Minimality via second variation for microphase separation of diblock copolymer melts

We consider a non local isoperimetric problem arising as the sharp interface limit of the Ohta-Kawasaki free energy introduced to model microphase separation of diblock copolymers. We perform a second order variational analysis that allows us to provide a quantitative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the nonlocal perimeter. Moreover we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from minimality for configurations close to the minimum in the $L^1$ topology .


Introduction
In this note we are interested in performing a second order analysis for a nonlocal isoperimetric problem arising as a variational limit of the Ohta-Kawasaki functional introduced for a density functional theory for microphase separation of A/B diblock copolymers.
Among the several mean field approximation theories proposed to model the phase separation in diblock copolymer melts, the one derived by Ohta and Kawasaki in [15] turns out to be one of the most promising from the mathematical point of view. Let Ω ⊂ R n be a domain representing the volume occupied by the polymeric material. The free energy can be written as a nonlocal functional of Cahn-Hilliard type as is the difference of the phases' volume fractions. The formulation (1.1) was first introduced in [14] and for a derivation of the Ohta-Kawasaki density function theory from the self-consistent mean field theory we refer the reader to [6] and the references therein.
As pointed out in [6], depending on the molecular structure of the polymers there are several regimes of phase mixture. Nevertheless the presence of an observable phase separation occurs in the so called intermediate or strong segregation regimes, wherein the domain size is much larger than the interfacial length. This suggests that most of the features of the model can be described, from a mathematical point of view, by looking at the sharp interface limit of E ε as the thickness ε of the diffuse interface tends to zero. Thus we are lead to study the minimizers of the following energy functional, that arises as the Γ-limit of E ε in the L 1 topology, where u : Ω → {−1, 1} is a function of bounded variation and |Du|(Ω) denotes its total variation in Ω. The functional (1.2) can be described in a more geometric way that turns out to be more suitable for our analysis. Indeed identifying the function u with the set E = {x ∈ Ω | u(x) = 1} and using the properties of G(x, y), we can rewrite E(u) as where P Ω (E) stands for the perimeter of E in Ω and v E is the solution of We are thus lead to study the variational problem We explicitly observe that the condition Ω u E dx = m is nothing but a volume constraint since |E| = 1 2 (m + |Ω|).
The problem (1.5) has been presented in [7] as a mathematical paradigm for the phenomenon of energy-driven pattern formation associated with competing short and long-range interactions. Recently an increasing interest has been devoted to the second order variational analysis of energy functionals which exhibit this competing behaviour. Indeed it has been successfully applied to prove stability and minimality criteria in several contexts (see for instance [1,5]). Directly related to our problem is the work by Ren and Wei and by Choksi and Sternberg (cfr. [16,17,18,19,20,7]). In a series of papers, as a first step toward the validation of the conjectured periodicity of global minimizers of (1.3), they calculate the second variation of (1.3) at a critical configuration and, among several other applications, they construct examples of periodical critical configurations and find conditions under which their second variation is positive definite.
A different approach is taken in [1], where the authors consider the problem of minimizing the functional (1.3) in the periodic case, i.e., when Ω is the n-dimensional flat torus. They prove the local minimality of critical configurations which second variation is positive definite. In this direction goes also our present work, whose aim is to prove that a similar minimality criterion holds true for the problem (1.5). Indeed we prove in Theorem 2.3 that if E is a sufficiently regular critical point of (1.3) such that the quadratic form ∂ 2 J(E) associated to the second variation of J at E is positively defined, then there exists a constant c 0 > 0 such that J(F ) ≥ J(E) + c 0 |F ∆E| 2 for any admissible set F ⊂ Ω sufficienlty close to E in the L 1 topology. This in particular implies that E is a local minimizer of J and in addition provides a quantitative estimate of the deviation from minimality for sets near to E. We remark that our result holds also in the case γ = 0 and thus we cover the case of volume constrained isoperimetric problem in a regular domain Ω. Some comments are in order on the differences between the periodic case (studied in [1]) and the Neumann boundary case. Indeed if on one side working with the Neumann setting dispenses us from several technicalities needed to deal with the translation invariance of the functional J, on the other side new delicate arguments need to be introduced to deal with problems which arise when E touches ∂Ω. We remark that in [1] the Neumann boundary case was considered only under a rather restrictive assumption that the set E does not intersect the boundary of Ω.
Finally we outline the structure of the paper. In the next section we introduce the notation and present the framework in order to precisely state our main results. In Section 3 we discuss some regularity results for ω-minimizers of the area functional and in particular a stability result for the regularity (cfr. Theorem 3.5) which will play an important role in the proof of the main theorem. Section 4 is devoted to the lengthy calculations of the second variation formula for regular sets satisfying the orthogonality condition on ∂Ω. The proof of the main result, Theorem 2.3, is divided in sections 5 and 6. The scheme follows a well established path (see for instance [1,5]). First we use the general second variation formula from Proposition 4.1 to prove the local minimality among regular sets which are close to the critical one in the W 2,p -topology. The final result is then proved by contradiction using a penalization argument and exploiting the regularity theory of ω-minimizers.

Preliminaries and statement of the results
In this section we set up the basic notation, recall some preliminary results and present the statements of the main results. Throughout the paper Ω ⊂ R n is assumed to be a bounded domain with C 3,αboundary, for some α > 0.
We say that a set E ⊂ Ω is C k,α regular, where k ≤ 3, if its relative boundary M (2.1) M := ∂E ∩ Ω is a C k,α -manifold with or without boundary if ∂E ∩ ∂Ω = ∅ or ∂E ∩ ∂Ω = ∅ respectively. Suppose E is C 1 regular and let X be C 1 -vector field in Ω which satisfies where ν M is unit normal of M . We define the associated flow Φ : We then define the first variation of (1.3) at E with respect to the field X (or flow Φ) by and the second variation by Formula for the first variation of (1.3) is well known. Proposition 2.1. Let E ⊂ Ω be a C 1 -regular set with Ω u E dx = m. Suppose that X satisfies (2.2) and (2.3). The first variation of (1.3) at E with respect to X can be written as Here ν * is the outward unit co-normal of M ∩ ∂Ω, i.e., normal to M ∩ ∂Ω and tangent to M , and ν Ω is the unit normal of Ω.
We say that C 1 regular set E ⊂ Ω is critical if the first variation is zero for every C 1 -vector field X which satisfies (2.2) and (2.3).
If we choose the vector field X such that it is compactly inside Ω, spt(X) ⋐ Ω, then the last term in the first variation is trivially zero. Therefore if E is critical it satisfies the Euler-Lagrange equation in a weak sense where the constant λ is Lagrange multiplier associated to the volume constraint. We may further deduce that also the second term in the first variation vanishes M∩∂Ω X, ν * dH n−2 = 0. We may use standard regularity theory for elliptic equations, as in [1], to deduce that every regular critical set is a C 1,α -manifold with boundary. Since v E solves (1.4) we obtain that v E ∈ C 1,β (Ω) for every β ∈ (0, 1). Standard Schauder estimate then implies that M is in fact C 3,β -manifold with boundary for every β ∈ (0, 1).
It is well known that the second variation of (1.3) at a regular critical point has a special structure and it can be written as a quadratic form. To present the formula we remark that the non-local part of (1.3) can be written more explicitly in terms of E by using Green's function. Let G : Ω × Ω → R be the Green's function with Neumann boundary conditions in Ω, The functional (1.3) can be written as Derivation of the second variation formula for critical sets can be found in [7] and [22], as it was noted after Remark 2.8 in [7]. To present the formula let E ⊂ Ω be a regular critical set and denote its relative boundary by M . The second variation of (1.3) at E with respect to X can be written as is defined by the trace operator. We will return to the subject of second variation in section 4 and the role of the function ϕ.
It is rather straightforward to show that if C 1 regular set E is a local minimizer of (1.3), i.e., there is δ > 0 such that J(F ) ≥ J(E) for every |F ∆E| ≤ δ, then the second variation is positive semi-definite for every ϕ ∈ H 1 (∂E) with ∂E ϕ dH n−1 = 0. The proof for this result is very similar to ([1], Corollary 3.4) and will be omitted here. As it was pointed out in the introduction, our main result Theorem 2.3, deals with the question whether we may conclude the stability from the formula (2.6).
Our goal is to prove that a regular critical point, i.e. a critical point with C 1 -boundary, with positive second variation is indeed a isolated local minimum.
for every ϕ ∈ H 1 (∂E) with ∂E ϕ dH n−1 = 0. Then E is a strict local minimum and there are c > 0 and δ > 0 such that for every F ∈ BV (Ω) with Ω u F dx = m and |F ∆E| ≤ δ.
In order to prove Theorem 2.3 we need to calculate the second variation at any regular set which satisfies the orthogonality condition, not just for critical ones. This is done in Proposition 4.1 and thus we obtain the formula (2.6) as a corollary.
It is well known that the functionals E ε Γ-converges in L 1 to E. As a corollary we obtain the following stability result.
Corollary 2.4. Assume E is a regular critical point of J with positive second variation and denote u E = χ E − χ Ω\E . There is ε 0 and a family {u ε } ε<ε0 of isolated local minimizers of E ε with constraint

Critical sets and Regularity of ω-minimizers
Let us briefly discuss about the regularity of local minimizers of (1.3). Let us assume that we know, a priori, that our local minimizer is C 1 regular then we may use the Euler-Lagrange equation (2.4) to deduce that it is in fact C 3,α regular, as we pointed out in the previous section. Since our attention is in local minimizers, not in critical points, we will use the minimality itself to obtain the first order regularity. This will be done by using the theory of ω-minimizers (of area). Definition 3.1. A set E ∈ BV (Ω) is an ω-minimizer with constants Λ > 0 and r > 0 if for every G ⊂ Ω with G∆E ⊂ B r (x 0 ) it holds P Br (x0) (E) ≤ P Br (x0) (G) + Λ|G∆E|.
We remark that the above definition differs slightly from the standard definition of ω-minimizer which appears in the literature e.g. in [11] and [23], where the measure of the symmetric difference |G∆E| is replaced by the volume of the ball |B r |. Definition 3.4 is clearly stronger and we may therefore apply known results from previously mentioned works. The motivation for the stronger definition is that for regular ω-minimizer we obtain a curvature bound.
Proposition 3.2. Suppose that E ∈ BV (Ω) is an ω-minimizer with constants Λ > 0 and r > 0 and that E is C 1 regular. Then there exists a bounded function f such that E satisfies the equation Proof. We may locally write M as a graph of a C 1 -function. Suppose that φ : D ⊂ R n−1 → R is such a function and let η ∈ C 1 0 (D). From the ω-minimizing property we obtain for every t it holds Divide by t and let t → 0 to obtain D ∇φ, ∇η Arguing exactly as in Proposition 7.41 in [3] we conclude that E satisfies (3.1) for some f with ||f || L ∞ ≤ Λ.
We return to the C 1 -regularity of ω-minimizers. The following regularity result in a smooth domain Ω follows from a work by Grüter [11].
However he proves the result first by showing that the solutions of the partitioning problem are ωminimizers. Then he uses a result from [12] to prove the C 1 -regularity by using the fact that mean curvature is L p -integrable with p > n which follows from the corresponding Euler-Lagrange equation.
Hence may apply this result to our ω-minimizers which by Proposition 3.2 have bounded mean curvature.
The interior regularity follows from a work by Tamanini [23]. As in [1] (Lemma 2.6) we have the Lipschitz continuity of the non-local part, i.e., for two measurable sets E, F ⊂ Ω we have that Here v E and v F are defined as in (1.4). We may use an argument from [11] to prove that local minimizers of (1.3) are ω-minimizers in the sense of Definition 3.4. Although the proof is an exact copy from [11] we include it for the convenience of the reader.
Proof. Let G ⊂ Ω be such that G∆E ⊂ B r (x 0 ) for some r > 0. We may assume that |G| ≤ |E| for in the case |G| ≤ |E| we argue similarly. Find a ball B r (y 0 ), disjoint with B r (x 0 ), such that We may use the isoperimetric inequality to estimate We may use the regularity theory for ω-minimizers and the equation (2.4) to obtain that every local minimizer of (1.3) C 3,α -regular outside a critical set Γ with dim H (Γ) ≤ n − 8. This motivates us to define regular critical sets as critical sets with no singularities.
It is well known that in higher dimension n ≥ 8 the singular set even of a minimal surface might not be empty and therefore the regularity for ω-minimizers is optimal. However by the flatness theorem of De Giorgi (see [10]) we obtain the full regularity if we know that the blow-up limit at every point on the minimal surface is flat enough. Since every blow-up limit of an ω-minimizer converges to a minimal cone we obtain the following convergence result which concludes the section. The result can be obtained by following [2], [13] and [21] as explained in [24] with a few modifications. See also [23].
In particular, ∂E k is C 1,α -regular when k is large.

Second variation formula
In this section we calculate the second variational of the functional (1.3). Our main task is to write the formula in a form where the quadratic structure appears. As in [1] we have to generalize the formulas from [7] and [22] and calculate the second variation at any regular set E, not necessarily critical, and with respect to any given flow.
We assume that E ⊂ R n is C 2 regular by which we mean that M = ∂E ∩ Ω is a C 2 -manifold with boundary. As in section 2 we consider a vector field X ∈ C 2 (Ω; R n ) which satisfies the tangent condition (2.2) and the associated flow Φ : We define the second variation at E with respect to the flow Φ as A major technical challenge are the calculations for the local part of the energy P Ω (·). We follow a slightly different path than ( [4], Proposition 3.9) where the proof contains calculations for the general second variation formula of the perimeter in R n . Instead of differentiating the first variation formula in Proposition 2.1 we will first calculate the second derivative of P Ω (E t ) as it is done e.g. in [10] and then use the divergence theorem in order to find the "right" formula. This saves us from differentiating the boundary term which appears in Proposition 2.1. Along the calculations it will become clear that we need to assume the orthogonality condition (2.5) on E in order to find the quadratic structure in its second variation. As we pointed out in the previous section the orthogonality condition appears in the regularity theory for ω-minimizers. It can thus be viewed as a part of regularity assumption on E.
We recall the divergence theorem on M where ν * is the unit co-normal of M ∩ ∂Ω, see Proposition 2.1. Imposing the orthogonality condition on E implies ν * = ν Ω on M ∩ ∂Ω. We may thus write the integration by parts formula for f ∈ C 1 (Ω) as Proposition 4.1. Let E ⊂ R n be C 2 regular set which satisfies the orthogonality condition (2.5) and denote M = ∂E ∩ Ω. Suppose that X satisfies the tangent condition (2.2) and denote X ν = X, ν M ν M and X τ = X − X ⊥ . The second variation of (1.3) at E with respect to X can be written as Here B ∂Ω stands for the second fundamental form of ∂Ω, i.e., the Hessian of the distance function from the boundary and |B M | 2 is the sum of the square of the principal curvatures of M .
Before the proof we remark that the above formula consists of the quadratic form (2.6) by defining ϕ := X, ν M and two additional terms. It turns out that both of these extra will terms vanish when E is critical. Indeed, since the minimizing problem (1.5) is volume-constrained it is natural to assume that the flow Φ preserves the volume From the volume constraint we obtain the first order condition d These calculations are standard and can be found e.g. in [7]. Therefore if E is a regular critical set, i.e., H + 4γv E is constant and the flow is volume preserving we obtain the quadratic form (2.6) with ϕ = X, ν M . Arguing exactly as in ( [1], Corollary 3.4) we deduce that every regular local minimizer of We give now the proof of the Proposition 4.1.
Proof of the Proposition 4.1. Denote the acceleration vector field by Z = ∂ 2 ∂t 2 Φ t t=0 = DX [X]. We treat the perimeter and the nonlocal part of the energy separately and denote For simplicity we write ν = ν M .
The value for F ′′ (0) is calculated in [7] in the periodic setting. However, as it was pointed out in Remark 2.8 in [7], the Neumann boundary case does not produce any new terms to it and the formula for F ′′ (0) is the same as in the periodic case. Recalling the condition (2.2) we have from [7] The value of E ′′ (0) is well known in the following form, [10] (Theorem 10.4), where we also used the orthogonality to conclude ν * = ν Ω on M ∩ ∂Ω. Notice that this is the second variation formula for the perimeter P Ω (E t ) given by any vector field X.
We have to manipulate (4.2) in order to find the quadratic structure in it. We proceed by writing The goal is to decompose every term in (4.2) by using X = X ν + X τ . We will first treat the term Since div M X ν = H X, ν the second term M (div M X) 2 dH n−1 can be written as For the third term we will use the equalities (D τ X ν ) 2 = X, ν 2 (Dν) 2 + (D τ X, ν · ν) ν ⊗ D τ X, ν and (D τ X τ )(D τ X ν ) = X, ν Dν D τ X τ . Hence, we deduce We treat the fourth term M H Z, ν dH n−1 by writing (here we denote Z τ = D(X τ )(X τ )) where in the last equality we have used div M X ν = H X, ν .
For the last term M∩Ω Z, ν Ω dH n−2 we notice that the tangent condition (2.2) on X implies X, ν Ω = 0 on M ∩ ∂Ω. In particular, D X, ν Ω · X = 0 on M ∩ ∂Ω. Hence the last term becomes From now on we will use the notation Dν Ω X, X = B ∂Ω (X, X).
We continue by treating the mixed terms in (4.8). We integrate by parts the third term in (4.8) and deduce For the fourth term in (4.8) we observe that, by the orthogonality condition, X τ vanishes on M ∩ ∂Ω and therefore integration by parts yields (4.10) Next we introduce a notation δ i for partial derivative on M , i.e., δ i g = ∂ xi g − Dg, ν ν i for any smooth function g. We use the facts δ j ν i = δ i ν j and where ∆ M = n i=1 δ i δ i is the Laplacian on M . Together with (4.9), (4.10) and (4.11) the formula (4.8) becomes Let us treat the last row in the previous formula. We claim that Indeed, the formula (4.13) is nothing but the second variation formula (formula (4.2)) of the perimeter with respect to the vector field X τ . As we noticed before X τ vanishes on M ∩ ∂Ω and therefore the flow associated by X τ leaves M unchanged. Hence, we have (4.13).

W 2,p -minimality
In this section we prove that a regular critical set E with a positive second variation is a strict local minimizer among sets which are regular and close to E in a strong sense (in W 2,p topology). This result is stated in Proposition 5.2 and is interesting in itself. The idea is to take an arbitrary set F near E and to construct a volume preserving flow Φ such that Φ(E, 1) = F . We use Proposition 4.1 to estimate the second variation at E t = Φ(E, t) at any time t ∈ [0, 1]. Notice that by the assumption on E we know that the second derivative of t → J(E t ) at t = 0 is strictly positive. We then use the fact that E t are close to E in W 2,p topology to deduce that the function t → J(E t ) is in fact strictly convex. We note that in order to use Proposition 4.1 all the sets have to satisfy the orthogonality condition.
We begin by a simple compactness argument. The proof is exactly the same as Lemma 3.6 in [1] and will therefore be omitted.
Lemma 5.1. Suppose that E is a critical point with positive second variation. There is c 0 > 0 such that for every ϕ ∈ H 1 (∂E) with ∂E ϕ dH n−1 = 0.
We define W 2,p and C 2 distances between regular sets E, F ⊂ Ω as Here is the main result of the section.
Proposition 5.2. Let E be as in Theorem 2.3 and p > n. There is δ > 0 and a constant c 1 > 0 such that for every F ⊂ Ω with |F | = |E| which satisfies the orthogonality condition and ||E, F || W 2,p ≤ δ it holds We first prove some technical lemmata and then give the proof of Proposition 5.2 at the end of the section. Most important of the lemmata is Lemma 5.3 where we construct a vector field X and a flow Φ which takes the given set E to a regular set F nearby. The proof is technical and long and it is therefore divided into three steps. The difficulty lies in the fact that the flow needs to satisfy both orthogonality and the volume constraint. We divide the proof by first constructing a primitive flow which satisfies the orthogonality condition and then, at the third step, we correct it so that finally also the volume constraint is satisfied. Lemma 5.3. Let E be as in Theorem 2.3. Suppose F is C 2 regular set which satisfies the volume constraint |F | = |E|, the orthogonality condition (2.5) and ||E, F || W 2,p ≤ δ, where p > n. When δ is small enough there is C 2 regular vector field X with uniformly bounded W 2,p -norm and associated flow such that at every t ∈ [0, 1] the set E t = Φ(E, t) satisfies the volume constraint |E t | = |E|, and the orthogonality condition (2.5). Moreover the flow takes E to F , i.e., E 1 = F . Proof.
Step 1: First we construct a primitive vector field Z which has the property that the associated flow Φ Z : Ω × (−t 0 , t 0 ) → Ω, is such that, at any t ∈ (−t 0 , t 0 ), the set Φ Z (E, t) satisfies the orthogonality condition. We will build the vector field only in a neighbourhood of the relative boundary M = ∂E ∩ Ω, which we denote by U . By neighbourhood of M we mean a connected set U which contains M and is open with respect toΩ. We begin by defining a function which plays the role of a signed distance function of the set E with respect to the domain Ω.
Since E is C 3,α regular and satisfies the orthogonality condition there is a C 3,α regular function d Ω,E : U → R which satisfies Such a function can be found e.g. by minimizing the energy with respect to Dirichlet boundary condition u = 0 on M and u = 1 on (∂U ∩ Ω) \ E. Here we assume without loss of generality that the boundary ∂U ∩ Ω meets ∂Ω orthogonally. We may choose d Ω,E in U \ E to be the minimizer of the above problem. One may then extend this function naturally to whole U .
It follows from the above conditions (i) − (iii) that for small values of t ∈ R the level sets form a foliation of U with C 3,α regular sets which meet ∂Ω orthogonally by the Neumann boundary condition (iii). We will define the primitive flow Φ Z such that For future purposes we need yet to construct the trajectories of the flow.
To that aim we construct a vector field Z ′ which defines the trajectories. We begin by defining To make sure that we will not loose the orthogonality in the forthcoming steps we correct Z 1 near the boundary ∂Ω. This will be done by "copying" the geometry of ∂Ω into its neighbourhood. We denote the neighbourhood of ∂Ω by U ′ .
Since ∂Ω is regular we may define the standard projection to ∂Ω, Π : U ′ → ∂Ω, as Π(x) = y x , where y x ∈ ∂Ω is the unique point for which Every point x ∈ U ′ can therefore be written as x = Π(x) − d(x, Ω)ν Ω (Π(x)) where ν Ω (y) is the outer normal of Ω at y ∈ ∂Ω. We define the field Z 2 such that Z 2 = Z 1 on the boundary ∂Ω ∩ U and The advantage of Z 2 is the following geometrical fact. If we know that the points x, y ∈ ∂Ω ∩ U lie on the same trajectory, then for small h > 0 also the points x − hν Ω (x) and y − hν Ω (y) lie on the same trajectory. We will use this fact frequently.
Finally we define Z ′ as where η ∈ C ∞ 0 (Ω) is a standard cut-off function such that η ≡ 0 near ∂Ω and η ≡ 1 outside U ′ . By choosing the cut-off function properly we can make sure that Z ′ is close to ∇d Ω,E , i.e., For every point x ∈ M we define the primitive flow x(t) such that the trajectories are given by Z ′ and x(t) is the unique point on the intersection of the level set {x ∈ Ω | d Ω,E (x) = t} and the trajectory passing through x. In other words x(·) is such that x(0) = x and We denote the flow by Φ Z (x, t) and the associated vector field by Z. Since Z ′ is close to ∇d Ω,E on M = {x | d Ω,E (x) = 0} we have that The flow Φ Z satisfies the orthogonality condition by construction. In fact the construction implies that every point y ∈ ∂Ω ∩ U can be written as y = Φ Z (x, t) for some t ∈ (−t 0 , t 0 ) and it holds This is a priori a stronger than the orthogonality condition since it implies that the vector DΦ Z (x, t)ν Ω (x) is the unit co-outer normal of the set Φ Z (M, t) and it equals ν Ω (Φ Z (x, t)).
To verify (5.4) write y = Φ Z (x, t) and notice that by the construction, for every small h > 0, the points x − hν Ω (x) and y − hν Ω (y) lie on the same trajectory. Therefore there is t h such that In particular, we deduce from (5.2) that d Ω,E (y − hν Ω (y)) = t h .
Notice that d Ω,E (y) = d Ω,E (Φ Z (x, t)) = t. Therefore by the orthogonality of the set {x | d Ω,E (x) = t} it holds Consequently t h = t + o(h) and (5.4) follows.
Step 2: Next we change Z to Y so that the new flow, denoted by Φ Y , satisfies Φ Y (E, 1) = F and the orthogonality is preserved.
In the neighbourhood U of M the map Φ Z | M×(−t0,t0) → U is C 3,α -diffeomorphism and every point y ∈ U can be written as y = Φ Z (x, t) for some x ∈ M and for t = d Ω,E (y). We may therefore define implicitly a projection π : U → M such that Φ Z (π(y), d Ω,E (y)) = y.
Since Z and d Ω,E are C 3,α regular, also π is C 3,α regular.
By the assumption on F there is C 2 -diffeomorphism Ψ : E → F with ||Ψ − Id|| W 2,p < δ. Without further mention we always assume δ > 0 to be small. We define a map S : M → M as S(x) = π(Ψ(x)).
The tangential differential of S is From the regularity of π and from the fact π(x) = x for x ∈ M we conclude that |D τ π(Ψ(x))| ≥ c. Since We define further a projection on F M F := ∂F ∩ Ω, π F : U → M F , as π F (y) = Ψ(S −1 (π(y))) which labels for every point y ∈ U a unique point z y on M F on the same trajectory. Trivially the map π F is constant along the trajectories of Φ Z and the value T F (y) := d Ω,E (π F (y)) denotes the time needed from M to M F along the trajectory passing through y. We define the vector field Y by and denote the associated flow by Φ Y . We note that we may write Φ Y explicitly as We denote by F t the set enclosed by Φ Y (M, t).
We have to make sure that at any time t ∈ (0, 1) the set F t satisfies the orthogonality condition. As in the previous step we will show that for any y ∈ Φ Y (M, t) ∩ ∂Ω it holds for some x ∈ M . This implies the orthogonality. To show (5.8) we use (5.7) to calculate Hence by (5.4) we need to show Since T F (y) = d Ω,E (π F (y)), (5.9) can be written as We claim that Dπ F (x)ν Ω (x) = ν Ω (π F (x)). Then (5.9) follows from the fact that d Ω,E satisfies the Neumann boundary condition ∂d Ω,E ∂ν Ω = 0.
To show Dπ F (x)ν Ω (x) = ν Ω (y) where y = π F (x), recall that π F is a map which labels for every x ′ ∈ U a unique point y ′ ∈ M F on the same trajectory. Therefore the points x − hν Ω (x), y − hν Ω (y) and π F (x − hν Ω (x)) all lie on the same trajectory. It follows from the orthogonality of F that and the claim follows.
At the end of the step we will write down some useful properties of the flow Φ Y and the field Y for the next step. First of all, it follows from the definition of Y (5.6), (5.3), (5.5) and the fact that Z is C 2 regular that In particular since T F is constant along the trajectories (5.10) yields and we may also estimate the divergence by t)), and Φ Y (x, 0) = x we may estimate, by differentiating the above equation for Φ Y twice and by using the second inequality in (5.10), that The above estimate together with (5.3) imply Step 3: We modify Y to X so that the new flow will finally be volume preserving. We note that the sets satisfy everything except the volume constraint but the final set |F 1 | = |F | has the right volume. The idea is to study the evolution of F t \ E and E \ F t . If the volumes of these two sets do not match we give the other one a little boost. More precisely we would like to find for every t ∈ [0, 1] a value s(t) such that The problem is that this would break the regularity of the flow.
To overcome this problem we choose a smooth function f defined on M such that that f ≃ χ M∩F .
To be more precise, we choose f such that it satisfies For every t ∈ [0, 1] we define a flowΦ t : and the set enclosed byΦ t (M, s) by F t,s . We denote the associated vector field byX t which we may write asX t (y) = f (π(y)) Y (y) where π is the projection on M defined in the beginning of step 2. We note that F t,0 = F t is the set given by the flow Φ Y . Moreover since f ≡ 1 on M ∩ F we have The idea is to correct the volume of F t such that the new set, denoted by E t , will satisfy the volume constraint. To that aim we define a function ϕ(t, s) := |F t,s | − |E|.
The correction value s(t) is then implicitly defined as ϕ(t, s(t)) = 0 and E t := F t,s(t) . To be more rigorous for every t ∈ [0, 1] we define s(t) as First we have to make sure that s(t) is well defined.
From (5.11) and (5.13) we conclude that the growth of the volume |F t \ E| do not degenerate By notation α ≃ β we mean that C −1 α ≤ β ≤ Cα for some constant C. In particular the above estimate implies (5.18) Since the vector fieldX t has the same regularity as Y we may estimate the growth of |E \ F t,s | for every t ∈ (0, 1) and every s ∈ (−t, 1 − t) exactly as we with (5.17) and obtain We note that T F (x) = 0 for x ∈ M ∩ ∂F . It is therefore clear that the above inequality and the condition (ii) on f imply To prove (5.16) we write ϕ(t, s) = |F t,s \E|−|E\F t,s |. From (5.14) it follows that |F t,s \E| = |F t+s \E|. Therefore (5.16) follows by choosing ε to be small enough. Notice that s(0) = s(1) = 0. This follows simply from the fact |F | = |E| and (5.16). To be sure that for every t the value s(t) indeed exists we claim that for t ∈ (0, 1) we have To that aim we again write ϕ(t, s) = |F t+s \ E| − |E \ F t,s |. Since F 0 = E and F 1 = F we conclude that |F t,−t \ E| = 0 and |F t,1−t \ E| = |F \ E|.
In order to conclude that t + s(t) defines a parametrization of the interval [0, 1] we show that for every t ∈ [0, 1] for some µ > 0. Similarly as we obtained (5.19) we use the regularity ofX t to deduce that for every t ∈ (0, 1) and s ∈ (−t, 1 − t) we have The above estimate, (5.17) and (5.20) yield Consequently we obtain (5.22) when ε > 0 is small enough.
We finally define the flow Φ : Ω × [0, 1] → Ω as and extend it smoothly to Ω. The sets E t are defined as By construction E t satisfies the volume constraint at any time t ∈ [0, 1]. It also satisfies the orthogonality condition, which can be checked similarly as it was done in (5.9) by using the Neumann boundary condition ∇f, ν Ω = 0 on M ∩ ∂Ω.
It is clear that the projections are C 2 regular with uniformly bounded W 2,p -norms. The vector field X can be written on F ∆E as (5.25) X(y) = (1 + s ′ (T (y)) f (π(y))) Y (y).
It is then clear that X is C 2 regular with uniformly bounded W 2,p -norm.
In the next lemma we study the regularity properties of the flow Φ and the vector field X constructed in the previous lemma. The reader is encouraged to skip it in the first reading. Lemma 5.4. Suppose that E, F , Φ and X are as in Lemma 5.3 and denote M t := ∂E t ∩ Ω. The vector field X ∈ C 2 (Ω; R n ) satisfies for some constant C.
Proof. We begin by making some remarks on the regularity of the flow Φ and the vector field X. Notice first that by (5.25) the vector field X can be written as where φ is a C 2 regular function with uniformly bounded W 2,p norm and C −1 ≤ φ ≤ C. The estimate |Y (y)| ≤ C|T F (y)| proved in (5.10) yields We recall that for x ∈ M , T F (x) denotes the time needed from x to M F along the primitive vector field Z and ||T F || W 2,p ≤ C||F, E|| W 2,p . Since T F is constant along the trajectories it follows from the above estimate that for every t ∈ [0, 1] Denote Φ t (·) = Φ(·, t) and by ν Mt the normal vector of M t . For every t ∈ [0, 1] the map Φ(·, t) is C 2 diffeomorphism. We may show, exactly as we did (5.13) in Lemma 5.3, that We proceed by showing (5.26). We use the same argument as we did with (5.18) and (5.19) and where we have used the regularity of the flow, i.e., |J M Φ t | ≤ C, (5.28) and (5.30). The previous calculation then yields The first estimate in (5.27) follows from (5.29) and (5.30) and from the W 2,p -regularity of the sets E t . In fact the proof is exatcly the same as the one of Lemma 7.1 in [1] and it will therefore be omitted here.
To prove the second inequality in (5.27) we estimate the divergence by using (5.12) Hence, the second inequality in (5.27) follows from (5.30).
In the following lemma we show by continuity that in a neighbourhood of a critical point with positive second variation the quadratic form remains positive (2.6). Notice that we have to compute the quadratic form for sets which are not critical. Lemma 5.5. Suppose that E is as in Theorem 2.3 and p > n. There is δ > 0 such that for any C 2 -regular set F with ||E, F || W 2,p ≤ δ and every ϕ ∈ H 1 (M F ) with MF ϕ dH n−1 = 0 it holds Here is defined in (2.6) and c 0 is the constant from Lemma 5.1.
Proof. We argue by contradiction and assume that there are F k with ||E, F || W 2,p → 0 and ϕ k ∈ H 1 (M k ) with M k ϕ k dH n−1 = 0, which by scaling we may assume to satisfy ||ϕ k || H 1 (M k ) = 1, such that where M k = ∂F k ∩ Ω. We recall the formula for ∂ 2 J(F k ) where ν k is the unit normal of F k . We will show that there is ϕ ∈ H 1 (M ) with ||ϕ|| H 1 (M) = 1 and M ϕ dH n−1 = 0 such that lim up to a subsequence. This and (5.31) then contradicts Lemma 5.1.
Since ||E, F || W 2,p → 0 there are C 2 -diffeomorphisms Φ k : E → F k such that ||Φ k − Id|| W 2,p → 0. By compactness there exists ϕ ∈ H 1 (∂E) with ||ϕ|| H 1 (∂E) = 1 and ∂E ϕ dH n−1 = 0 such that, up to a subsequence, where M = ∂E ∩ Ω. In particular ϕ k • Φ k → ϕ strongly in L 2 (M ). We also conclude that ν k • Φ k → ν uniformly on M where ν is the unit normal of E. Therefore by the weak lower semicontinuity we obtain the convergence of the first term in (5.32) We proceed by observing the following convergences Indeed, the first one follows trivially from the W 2,p -convergence of F k and the second one follows from the uniform C 1,α -regularity given by the equation (1.4). Therefore we obtain the convergence of the second and the last term in (5.32) Therefore the estimates (5.34) and (5.26) from Lemma 5.4 yield and the claim follows. Hence, we need to proof (5.34).
The formula for d 2 dt 2 J(E t ) in Proposition 4.1 can be written as is the quadratic form defined in (2.6) and ϕ t = X, ν t as before. Notice that since the flow preserves the volume we have When δ > 0 is small enough Lemma 5.5 yields We proceed by showing that when δ > 0 is small enough we have . Therefore when δ > 0 is small we have for small ε > 0 where the last inequality follows from Sobolev inequality and the fact that p > n. The estimate (5.36) then follows from (5.27).
Similarly we prove that when δ > 0 is small we obtain The volume constraint |F t | = |E| yields Mt div(X) X, ν t dH n−1 = 0 as it was discussed earlier after Proposition 4.1. As before using the fact that E solves (2.4) we obtain by Hölder and Sobolev inequalities for any ε > 0 when δ > 0 is small enough. The estimate (5.37) follows from (5.27).

Proof of the main theorem
In this section we prove Theorem 2.3. As it was mentioned in the introduction we will effectively use the regularity of ω-minimizers to rule out those competing sets which are not regular. This idea goes back to the work by Cicalese and Leonardi [8] and by Fusco and Morini [9].
We begin with a simple lemma. On the other hand by the assumptions on E we may construct a C 1 vector field X on Ω such that X = ν E on M = ∂E ∩ Ω, X, ν Ω = 0 on ∂Ω and ||X|| L ∞ ≤ 1. Therefore We remark that the above proof also yields that E is an ω-minimizer in the sense of Definition 3.4. The next lemma states that the convergence of mean curvatures implies W 2,p convergence for any p > 1. It is very similar to Lemma 7.2 in [1].
Lemma 6.2. Let p > 1. Suppose that E is as in Theorem 2.3 and there are diffeomorphisms Φ k : E → F k such that ||Φ k − Id|| C 1,α → 0 and Then there are diffeomorphismsΦ k : E → F k such that ||Φ k − Id|| W 2,p → 0.
Sketch of the proof. Since F k are C 1,α manifolds with boundary they can be presented locally as a graph of C 1,α functions. Away from the relative boundary ∂F k ∩ Ω the convergence follows as in ([1], Lemma 7.2). On the boundary we follow a standard argument and flatten the boundary ∂Ω locally to half space and then use similar elliptic regularity estimate as in the interior case.
We are now ready to prove the main result of the paper. As in [1,5] the proof is divided into two steps where we first prove that E is a local minimizer with respect to Hausdorff distance and then sharpen the result to obtain the local minimality with respect to L 1 distance with the quantitative estimate.
Proof of Theorem 2.3.
We will use the signed distance function d Ω,E (·) of E on Ω constructed in the first step of Lemma 5.3.
Denote the sets {x ∈ Ω | d Ω,E (x) < 1/k} ∪ E by U k . The reader needs only to know that these sets are C 3 regular and satisfy (i) I 1/k ′ (E) ⊂ U k , for some k ′ (ii) ||U k , E|| C 2 → 0 in a sense of definition (5.1) and (iii) U k satisfies the orthogonality condition (2.5).
As in [1] we replace the contradicting sequence (E k ) by (F k ) which minimizes (6.1) J(F ) with constraint |F | = |E| and F ⊂ U k .
By the contradiction assumption J(F k ) < J(E) for every k and obviously F k → E in Hausdorff topology. Let us show that F k are ω-minimizers with uniform constants Λ and r. To that aim let G ⊂ Ω be a regular set such that G∆F k ⊂ B r (x 0 ). Let us divide G into two parts G ∩ U k and G ∪ U k .
Since F k minimizes (1.3) in U k with a volume constraint we obtain as in the proof of Proposition 3.4 that (6.2) P Br (x0) (F k ) ≤ P Br (x0) (G ∩ U k ) + C|(G∆F k ) ∩ U k | for some C. Moreover, since U k are uniformly C 3 regular and satisfy the orthogonality condition they are ω-minimizers, as we remarked after the proof of Lemma 6.1, and therefore (6.3) P Br (x0) (U k ) ≤ P Br (x0) (G ∪ U k ) + C|G \ U k | , for some C > 0. Since P Br (x0) (G ∩ U k ) + P Br (x0) (G ∩ U k ) ≤ P Br (x0) (G) + P Br (x0) (U k ) the estimates (6.2) and (6.3) yield P Br (x0) (F k ) ≤ P Br (x0) (G) + Λ|F k ∆G| for some large Λand the ω-minimality is proved. By the ω-minimizing property of F k and by Theorem 3.5 we conclude that F k → E in C 1,α and that F k satisfy the orthogonality condition (2.5).
Denote the relative boundaries as usual by M = ∂E ∩ Ω and M k = ∂F k ∩ Ω. The Euler-Lagrange equation for F k reads as where γ k are some remainder terms with γ k → 0 as k → ∞. Since F k are C 1 regular Proposition 3.2 yields ||H ∂F k || L ∞ ≤ Λ. Moreover the equation (1.4) implies