On the regularity of stationary points of a nonlocal isoperimetric problem
In this article we establish -regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain . In particular, stationary points satisfy the corresponding Euler-Lagrange equation classically on the reduced boundary. Moreover, we show that the singular set has zero -dimensional Hausdorff measure. This complements the results in  in which the Euler-Lagrange equation was derived under the assumption of -regularity of the topological boundary and the results in  in which the authors assume local minimality. In case has non-empty boundary, we show that stationary points meet the boundary of orthogonally in a weak sense, unless they have positive distance to it.
The main goal of this work is to establish -regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem, and estimate the size of its singular set. More precisely, we consider the following functional
where is a domain (open, connected) of class , is a bounded set of finite perimeter in , , , and denotes a symmetric “kernel” (see below for precise assumptions on ). The reader should think of as the Green’s function of the Laplace operator with Neumann boundary condition in or the Newtonian potential in case .
Physically, the first term in (1) models surface tension an thus its minimization favors clustering, whereas the second term can be used to model a competing repulsive term. The third term can be used to model additional external forces, cf. . The functional is often referred to as the sharp-interface Ohta-Kawasaki energy  in connection with di-block copolymer melts. Minimizers of under a volume constraint describe a number of polymer systems [6, 22, 24] as well as many other physical systems [3, 7, 13, 18, 22] due to the fundamental nature of the Coulombic term. Despite the abundance of physical systems for which (1) is applicable, rigorous mathematical analysis for the case is fairly recent. We refer to the introduction of  for more details and an account of the results about this functional.
Regularity for (local) minimizers of under a volume constraint was established by Sternberg and Topaloglu . Sternberg and Topaloglu showed that any local minimizer of in a ball is a so called -minimizer of perimeter in the sense that
for some some . Standard results (see for example [11, 21, 19]) imply that the reduced boundary is of class and that the singular set has Hausdorff dimension at most . Standard elliptic regularity theory then implies higher regularity.
For stationary points of , which are not a priori minimizing in any sense, these methods are no longer available. To this end Röger and Tonegawa [25, Section 7.2] proved -regularity of the reduced boundary of stationary points of that arise as the limit of stationary points of the (diffuse) Ohta-Kawasaki energy with parameter going to zero. They also showed that in this case the singular set has Hausdorff dimension at most .
Our main result (Theorem 1.2) removes this special assumption. In particular, we do not require any minimality assumptions. As part of our proof we establish a weak measure theoretic form of the Euler-Lagrange equation for arbitrary stationary points of under very weak regularity assumptions (we only require the set to have finite perimeter). The Euler-Lagrange equation for stationary points of has previously been derived by Choksi and Sternberg , however assuming -regularity of the topological boundary.
An application of our main result will be used in  which studies the asymptotics of stationary points of the Ohta-Kawaski energy and its
diffuse interface version.
In order to state our main result we need to introduce some notation and specify our hypotheses:
For a given domain with -boundary we consider two classes of sets.
where . A stationary point of in or is then defined as follows.
Definition 1.1 (stationary point of ).
We now specify the assumptions that we impose on the function appearing in (1).
Firstly, we let be the fundamental solution of the Laplace operator given by
Here . We assume that
where is a symmetric corrector function. I.e.
for all . Here we interpret to be zero for unbounded domains . In case is bounded is a Neumann Green’s function of the Laplace operator. In case we also allow that for , where .
For a bounded Borel set we define
to be the potential of associated to the kernel .
By standard elliptic theory we have .
Our main result reads as follows.
Let be a stationary point of the functional in or with and as above. Then the reduced boundary is of class for all . In particular, the equation
holds classically on where is the mean curvature111by our convention the mean curvature is chosen such that the boundary of the unit ball in has positive mean curvature equal to of , is a Lagrange multiplier, and is the potential arising from , given by (3). (When is a stationary point in the class , then .) The measure is weakly orthogonal to in the sense that
for all with on .
Moreover, the singular set is a relatively closed subset of which satisfies .
The estimate on the singular set in Theorem 1.2 is optimal. This can already be seen in the case . E.g. let and set . Then is a stationary point of the perimeter functional with singular set .
Our paper is organized as follows. In Section 2 we introduce our notation and review the basic theory of rectifiable varifolds and sets of finite perimeter, and present Allard’s regularity theorem and De Giorgi’s structure theorem for the reader’s convenience. In Section 3 we prove some preliminary results that are needed in order to prove Theorem 1.2. In Section 4 we prove Theorem 1.2. In Section 5 we include, for convenience, the regularity for local minimizers of near boundary points . This has already been proven independently by Julin and Pisante [17, Theorem 3.2].
2 Notation and preliminaries
Throughout this work we assume that , , is a domain (open, connected) of class (although the regularity assumption on the boundary is only needed when we consider vector fields that do not have compact support inside ). In this section we introduce our notation and summarize basic results from geometric measure theory that are needed in the sequel. For more details on the subject we refer the reader to [8, 11, 19, 26].
2.1 Varifolds and Allard’s regularity theorem
Here we collect basic definitions for varifolds and state Allard’s regularity theorem. An -measurable set is called countably -rectifiable if
where , , are -dimensional submanifolds of class and . For a vector field we can define the tangential divergence of by setting
for , which is well-defined -a.e. on . Here , where is an orthonormal basis of the tangent plane of at the point .
For the purpose of this article we use the following pragmatic definition of rectifiable -varifolds, which usually has to be deduced from the definition (we refer to  for details):
A rectifiable -varifold in is a Radon measure on such that
where is a countably -rectifiable set and where the multiplicity function is such that -a.e. on .
The first variation of with respect to is given by
which by [26, §16] is equal to . Here denotes the image varifold given by , and where denotes the flow of .
We say that has generalized mean curvature in if
where is a locally -integrable function on with values in . We remark that using the Riesz representation theorem such an exists if the total variation is a Radon measure in and moreover is absolutely continuous with respect to (see  for details).
We make the trivial but important remark that a rectifiable -varifold in that has finite total mass naturally defines a rectifiable -varifold in .
A fundamental result in the theory of varifolds is the following regularity theorem due to Allard  (see also [26, Chapter 5] for a more accessible approach) that holds for rectifiable -varifolds in . We use the following hypotheses.
Theorem 2.1 (Allard’s Regularity Theorem).
More precisely, there is a linear isometry of and a function with , , and
2.2 Sets of finite perimeter
Let be a Borel set. We say that has finite perimeter in if
The Riesz representation theorem implies the existence of a Radon measure on and a -measurable vector field with -a.e. such that
The vector valued measure is sometimes referred to as the Gauss-Green measure of (with respect to ). For the total perimeter of the set in we have
In the case that is of class , we have
In particular, we have for every point
For a generic set of finite perimeter, the reduced boundary of in is defined as those such that the above limit on the right hand side exists and has norm . The Lebesgue-Besicovitch differentiation theorem implies that . The vector field defined by the equation (7) on (and set to elsewhere), is called the measure theoretic outer unit normal of . For more details on sets of finite perimeter we refer to [11, 26, 8].
Theorem 2.3 (De Giorgi’s structure theorem).
Suppose has finite perimeter in . Then is countably -rectifiable. In addition for all
where is the volume of the unit ball in . (i.e. the limit exists and is equal to .) Moreover, .
De Giorgi’s structure theorem in particular shows that every set of finite perimeter defines - through its generalized surface measure - a rectifiable -varifold of multiplicity on .
Let be of finite perimeter in . If is Lipschitz regular, one can define the (inner) trace of on . For details we refer to [8, Chapter 5.3]. For every vector field we have
This implies that is also a set of finite perimeter as a subset of with
As a finite perimeter set in , also has a Gauss-Green measure which we shall denote by . Obviously and .
Since sets of finite perimeter are equivalence classes of sets, one needs to choose a good representative in order to talk about their regularity properties. W.l.o.g. (see [11, Proposition 3.1] for details) we will always assume that any finite perimeter set at hand satisfies the following properties:
2.3 The first variation of perimeter
where is the tangential divergence of the vector field with respect to :
which obviously agrees with the definition of tangential divergence with respect to . Hence, the expression (10) equals the first variation of the varifold with respect to .
In order to investigate the behavior of stationary points of at the boundary of we need to allow for more general variations (as already appearing in Definition 1.1). By the regularity assumption on it follows (cf. ) that (and ) for the flow of any vector field such that on . Since we see that the formula (10) still holds for such vector fields .
3 Preliminary results
Proposition 3.1 (First variation of nonlocal perimeter).
Let and let with on be a vector field with corresponding flow . Then
The first variation of perimeter is equation (10). It remains to compute the first variation of the nonlocal term; the computation of the first variation of the third term is similar but easier. By the change of variables formula it holds that
Hence, we compute using (11) and the assumptions on which allow us to differentiate under the integral
We cannot directly apply the divergence theorem to the first term of (3) since is only -rectifiable and is not of class near . We can get around this technical obstacle by applying the results of , but we present a simple argument which suffices in our case. Since is of class (for this argument Lipschitz is enough) we have (see Section 3) that is a set of finite perimeter in . By [11, Theorem 1.24] we can approximate in the support of by smooth sets such that
We may apply the Lebesgue dominated convergence theorem to conclude that
Moreover, we have
We may now apply the divergence theorem and we have for a.e.
The second term on the right hand side of (3) can be estimated by for some , and hence goes to zero as . On the other hand, we have
Using (3) and applying Fubini’s theorem we arrive at
The claim now follows from the fact that is tangential to . ∎
There exists a vector field such that .
Assume by contradiction that for every vector field : . Then by Du Bois-Reymond’s lemma  we conclude that
where we used that is connected. Hence,
This contradicts the assumption that , proving the claim. ∎
Proposition 3.3 (Euler-Lagrange equation of non-local perimeter).
Let be a stationary point of in or . Then there exists a real number such that has a generalized mean curvature vector
and such that is weakly orthogonal to . That is, for every vector field with on the following variational equation is true:
For stationary points in , we have .
Step 1: Construction of the local variation.
The case of variations in is an immediate consequence of Proposition 3.1. For the case of let be a vector field such that . The existence of such a vector field is guaranteed by Lemma 3.2. Let be the flow of and the flow of . For set
Then , and . The implicit function theorem ensures the existence of an open interval containing and a function such that
is a 1-parameter family of -diffeomorphisms of and thus defines a volume preserving variation of in .
Step 2: Computing the first variation.
The fact that is a stationary point in the class then implies from Proposition 3.1
On the other hand, we have
where , and where we used the divergence theorem on the last line. Therefore, setting , we have
for every vector field with on . ∎
4 Proof of the Theorem 1.2
Firstly, notice that the weak orthogonality of and is included in Proposition 3.3.
We want to apply Allard’s regularity theorem (here Theorem 2.1) to establish the regularity of the reduced boundary . We verify the necessary hypotheses:
By De Giorgi’s structure theorem (here Theorem 2.3) and Remark 2.4 we have that is a multiplicity- rectifiable -varifold. Moreover, for each point we have that . Now, we choose any point . W.l.o.g., after possibly translating and rotating the set , we may assume that and . We fix any and pick to be as in the statement of Theorem 2.1. Since we can find a small radius such that
Proposition 3.3 implies that has generalized mean curvature in , given by
for some constant . We have
With Hölder’s inequality and (19) we get
which is less that provided . Thus the hypotheses (h) are satisfied and Theorem 2.1 implies the existence of a function of class , , such that , , and . Moreover, our orientation assumption on implies that for some open interval .
Now let , where , , is the -th-standard basis vector, and where is a cut-off function such that for every .
Then recalling that , we have where is the -th component of the normal vector, and where is the gradient in . Since is the graph of , and by our orientation assumption, we have that . Using the area formula, equation (18) becomes
Equation (20) is the weak form of the prescribed mean curvature equation. Since by Theorem 2.1 the gradient of is locally uniformly bounded in and since the right hand side of (20) is of class , interior Schauder estimates (see ) and bootstrapping imply local regularity of the function . Thus (20) holds pointwise, and since was arbitrary we have
where is the classical mean curvature of the surface .
4.1 On the size of the singular set
By a direct consequence the monotonicity formula, see [26, Corollary 17.8], we have that exists and that for every point