7), ( 8), ( 9) and (42) to infer where we denoted L α , L β the greatest modulus of Hölder continuity of the data a ij , b ij , a i , b i , a, b defined in ( 6) and ( 7).Now we use Hölder's inequality and Lemma 2 to estimate Merging the last estimate in (52) we deduce If we choose now z ∈ ∂B r/2 , recalling that u(z) = v(z) we deduce where we used the fact that osc(v, B r/2 ) ≤ osc(u, ∂B r/2 ) + C(n, ν, N, L)r (see [16,Lemma 8.4]).
Analogously we can estimate the other differences in (55), deducing Reasoning in a similar way as in (53), we can apply the higher integrability for v given by Lemma 3 and infer ˆBr/2 ∩E |∇v| 2 dx ≤ C(n, ν, N, L)ε Therefore we obtain Finally, collecting (50), ( 51), ( 54) and ( 56), if we choose ε 0 such that ε for some constant C = C n, ν, N, L, α, β, L α , L β ∇u L 2 (Ω) .On the other hand v is the solution of a uniformly elliptic equation with constant coefficients, so we have Hence we may estimate, using (57) and (58), We are left with the case (iii).Let H be the half-space from our assumption and let us denote accordingly Let us point out that v H solves the Euler-Lagrange equation Therefore we are in position to apply Lemma 4 to the function v H . Indeed, from the Hölder continuity of u (see Remark 1) we deduce that the restrictions of A 0 and B 0 onto H ∩ B r and B r \ H respectively are Hölder continuous.We can conclude using also (42) that there exist two constants In addition, using the ellipticity condition of A 0 we can argue as in (49) to deduce using also the fact that v H satisfies (59), One more time we can prove that u is an ω-minimizer of F 0 .We start as above writing We can estimate the differences F 0 (u) − F (E, u) and F (E, v H ) − F 0 (v H ) exactly as before using this time the higher integrability given in Lemma 3. We conclude that for some constant C = C n, ν, N, L, α, β, L α , L β ∇u L 2 (Ω) .From the last estimate we can conclude the proof as before using (60) and (61).

Energy density estimates
This section is devoted to prove a lower bound estimate for the functional F (E, u; B r (x 0 )).Points i) and ii) of Lemma 5 are the main tools to achieve such result.We shall prove that the energy F decays "fast" if the perimeter of E is "small".In this section we will use a scaling argument.
Thus, if F ⊂ R n is a set of finite perimeter with F ∆E r ⊂⊂ B 1 and ṽ ∈ H 1 (B 1 ) is such that ṽ − u r ∈ H 1 0 (B 1 ), then where F := x 0 + r F and v(x) = r 1 2 ṽ x−x0 r , for x ∈ B r (x 0 ).Lemma 7. Let (E, u) be a (Λ, α)-minimizer in Ω of the functional F defined in (1).For every τ ∈ (0, 1) there exists Proof.Let τ ∈ (0, 1) and B r (x 0 ) ⊂ Ω.Without loss of generality, we may assume that τ < 1 2 .We may also assume that x 0 = 0, and r = 1 by scaling E r = E−x0 r , u r (y) = r − 1 2 u(x 0 + ry) for y ∈ B 1 , and replacing Λ with Λr γ .Thus, we have that (E r , u r ) is a (Λr γ , α)-minimizer of F r in Ω−x0 r .For simplicity of notation we can still denote E r by E, u r by u and then, recalling that F = r n−1 F r and γ = nα − (n − 1), we have to prove that there exists Note that, since P (E; B 1 ) < ε 1 , by the relative isoperimetric inequality, either |B 1 ∩ E| or |B 1 \ E| is small and thus Lemma 5 can be applied.Assuming that |B 1 \ E| ≤ |B 1 ∩ E| and using the relative isoperimetric inequality we can deduce that If we choose as a representative of E the set of points of density one, we get, by Fubini's theorem that Combining these inequalities, we can choose ρ ∈ (τ, 2τ ) such that Now we set F = E ∪ B ρ and observe that If we choose (F, u) to test the (Λr γ , α)-minimality of (E, u) we get Then getting rid of the common terms we obtain ≤ τ n+1 we have from ( 62) Then, we choose to obtain, using Lemma 5 and growth conditions ( 8), ( 9 Finally, we recall that ρ ∈ (τ, 2τ ) to conclude, using the previous estimates, From this estimate the result easily follows applying again Lemma 5.
In the sequel we will assume that the representative of the set E is choosen in such a way that the topological boundary ∂E concides with the closure of the reduced boundary, that is ∂E = ∂ * E, (see also [24] Proposition 12.19).
For h = 0, using Lemma 7 with 3 and 2C 5 C 3 σ γ < ε 1 (τ ), we get: In order to prove the induction step we have to ensure to be in position to apply Lemma 7, that is by proving smallness of the perimeter.In such regard, let us observe that, by the definition of F (E, u; B ρ ) and the growth condition given in (13), for any B ρ ⊂ Ω.

Compactness for sequences of minimizers
In this section we basically follow the path given in [24,Part III].We start proving a standard compactness result.
Lemma 8 (Compactness).Let (E h , u h ) be a sequence of (Λ h , α)-minimizers of F in Ω such that sup h F (E h , u h ; Ω) < ∞ and Λ h → Λ ∈ R + .There exist a (not relabelled) subsequence and a (Λ, α)minimizer of F in Ω, (E, u), such that for every open set U ⊂⊂ Ω, it holds Finally, if we assume also that ∇u h ⇀ 0 weakly in L 2 loc (Ω, R n ) and Λ h → 0, as h → ∞, then E is a local minimizer of the perimeter, that is Proof.We start observing that, by the boundedness condition on F (E h , u h ; Ω), we may assume that u h weakly converges to u in H 1 (U ) and strongly in L 2 (U ), and ½ E h converges to ½ E in L 1 (U ), as h → ∞.By lower semicontinuity we are going to prove the (Λ, α)-minimality of (E, u).Let us fix B r (x 0 ) ⊂⊂ Ω and assume for simplicity of notation that x 0 = 0. Let (F, v) be a test pair such that F ∆E ⊂⊂ B r and supp(u − v) ⊂⊂ B r .We can handle the perimeter term as in [24], that is, eventually passing to a subsequence and using Fubini's theorem, we may choose ρ < r such that, once again, F ∆E ⊂⊂ B ρ and supp(u − v) ⊂⊂ B ρ , and, in addition, Now we choose a cut-off function ψ ∈ C 1 0 (B r ) such that ψ ≡ 1 in B ρ and define to test the minimality of (E h , u h ).Thanks to the (Λ h , α)-minimality of (E h , u h ) we have ˆBr The mismatch term h )) appears because F is not in general a compact variation of E h .Nevertheless we have that ε h → 0 because of the assumption (66) (see also [24,Theorem 21.14]).Now we use the convexity of F and G with respect to the z variable to deduce ˆBr where the last two terms in the previous estimate tend to zero as h → ∞.Indeed, the term ∇ψ(v −u h ) strongly converges to zero in L 2 , being u = v in B r \ B ρ and the first part in the scalar product weakly converges in L 2 .Then using again the convexity of F and G with respect to the z variable we obtain, for some infinitesimal σ h , ˆBr Finally, we combine (67) and (68) and pass to the limit as h → +∞, using the lower semicontinuity on the left-hand side.For the right-hand side we observe that and we use also the equi-integrability of {∇u h } h to conclude, Letting ψ ↓ ½ Bρ we finally get ˆBρ and this proves the (Λ, α)-minimality of (E, u).
To prove the strong convergence of ∇u h to ∇u in L 2 (B r ) we start observing that by ( 67) and (68) applied using (E h , u) to test the (Λ, α)-minimality of (E h , u h ) we get Then from the equi-integrability of {∇u h } h in L 2 (U ) and recalling that ½ E h → ½ E in L 1 (U ), we obtain lim sup h→∞ ˆBr The opposite inequality can be obtained by semicontinuity.Thus we get From the ellipticity condition in (8) we infer, for some σ h → 0, Passing to the limit we obtain lim h→∞ ˆBr ψ|∇u h − ∇u| 2 dx = 0.
Finally testing the minimality of (E h , u h ) with respect to the pair (E, u) we also get With a usual argument we can deduce u h → u in W 1,2 (U ) and P (E h ; U ) → P (E; U ), for every open set U ⊂⊂ Ω.The topological information stated in (64) and (65) follows as in [24,Theorem 21.14] because it does not depend on the presence of the integral bulk part.

Height bound and Lipschitz approximation
In the following for R > 0 and ν ∈ Ë n−1 we will denote the cylinder centered in x 0 with radius R oriented in the direction ν.
The cylinder of radius R oriented in the direction e n with height 2 will be denoted as , In addition we introduce some usual quantities involved in regularity theory Definition 6.Let E be a set of locally finite perimeter, x ∈ ∂E, r > 0 and ν ∈ Ë n−1 .We define: • the cylindrical excess of E at the point x, at the scale r and with respect to the direction ν, as • the spherical excess of E at the point x, at the scale r and with respect to the direction ν, as • the spherical excess of E at the point x and at the scale r, as e(x, r) := min ν∈Ë n−1 e(x, r, ν).In the following, for simplicity of notation we will denote The following height bound lemma is a standard step in the proof of regularity because it is one of the main ingredients to prove the Lipschitz approximation theorem.The results contained in this section are a consequence of the compactness lemma, the density lower bound and the lower semicontinuity of the excess.In the statement of these results we assume that (E, u) is a (Λ, α)-minimizer of F .However the minimality is not used except to ensure compactness and the density lower bound.
Proof.The proof of this lemma is almost identical to the one in [24,Theorem 22.8].Indeed, it follows from the density lower bound (see Theorem 4), the relative isoperimetric inequality and the compactness result proved in the previous section.
Proceeding as in [24], we give the following Lipschitz approximation lemma, which is a consequence of the height bound lemma.Its proof follows exactly as in [24,Theorem 23.7].It is a foundamental step in the long journey to the regularity because it provides a connection between the regularity theories for parametric and non-parametric variational problems.Indeed we are able to prove for (Λ, α)-minimizers that the smallness of the excess guaranties that ∂E can be locally almost entirely covered by the graph of a Lipschitz function.
Theorem 5 (Lipschitz approximation).Let (E, u) be a (Λ, α)-minimizer of F in B r (x 0 ).There exist two positive constants ε 3 and C 8 , depending on ∇u L 2 (Br(x0)) , such that if x 0 ∈ ∂E and e(x 0 , r, e n ) < ε 3 , then there exists a Lipschitz function f : R n−1 → R such that sup where Γ f is the graph of f .Moreover,

Reverse Poincaré inequality
In this section we shall prove a reverse Poincaré inequality.This is the counterpart for (Λ, α)minimizers of the well-known Caccioppoli inequality for weak solutions of elliptic equations.The proof of the results of this section can be obtained as in the case of Λ-minimizers of the perimeter (sse [24,Section 24]).For the sake of completeness we give here the main steps of the proof underlining the minor changes.We will need first a weak form.
Proof.We may assume z = 0.
Step 1: The set function defines a Radon measure on R n−1 , concentrated on D 2 .
Step 2: Since E is a set of locally finite perimeter, by [24,Theorem 13.8] there exist a sequence {E h } h∈N of open subsets of R n with smooth boundary and a vanishing sequence as h → +∞, where I ε h (∂E) is a tubolar neighborhood of ∂E with half-lenght ε h .By Coarea formula we get Moreover, provided h is large enough, by ∂E h ⊂ I ε h (∂E), we get: Therefore, given λ ∈ 0, 1 4 and |c| < 1 4 , we are in position to apply [24,Lemma 24.8] to every E h to deduce that there exists I h ⊂ 2 3 , 3  4 , with |I h | ≥ 1 24 , and, for any r ∈ I h , there exists an open subset F h of R n of locally finite perimeter such that We will write F k in place of F h k .Now we test the (Λ, α)-minimality of (E, u) in C 4 with (G k , u), where . By [24, (16.33)] we infer: Thus, since ζ is nondecreasing and r ≥ 2 3 , by (71) we deduce that Letting k → +∞, (69) implies that P (E h(k) ; K s ) → P (E; K s ) and therefore for any λ ∈ 0, and thus (72) holds true for λ > 0, provided we choose c(n) ≥ 4. Minimizing over λ, we get the thesis.
Proof.Up to replacing (E, u) with E−x0 r , r − 1 2 u(x 0 + ry) , see Lemma 6, we can assume that (E, u) is a (Λr γ , α)-minimizer of F r in C 4 , 0 ∈ ∂E and, by [24,Proposition 22.1], Applying [24,Lemma 22.10 and Lemma 22.11], we get that then our aim is to show for any c ∈ R. Actually it suffices to prove it only for |c| < 1 4 ; indeed, for |c| ≥ 1 4 , we have: Step 2: the set function , defines a Radon measure on R n−1 , concentrated on D 2 .We apply Lemma 10 to E in every cylinder K s (z) with z ∈ R n−1 and s > 0 such that where Multiplying by s 2 and using an approximation argument to remove the second assumption in (73), we obtain: for D 2s (z) ⊂ D 2 , where we used that s < 1.In order to prove the thesis, we use a covering argument by setting We cover D s (z) by finitely many balls {D z k , s 4 } k∈{1,..., Ñ } with centers z k ∈ D s (z).Of course, this can be done with Ñ ≤ Ñ (n), for some Ñ (n) ∈ N. Hence, by the sub-additivity of ζ and (74) for s 4 , since D s (z k ) ⊂ D 2 , we have: Passing to the supremum for D 2s (z) ⊂ D 2 we infer that Qh and thus Q ≤ c(n, N, L, α)h.In both cases we obtain: which leads to the thesis.

Energy first variation
In this section we deduce a kind of Taylor's expansion formula, with respect to a parameter t ∈ R, for the energy quantity involved in the definition of (Λ, α)-minimizer, under a "small" domain perturbation of the type Φ t (x) = x + tX(x).
We start with the energy of the rescaled functional F r .For the sake of simplicity we will denote with A 1 (x, s) the matrix whose entries are a hk (x, s), A 2 (x, s) the vector of components a h (x, s), A 3 (x, s) = a(x, s) and similarly for B i , i = 1, 2, 3. Then we define Theorem 7 (First variation of the bulk term).Let u ∈ H 1 (B 1 ) and let us fix X ∈ C 1 0 (B 1 ; R n ).We define Φ t (x) := x + tX(x), for any t > 0. Accordingly we define where L α is defined in (7).
Proof.Taking into account that where we set for i = 1, 2, 3. From the previous identity, by subtracting the term ˆB1 F r (x, u, ∇u) + ½ E (x)G r (x, u, ∇u) dx, we gain: Let us estimate separately the two terms I 1 , I 2 on the right-hand side.By the Hölder continuity of the data with respect to the first variable given in ( 7) and Young's inequality we get Regarding I 2 we have that where C = C(N, L, ∇X ∞ ).From the last estimates the thesis easly follows.
Theorem 8 (First variation of the perimeter).If A ⊂ R n is an open set, E ⊂ R n is a set of locally finite perimeter and Φ t (x) := x + tX(x) for some fixed X ∈ C 1 0 (A; R n ), then where the tangential divergence of X, div E X : The last result we will use in the sequel concerns the penalization term (see [24,Lemma 17.9]).
Theorem 9. Let A ⊂ R n be an open set, E ⊂ R n be a set of locally finite perimeter and Φ t (x) := x + tX(x), for some fixed X ∈ C 1 0 (A; R n ), be a local variation in A, i.e. {x = Φ t (x)} ⊂ K ⊂ A, for some compact set K ⊂ A and for |t| < ε 0 .Then where C is a positive constant.
Proof.Without loss of generality we may assume that τ < 1 8 .Let us rescale and assume by contradiction that there exist an infinitesimal sequence {ε h } h∈N ⊆ R + , a sequence {r h } h∈N ⊆ R + and a sequence {(E h , u h )} h∈N of (Λr γ h , α)-minimizers of F r h in B 1 , with equibounded energies, such that, denoting by e h the excess of E h and by D h the rescaled Dirichlet integral of u h , we have e h (0, 1) = ε h , D h (0, 1) + r γ h ≤ M ε h and e h (0, τ ) > C 10 (τ 2 e h (0, 1) + D h (0, 2τ ) + (τ r h ) γ ), with some positive constant C 10 to be chosen.Up to rotating each E h we may also assume that, for all h ∈ N, Step 1. Thanks to the Lipschitz approximation theorem, for h sufficiently large, there exists a 1-Lipschitz function f We define and we assume, up to a subsequence, that {g h } h∈N converges weakly in H 1 (D 1 2 ) and strongly in L 2 (D 1  2 ) to a function g.We prove that g is harmonic in D 2 ), by weak convergence we have Using the Lipschitz continuity of f h and the third inequality in (76), we infer that the second term in the previous equality is infinitesimal: Therefore, we should prove (77).We fix δ > 0 so that spt φ × [−2δ, 2δ] ⊂ B 1 2 and choose a cut-off function ψ : R → 1 AND 2 Applying Theorem 7 and Theorem 9 in the right-hand side we get for some C = C(N, L, L α , α, Λ, X ∞ , ∇X ∞ ).Then, using the second assumption in (75), we obtain We want apply now Theorem 8 on the left-hand side.For this reason let us observe that by Lemma 9, for h large enough, |x n | < δ for every x ∈ ∂E h , so that ψ ′ = 0 and then we can write Therefore, applying Theorem 8, we obtain and then inserting this equality in (78) we deduce, Finally, if we replace φ by −φ, we deduce dividing by ε h then recalling that α > n−1 n ≥ 1 2 we deduce lim Since by the second inequality in (76) we have then by (79) and the area formula, we infer This proves that g is harmonic.
Step 2. The proof of this step now follows exactly as in [15] using the height bound lemma and the reverse Poincaré inequality.We give here the proof for the sake of completeness.
By the mean value property of harmonic functions, Lemma 25.1 in [24], Jensen's inequality, semicontinuity and the third inequality in (76) we deduce that On one hand, using the area formula, the mean value property, the previous inequality and setting we have lim sup On the other hand, arguing as in Step 1, we immediately get from the height bound lemma and the first two inequalities in (76) that Hence we conclude that lim sup We claim that the sequence {e h (0, 2τ, ν h )} h∈N is infinitesimal; indeed, by the definition of excess, Jensen's inequality and the third inequality in (76) we have Therefore, applying the reverse Poincaré inequality and (80), we have for h large that e h (0, τ ) ≤ e h (0, τ, ν h ) ≤ C 9 ( Cτ 2 e h (0, 1) + D(0, 2τ ) + (2τ r h ) γ ), which is a contradiction if we choose C 10 > C 9 max{ C, 2 γ }.

Proof of the main theorem
The proof works exactly as in [15].We give here some details to emphasize the dependence of the constant ε appearing in the statement of Theorem 1 from the structural data of the functional.The proof is divided in four steps.
Step 1.We show that for every τ ∈ (0, 1) there exists ε 5 = ε 5 (τ ) > 0 such that if e(x, r) ≤ ε 6 , then D(x, τ r) ≤ C 4 τ D(x, r), where C 4 is from Lemma 5. Assume by contradiction that for some τ ∈ (0, 1) there exist two positive sequences (ε h ) h and (r h ) h and a sequence (E h , u h ) of (Λr γ h , α)-minimizers of F r h in B 1 with equibounded energies such that, denoting by e h the excess of E h and by D h the rescaled Dirichlet integral of u h , we have that 0 ∈ ∂E h , e h (0, 1) = ε h → 0 and D h (0, τ ) > C 4 τ D h (0, 1). (81) Thanks to the energy upper bound (Theorem 3) and the compactness lemma (Lemma 8), we may assume that E h → E in L 1 (B 1 ) and 0 ∈ ∂E.Since, by lower semicontinuity, the excess of E at 0 is null, E is a half-space in B 1 , say H.In particular, for h large, it holds where ε 0 is from Lemma 5, which gives a contradiction with the inequality (81).
In particular e(x, τ k 0 r 0 ) ≤ τ 2σk 0 and, arguing as in [15], we obtain that for every x ∈ ∂E ∩ V and 0 < s < t < r 0 it holds for some constant c = c(n, τ 0 , r 0 ), where The previous estimate first implies that Γ ∩ U is C 1 .By a standard argument we then deduce again from the same estimate that Γ ∩ U is a C 1,σ -hypersurface.Finally we define Γ := ∪ i (Γ ∩ U i ), where (U i ) i is an increasing sequence of open sets such that U i ⊂⊂ Ω and Ω = ∪ i U i .
Step 4. Finally we are in position to prove that there exists ǫ > 0 such that Being the argument rather standard, Setting Σ = x ∈ ∂E \ Γ : lim r→0 D(x, r) = 0 , by Lemma 2 we have that ∇u ∈ L 2s loc (Ω) for some s = s(n, ν, N, L) > 1 and we have that dim H x ∈ Ω : lim sup r→0 D(x, r) > 0 ≤ n − s.

2 :
Regularity of D c w. First of all observe that ∇ τ (D c w) = D c (∇ τ w) − G, e n .This implies by Step 1 that the tangential derivatives of D c w belong to L 2 loc (B r (x 0 )).Furthermore we can estimate directly by definition of D c w: |∇ n (D c w)| ≤ c(n, N )|∇∇ τ w|, which implies again by Step 1 |∇D c w| ≤ c(n, N )|∇∇ τ w|.
Assume that there exist a sequence (λ h ) h∈N such that λ h → ∞ as h → ∞ and a sequence of configurations (E h , u h ) minimizing F λ h and such that u h = u 0 on ∂Ω and |E h | = d for all h ∈ N. Let us choose now an arbitrary fixed E 0 ⊂ Ω with finite perimeter such that |E 0 | = d.Let us point out that |∇v| 2 dy + C G r n+σ .