Regularity of Almost-Minimizers of Hölder-Coefficient
Surface Energies
Abstract.
We study almost-minimizers of anisotropic surface energies defined by a Hölder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally Hölder continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.
1. Introduction
The Plateau problem is a classical geometric variational problem. It consists in minimizing surface area among all surfaces with a certain prescribed boundary. The analogous physical phenomenon occurs in soap films as they seek to minimize surface tension, an equivalent to minimizing surface area. The existence and regularity of solutions to the Plateau problem has been the subject of study in a variety of settings and continues to be a centerpiece of much mathematical research (to name a few, see [Dou31, Rad30, DG60, Rei60, All72, Tay76a, HP16, DLGM17, KMS19]). A natural generalization of the Plateau problem is to study minimizers of surface energies other than surface area. Anisotropic surface energies are those which depend on the normal direction to the surface and possibly the spatial location of the surface as well. This means that the energy assigned to a surface depends not only on its geometry but also on how and where the surface sits in space. Such anisotropic energies arise in physical phenomena such as the formation of crystals and in crystalline materials.
Almgren was the first to study regularity of minimizers to anisotropic variational problems in his paper [Alm68]. This initial work as well as much of the subsequent work in the area was done in the setting of varifolds and currents with many of the results applying to surfaces of arbitrary codimension but with rather strong regularity assumptions on the integrands of the anisotropic energies.
In this paper we work in the setting of sets of locally finite perimeter and study the existence and regularity of minimizers of anisotropic surface energies of the form
| (1.1) |
where is a uniformly elliptic, Hölder continuous matrix-valued function, is a set of locally finite perimeter in , and is an open set. Here denotes the -dimensional reduced boundary of and denotes its outward unit normal vector. We note that Hölder continuity is a rather weak regularity assumption and the previously known regularity results for general integrands do not apply (see the discussion below). Our main regularity result applies to almost-minimizers which are sets of locally finite perimeter in satisfying the minimality condition
| (1.2) |
whenever , , and (see Section 2 for full definitions and notation).
Theorem 1.1 (Regularity of almost-minimizers).
Let and be an open set in . Suppose is the anisotropic energy given by (1.1) for a uniformly elliptic, Hölder continuous matrix-valued function with Hölder exponent . If is a -almost-minimizer of in , that is, it satisfies (1.2), then is a -hypersurface which is relatively open in , while the singular set of in ,
| (1.3) |
satisfies the following:
-
(i)
if , then is empty;
-
(ii)
if , then has no accumulation points in ;
-
(iii)
if , then for .
A regularity result of the form of Theorem 1.1 was first proved by De Giorgi in [DG60] for minimizers of surface area. De Giorgi worked within the framework of sets of locally finite perimeter which he had introduced and shown to be equivalent to the earlier notion of Caccioppoli sets. Shortly thereafter Reifenberg also proved a similar regularity result for minimizers of surface area in [Rei60, Rei64a, Rei64b]. In [Tam82, Tam84], Tamanini extended De Giorgi’s result to almost-minimizers of perimeter satisfying the minimality condition , proving -regularity at points in the reduced boundary for each . In fact, his result applies with a more general error term.
The anisotropic surface energies treated by Almgren in [Alm68] are given in terms of the integral of a bounded, continuous, elliptic integrand over the surface. Here denotes the spatial variable and denotes the directional variable. Almgren proved that if is for some , then minimal surfaces with respect to are -regular almost everywhere. Bombieri extended this to the case by showing in [Bom82] that if is , then minimal surfaces with respect to are -regular almost everywhere. In [SS82], Schoen and Simon provided an alternate proof of this type of regularity result with weakened hypotheses. They showed that if is Lipschitz in the spatial variable and in the directional variable , then minimizers are -regular almost everywhere for any .
A characterization of the singular set for codimension one oriented hypersurfaces as in Theorem 1.1 was proved in the case of the area integrand in a series of papers by various authors. Miranda proved in [Mir65] that -measure of the singular set is zero. The rest of the results deal with the Bernstein problem which asks about the existence of global minimizers of surface area in . Fleming and Almgren proved some intermediate results of nonexistence in singular minimizing cones in and , respectively, in [Fle62, Alm66]. The next result was by De Giorgi in [DG65] where he showed that the non-existence of a singular minimal cone in implies non-existence in . Simons showed the non-existence of singular minimal cones in dimensions in [Sim68] and Bombieri, De Giorgi, and Giusti demonstrated in [BDGG69] that Simons’ cone
| (1.4) |
is a singular minimal cone in with singular set . Federer concluded in [Fed70] by proving the Hausdorff dimension of the singular set is less than or equal to . In the anisotropic case, it was shown in [SSA77] that -measure of the singular set is zero for elliptic integrands which are .
Surface energies of the particular form of (1.1) first appeared in the paper [Tay76b] by Jean Taylor. This is a follow-up paper to her celebrated paper [Tay76a] in which she proves that the structure of singularities of soap-like minimal surfaces in are exactly as conjectured by the experimental physicist Joseph Plateau. In [Tay76b], she proves that minimizers of in are locally at regular points and possess a singular set with the same general structure as in the case of surface area minimizers. Taylor worked with varifolds as her notion of surface and only with -dimensional surfaces in . This enabled her to utilize the classification of -dimensional surface area minimizing cones in . Such a classification is not known in higher dimensions. Note that the singularities dealt with by Jean Taylor cannot occur within our setting of sets of locally finite perimeter as -dimensional minimizing cones in come from non-oriented surfaces. This is why, for instance, we do not have singularities when , even though there are singularities in lower dimensions when working with varifolds.
Allard’s work in [All72] established some important results for the Plateau problem in the setting of varifolds, some of which have been generalized to the anisotropic setting. Allard first proved that a varifold with bounded first-variation is rectifiable. He then proved regularity by showing that if there are -type bounds on the generalized mean-curvature of for large enough (depending on the dimension of ), then is locally for some (depending on and the dimension of ) outside a closed singular set of measure zero. A recent breakthrough was made in the setting of anisotropic integrands in [DPDRG18] to prove rectifiability. There, the authors were the first to successfully compute the first-variation with respect to an anisotropic integrand . Using this they showed that if is an elliptic -integrand satisfying the so-called atomic condition (equivalent to ellipticity in codimension one), then a varifold whose anisotropic first-variation is locally bounded is indeed rectifiable. Further regularity is currently not known as the monotonicity formula which is essential in Allard’s regularity arguments does not exist for general integrands as demonstrated in [All74]. Much of the related relevant literature in contained in [DLDRG19, DPDRG16, DR18, DDG20, DRK20].
Another related problem of interest is volume constrained minimization. Regularity is known in the case of volume constrained perimeter minimizers [GMT83] and some results are known in anisotropic settings [Par84, LS20].
Let us briefly describe the organization of this paper. We start in Section 2 by providing the essential definitions pertaining to sets of locally finite perimeter, our anisotropic surface energies, and almost-minimizers. In Section 3 we follow the Direct Method of the Calculus of Variations to establish the existence of minimizers to our formulation of the anisotropic Plateau problem. The rest of the paper is devoted to the study of the regularity of almost-minimizers and to the characterization of the singular set. In Section 4 we cover a key change of variable that allows us to assume (the identity matrix) at a given point , as well as prove many important properties of almost-minimizers. These include an almost-monotonicity formula, Theorem 4.7, volume and perimeter bounds, Proposition 4.10, and compactness of the class of almost-minimizers, Proposition 4.13. Next in Section 5 we define the excess, (5.4), an important notion in regularity theory, and recall some of its properties. There we also state the height bound, Proposition 5.9, which allows us to control the height of the boundary of an almost-minimizer given a small excess assumption. Following this we show in Section 6 that a small excess assumption together with the assumption allows us to find a Lipschitz function that well approximates and is ‘almost-harmonic’ with a controlled error, Theorem 6.1. In Section 7 we prove a reverse Poincaré inequality, Theorem 7.1, which in Section 8 we combine with a harmonic approximation of the Lipschitz function from Section 6 to prove a tilt-excess decay result, Theorem 8.3. Finally, in Section 9 we use this and an iteration argument to prove our main regularity result, Theorem 9.2. We conclude the paper in Section 10 by using blow-up analysis and a Federer reduction argument to prove the characterization of singular set, Theorem 10.1.
2. Preliminaries
We will work in for a fixed . The open ball centered at of radius is defined by
| (2.1) |
where denotes the standard Euclidean norm and we write for . We denote the volume of the -dimensional ball by .
Like De Giorgi and Tamanini, we shall also work with sets of locally finite perimeter following much of the notation and definitions given in the insightful expository book by Maggi [Mag12]. Throughout this paper we will follow a scheme inspired by the one presented there.
A Lebesgue measurable set is said to be of locally finite perimeter if there exists an -valued Radon measure (called the Gauss-Green measure of ) such that the Gauss-Green formula
| (2.2) |
holds. The induced total-variation measure is called the perimeter measure of and is denoted by . The set is said to be of finite perimeter if . The set of those -a.e. for which
| (2.3) |
is called the reduced boundary of and is denoted by . The measure-theoretic outer unit normal to is then defined to be the measurable function given by
| (2.4) |
The De Giorgi structure theorem states that is -rectifiable and that where denotes the -dimensional Hausdorff measure. We may modify a set of locally finite perimeter on and/or up to a set of Lebesgue measure zero without changing its perimeter measure. As a consequence, the topological boundary of a generic set of locally finite perimeter may be quite messy and might not be well related to . However, we may always modify our set of locally finite perimeter so that without changing its perimeter measure, in which case (see [Mag12, Remark 16.11, Remark 15.3]). When discussing boundary regularity of a set of locally finite perimeter we shall always choose this representative of .
2.1. Anisotropic surface energies with Hölder coefficients
Now let’s provide precise definitions for the anisotropic energies and almost-minimizers we will study. Denote by the set of real -matrices equipped with the operator norm . Let be a bounded, measurable function on that takes values in . We say that is symmetric if for all , where denotes the matrix transpose. We say that is uniformly elliptic if there exist constants such that
| (2.5) |
for all , where denotes the standard Euclidean inner product. We say that is Hölder continuous with exponent if
| (2.6) |
and call the Hölder seminorm of . In particular,
| (2.7) |
holds for all .
Definition 2.1 (-surface energy).
Let be uniformly elliptic, and Hölder continuous. Given a set of locally finite perimeter in and a Borel set , we define the -surface energy of in by
| (2.8) |
Note that defines a Borel measure on and we will often denote by .
Remark 2.1 (Symmetry of ).
We may assume without loss of generality that is symmetric which we do throughout this paper. We may make this assumption as the equality holds for all . Hence we can always symmetrize without changing the values of .
Remark 2.2 (Ellipticity).
The integrand is elliptic in the sense of Almgren in [Alm68]. In our setting this means that for every bounded set there is a constant such that for every set of locally finite perimeter , half-space , and ,
| (2.9) |
whenever , . Here denotes the energy associated to the frozen integrand . As Almgren notes, this notion is equivalent to uniform convexity in codimension one as is our case by uniform ellipticity of and (2.9) holds with . Ellipticity ensures that half-spaces are the unique minimizers when compared with their compactly contained variations.
Remark 2.3 (Hölder continuity of integrand of ).
The integrand is Hölder continuous with respect to the spatial variable , that is,
| (2.10) |
for all with . This follows from (2.7) combined with the useful inequality
| (2.11) |
for all with . Note our regularity assumption is much weaker than in [Alm68] where he assumes the integrand is for some and weaker than the assumption in [SS82] where they assume the integrand is Lipschitz in .
Remark 2.4 (Comparability to perimeter).
is comparable to since it follows for all Borel sets that
| (2.12) |
by the uniformly ellipticity of . When equals the identity matrix we have the isotropic case .
Remark 2.5.
The complement of a set of locally finite perimeter is also a set of locally finite perimeter with and so .
2.2. Notions of almost-minimizers
We are interested in studying the boundary regularity of those sets of locally finite perimeter which are almost-minimizers of the -surface energy in an open set when compared to their local compactly contained variations. Recent work addressing regularity of almost-minimizers for other variational problems can be found in [STV18, dQT18, DESVGT19, JPSVG20] and the notions of almost-minimizers we consider are similar.
Fix universal constants , , , and , and let be a symmetric, uniformly elliptic, and Hölder continuous with respect to , , and and fix an open set in .
Definition 2.2 (-almost-minimizer of ).
We say a set of locally finite perimeter in is a -(additive) almost-minimizer of in at scale if and
| (2.13) |
whenever where is a set of locally finite perimeter, , and .
When (2.13) holds with , we say that is a local minimizer of in at scale , and when (2.13) holds for all scales , we say that is a minimizer of in . Typically we will omit the descriptor additive when discussing almost-minimizers. However, we will include it when we wish to highlight the difference from the following alternative notion of almost-minimality.
Definition 2.3 (-multiplicative almost-minimizer of ).
We say a set of locally finite perimeter in is a -multiplicative almost-minimizer of in at scale if and
| (2.14) |
whenever where is a set of locally finite perimeter, , and .
Note that Taylor worked with this notion of multiplicative almost-minimizer in [Tay76b] but handled a more general error term. We now show that multiplicative almost-minimizers are also additive almost-minimizers. To prove this, we need an upper bound for perimeter bounds of multiplicative almost-minimizers at points in the topological boundary. Whenever we write we mean a constant (which may change from line to line) that depends only on the universal constants and an upper bound for , but does not depend on or . If we wish to specify dependence on fewer constants and write for example, for constants that only depend on .
Lemma 2.1.
There exists a positive constants with the following property. If is a -multiplicative almost-minimizer of in at scale , then for every with ,
| (2.15) |
Proof.
Since is Radon, for a.e. . Choose one such radius and for consider the comparison set in . Then . It follows from comparability to perimeter (2.12) and the multiplicative almost-minimality of that
| (2.16) |
Hence
| (2.17) |
since . Sending and noting gives
| (2.18) |
By density of these radii, this holds for all . ∎
Proposition 2.2 (Multiplicative almost-minimizers are (additive) almost-minimizers).
If is a -multiplicative almost-minimizer of in at scale , then for each open set , there is a constant such that is a -(additive) almost-minimizer of in at scale .
Proof.
Let , , and . Suppose is a -multiplicative almost-minimizer of in at scale . The minimality condition is trivially satisfied if or . So suppose and . Then there is . So by Lemma 2.1, which applies since and , we have . Hence by comparability to perimeter (2.12) we have . It follows that
| (2.19) |
for some . ∎
Thus Proposition 2.2 implies that any interior regularity results for (additive) almost-minimizers shall also apply to multiplicative almost-minimizers. We shall focus on proving a regularity theorem for (additive) almost-minimizers and shall henceforth only work with (additive) almost-minimizers which we simply refer to as almost-minimizers.
3. Existence of Anisotropic Minimizers
Our first order of business is to establish existence of solutions to the anisotropic Plateau problem for . The existence of anisotropic minimizers in the setting of varifolds and currents is known in general in the framework of varifolds and currents (see [Fed69, Chapter 5]) which should imply existence of minimizers of in the framework of sets of locally finite perimeter. However, for completeness, we present our own full proof of this result in our setting. Additionally, the lower semicontinuity result of Proposition 3.4 will prove useful at several places in the regularity portion of our paper.
Let be a symmetric, uniformly elliptic, continuous function on with values in (we do not need Hölder continuity to show existence of minimizers) and consider the -surface energy. Fix an open bounded set and a set of finite perimeter in . The anisotropic Plateau problem for in with boundary data is to show that the infimum
| (3.1) |
is attained (Cf. [Mag12, (12.29)]). That is, we minimize in among those sets of finite perimeter which agree with outside of .
To show that (3.1) is achieved by a set of finite perimeter, we follow the Direct Method of the Calculus of Variations. This consists of taking a sequence of competitors such that , using a key compactness result in an appropriate topology to extract a subsequence converging to some competitor satisfying , and applying lower semicontinuity of with respect to the convergence in the chosen topology which shows that equals the infimum in (3.1).
3.1. Compactness of sets of locally finite perimeter
The first key ingredient of the Direct Method is compactness of our class of admissible competitors. One of the primary reasons that sets of locally finite perimeter provide a suitable setting to work on geometric variational problems is that they possess compactness with respect to local convergence of sets. Let’s recall the definition of this convergence and a known compactness theorem for sets of locally finite perimeter.
We say that a sequence of sets of locally finite perimeter in converges locally to (and write ) if
| (3.2) |
for each compact , and say converges to (and write ) if
| (3.3) |
Recall that and that denotes Lebesgue measure on .
Theorem 3.1 (Compactness from perimeter bounds, [Mag12, Theorem 12.26]).
If and are sets of finite perimeter in , with
| (3.4) |
then there exist a set of finite perimeter in and indices as , with
| (3.5) |
3.2. Lower semicontinuity of
The second key ingredient of the Direct Method is to show lower semicontinuity of the -surface energy. Here we have some work to do and start with a couple lemmas. The first lemma deals with lower semicontinuity when is constant, while the second one is a technical lemma we need in the proof when is no longer constant.
Lemma 3.2 (Lower semicontinuity for constant ).
If is a constant, uniformly elliptic matrix, and and are sets of locally finite perimeter with , then for any open set ,
| (3.6) |
Proof.
By Remark 2.1 we may assume is symmetric and by uniform ellipticity its eigenvalues are positive. So by the spectral theorem we can write where is a diagonal matrix with the eigenvalues of and where is the matrix of corresponding orthonormal eigenvectors. Setting , we have with symmetric since . So . Define -valued Radon measures on ,
| (3.7) |
It then follows that because, given any , we have and thus
| (3.8) |
By lower semicontinuity of the total variation of weak-star convergent vector-valued Radon measures ([Mag12, Proposition 4.19]), we have
| (3.9) |
which concludes the proof. ∎
Lemma 3.3.
Let and be Radon measures on and such that
. Then the following two statements hold:
-
(i)
If for any open set , then
(3.10) for any open set in .
-
(ii)
If for any compact set , then
(3.11) for any compact set in .
Proof.
Let and choose such that and for . This is possible since is Radon and so for at most countably many . Set
| (3.12) |
and note that the ’s are open and the ’s are compact.
Proof of : Assume the hypothesis and let be an open set. Observe that
| (3.13) |
since for . Since on and for , we have
| (3.14) |
where we used the property that for any sequences . Note that on and so
| (3.15) |
since . Sending completes the proof of .
Proof of : Assume the hypothesis and let be a compact set. Recalling , observe that
| (3.16) |
since for , and
| (3.17) |
It follows that
| (3.18) |
where we used the property that for any sequences . Since on and , we have
| (3.19) |
Note that on and so
| (3.20) |
where we used for . Sending completes the proof of . ∎
With these lemmas in hand, we are now ready to state and prove the lower semicontinuity of .
Proposition 3.4 (Lower semicontinuity of ).
Let be a symmetric, uniformly elliptic, continuous function on with values in . Suppose is a sequence of sets of locally finite perimeter in and is Lebesgue measurable, with
| (3.21) |
for every compact set in . Then is a set of locally finite perimeter in with
| (3.22) |
and for any open set in ,
| (3.23) |
Proof.
That is of locally finite perimeter and follow from [Mag12, Proposition 12.15]. Thus we need only to prove the lower semicontinuity.
First assume that is bounded. By taking a subsequence of , we may assume up to relabeling that
| (3.24) |
Note this subsequence depends on but this is not an issue. Since for every compact set , there is a further subsequence and a Radon measure such that as (see [Mag12, Remark 4.35]).
Let be open and fix . Since is uniformly continuous on , there exists such that for any , we have whenever . Thus by the inequality (2.11), for any ,
| (3.25) |
whenever and .
Since is compactly contained in and , there exist finitely many balls each of radius and center which cover . Take a partition of unity with such that on and elsewhere. It follows that
| (3.26) |
where in the last inequality, for each , we applied part of Lemma 3.3 to and the measures and which by Lemma 3.2 satisfies the lower semicontinuity hypothesis. By part of Lemma 3.3, applied to ,
| (3.27) |
for each , and so
| (3.28) |
since and . It follows that
| (3.29) |
since, as above,
| (3.30) |
by and . Letting , we obtain . Approximating by from below and using monotone convergence, we obtain .
For the case when is unbounded, we have for every bounded open set . We conclude by approximating from below by bounded open sets and using monotone convergence. ∎
3.3. Existence theorem of minimizers for
We now show that the anisotropic Plateau problem for given by (3.1) has a solution. We follow a similar approach as [Mag12, Theorem 12.29].
Theorem 3.5 (Existence of minimizers for the anisotropic Plateau problem for ).
Let be a uniformly elliptic, continuous function on with values in , let be a set of finite perimeter in , and let be an open bounded set. There exists a set of finite perimeter in with such that from (3.1). In particular, is a minimizer of in .
Proof.
Let be a sequence of sets of finite perimeter in with such that as and . Consider . Noting that by [Mag12, Theorem 16.3], (in particular, by [Mag12, Exercise 16.5]),
| (3.31) |
Hence . Choose with so that . By Theorem 3.1, there is a set of finite perimeter and as such that . Up to modifying by a set of measure zero . Set . Then and note that . Hence since as . Finally, observe that
| (3.32) |
Consequently, by Proposition 3.4,
| (3.33) |
Thus .
Suppose , , and . Then and so . Since , we have . Hence . ∎
4. Basic Properties of Almost-Minimizers
In this section we begin our journey toward proving regularity of almost-minimizers by proving some fundamental properties that almost-minimizers possess and which play a crucial role in our excess-decay argument.
4.1. Invariance under an affine change variable
One of the key ideas that allows us to adapt the standard excess-decay arguments for perimeter minimizers to the setting is a certain change of variable.
If is a constant matrix, then by symmetry we can orthogonally diagonalize and write , where is a diagonal matrix with the eigenvalues of and where is the matrix of corresponding orthonormal eigenvectors. By ellipticity the eigenvalues of are bounded below and above by the positive constants and . Setting , we have . Note that and are symmetric since . In the coordinate system of , the matrix is diagonal and so almost-minimizers of can be viewed as almost-minimizers of perimeter when deformed by the change of variable (see Proposition 4.1 below). Of course this change of variable preserves any regularity of almost-minimizers and we know by Tamanini’s work in [Tam84] that almost-minimizers of perimeter are Hölder continuously differentiable.
If varies Hölder continuously, then almost-minimizers of cannot simply be viewed as almost-minimizers of perimeter since deformation varies from point to point. However, philosophically it is reasonable to expect a similar amount of regularity since the deformation varies Hölder continuously. In subsequent sections we will prove decay estimates for the excess at points with small excess on some ball or cylinder. In the proofs of these estimates it will be convenient to be able to assume that , allowing us to think of as a perturbation of perimeter at the point . In order to make this assumption, we shall do the following change of variable which was similarly used in [DESVGT19, JPSVG20] for almost-minimizers of other types of functionals involving coefficients . As in the constant case, for each fixed we can write , where is a diagonal matrix with the eigenvalues of and where is the matrix of corresponding orthonormal eigenvectors. Setting , we have that and are symmetric since and satisfy
| (4.1) |
In particular, and .
Define the affine change of variable at by
| (4.2) |
and define
| (4.3) |
Note that , , while is symmetric, uniformly elliptic with constants and Hölder continuous with exponent and Hölder seminorm . The uniform ellipticity constants follow from
| (4.4) |
and the bound on the Hölder norm follows from estimate that for all there holds
| (4.5) |
Thus constants for depend on the same universal constants as .
The ellipsoid at of radius is defined by
| (4.6) |
We use for our notation as this is the Wulff shape, introduced in [Wul01], for the integrand . The ellipsoid has axial directions corresponding to the eigenvectors of and axial lengths corresponding to the eigenvalues scaled by a factor of . Since the eigenvalues of are bounded between and , we have
| (4.7) |
We now prove the invariance of almost-minimizers under the change of variable and refer readers to the change of variable formula given in Proposition A.1 in Appendix A.
Proposition 4.1 (Invariance of almost-minimizers under the change of variable ).
If is a -almost-minimizer of in at scale , then is a -almost-minimizer of in at scale .
Proof.
Suppose for some and (here we write as an arbitrary competitor for whose image under will be a competitor for ). Applying Proposition A.1 with , noting that and since is symmetric, we have
| (4.8) |
Note that and so . Hence
| (4.9) |
Likewise, . Note that
| (4.10) |
Thus by the minimality condition. This simplifies to . It then follows that
| (4.11) |
as desired. ∎
Hence any of the properties or estimates we prove for -almost-minimizers also hold for the set (with any bounds or estimates having modified constants but which depend only on the same universal constants). Working with will allow us to assume in the proof of many of our estimates and will in turn allow us to prove additional properties and estimates for general -almost-minimizers.
As previously mentioned, whenever we write we mean a constant (which may change from line to line) that depends only on the universal constants , and upper bounds for , but does not depend on the set or the point . In cases where we wish to emphasize that a constant depends on fewer constants such as, for example, on the dimension only, we write .
4.2. Scaling of the energy
In Section 7 we will use the scaling of the energy to simplify and work at scale instead of of scale and in Section 10 we will utilize blow-up analysis to study the singular set almost-minimizers. The blow-ups of a set at a point and scale are defined by
| (4.12) |
where is the map defined by
| (4.13) |
We denote the inverse of by , that is, . Given a matrix-valued function , we denote by the matrix-valued function
| (4.14) |
(this is not to be confused with from the previous subsection). Note that .
Proposition 4.2 (Scaling of ).
If is a set of locally finite perimeter in , , , then
| (4.15) |
for Borel sets . In particular, if is a -almost-minimizer of in at scale , then is a -almost-minimizer of in at scale .
Proof.
We apply Proposition A.1 with the change of variable and integrand . Then , , and . So and it follows that
| (4.16) |
Now, let be a set of locally finite perimeter in with for and . Then . Note with and . Applying (4.2) to and using the almost-minimality of in at scale , we have
| (4.17) |
that is, is an -almost-minimizer of in at scale . ∎
4.3. Comparison sets
To utilize the almost-minimality condition we will often construct competitors by modifying inside an open set. The following proposition allows us to do this.
Proposition 4.3 (Comparison sets by replacements).
If and are sets of locally finite perimeter in and is an open set of finite perimeter in such that
| (4.18) |
then the set defined by
| (4.19) |
is a set of locally finite perimeter in . Moreover, if and is open, then
| (4.20) |
Proof.
In the proof of [Mag12, Theorem 16.16] the decomposition, see (16.35),
| (4.21) |
is proved. Since all of the measures on the right-hand side are concentrated on disjoint sets and since the measures and are equal and , we have
| (4.22) |
by additivity of . By and ,
| (4.23) |
Likewise, and and so
| (4.24) |
4.4. Volume and perimeter bounds and the almost-monotonicity formula
One important property which almost-minimizers of possess is bounds on both the volume and the perimeter of on balls centered at points in their topological boundary. Recall that we require for almost-minimizers. The full set of estimates is given in Proposition 4.10 but we have some work to do to prove this. The first step is showing the upper bound on perimeter.
Define the perimeter density ratio of at by
| (4.25) |
and perimeter density of at by
| (4.26) |
whenever the limit exists.
Lemma 4.4 (Upper perimeter bound).
There exists a positive constant with the following property. If is a -almost-minimizer of in at scale , then for every with ,
| (4.27) |
Proof.
Consider the function defined by . Note that is increasing, for a.e. by the coarea formula, and for a.e. because is a Radon measure. Let be one of the a.e. radii that satisfies both and . For consider the comparison set in . Then . It follows from comparability to perimeter and the almost-minimality that
| (4.28) |
and so
| (4.29) |
since . Sending yields the inequality
| (4.30) |
This, together with and , gives . ∎
To obtain the lower perimeter bound for almost-minimizers of , we shall adapt an argument given by Tamanini for almost-minimizers of perimeter in [Tam82, Tam84] which makes use of an almost-monotonicity formula. Monotonicity formulas are often times a valuable tool in regularity theory. For example, the monotonicity of density ratios for minimizers of surface area is heavily relied upon in [All72, Tay76a] as well as in many other papers. By this we mean the fact that if is a perimeter minimizer in , , then the density ratio
| (4.31) |
is monotonically increasing in (see, for example, [Mag12, Theorem 17.16]). In [All74], Allard demonstrated for integrands depending solely on the direction variable (and not on the spatial variable ) that monotonicity formulas exist if and only if the integrand is a linear change of variable from the area integrand. Under the change of variable we have so that our sets satisfy the condition for almost-minimality of perimeter when making comparisons on balls centered at as shown in Lemma 4.5 below. A key observation is that we only need these comparisons to apply the standard cone-competitor argument to obtain an almost-monotonicity formula as we do in Lemma 4.6.
Lemma 4.5.
There exists a positive constant with the following property. If is a -almost-minimizer of in at scale , , and , then
| (4.32) |
whenever and .
Proof.
Let and . If , then (4.32) trivially holds true. So consider the case when .
Lemma 4.6.
There exists a positive constant such that the following holds. If is a -almost-minimizer of in at scale with and , then the function
| (4.36) |
is monotonically increasing on where .
Proof.
Without loss of generality assume and write . Define the function by . is increasing and hence differentiable for a.e. . Thus it suffices to prove
| (4.37) |
which can be rewritten as
| (4.38) |
The idea of the proof of (4.38) is to construct cone competitors over with vertex at for each to use in the comparison inequality (4.32). To do this we will need to approximate by open sets with smooth boundary and construct the cone competitors for the approximating sets.
By [Mag12, Theorem 13.8], there is a sequence of open sets with smooth boundary in such that and . For now hold fixed. The set is relatively open in for every . By Sard’s lemma,
| (4.39) |
Consider the cones with vertex over ,
| (4.40) |
For the a.e. such that (4.39) holds we have that is a set of locally finite perimeter in with
| (4.41) |
For such that (4.39) holds, the coarea formula for -dimensional rectifiable sets (see [Mag12, Theorem 18.8]) on with yields
| (4.42) |
since . Note that for we have
| (4.43) |
and hence for such that (4.39) holds we have
| (4.44) |
Consider a radius such that for all both (4.39) and hold (and consequently (4.44) as well). This true for a.e. since and are Radon measures and by Sard’s lemma. Consider the comparison sets . Let be such that . By (16.32) of [Mag12] we have
| (4.45) |
Since , applying Lemma 4.5 gives
| (4.46) |
which by subtracting from each side together with (4.44) simplifies to
| (4.47) |
Sending gives
| (4.48) |
This inequality holds for a.e. and integrating over the interval yields
| (4.49) |
Applying the coarea formula for -dimensional rectifiable sets ([Mag12, Theorem 18.8]) on with , and , gives
| (4.50) |
since . Thus combining (4.49) and (4.50) gives
| (4.51) |
By and , sending gives
| (4.52) |
Dividing by and sending at points of differentiability of yields (4.38) as desired. ∎
Now we are able to use a perturbation argument and the change of variable to obtain an almost-monotonicity formula when is not assumed to equal .
Theorem 4.7 (Almost-monotonicity formula).
There exists a positive constant
with the following property. If is a -almost-minimizer of in at scale , then for every , we have
| (4.53) |
whenever where .
Proof.
Applying Lemma 4.6 to and gives that is monotone increasing on where . The change of variable applied to gives . This and the bound imply that
| (4.54) |
is monotone increasing on . Note and . Given , setting , it follows that
| (4.55) |
Hence and so . Thus (4.54) is monotone increasing on . By (2.11) we have
| (4.56) |
and so for by (4.7). It follows that
| (4.57) |
where we used that by the upper perimeter bound (4.27). Similarly, we have
| (4.58) |
Combining this last inequality and (4.4) with (4.54) and yields
| (4.59) |
as desired. ∎
For , define the -density ratio of at by
| (4.60) |
and the -density of at by
| (4.61) |
when the limit exists.
Corollary 4.8 (Existence of densities).
If is a -almost-minimizer of in at scale , then for every the density
| (4.62) |
exists.
Proof.
For every we have by almost-monotonicity that . Taking the as followed by the as yields
| (4.63) |
Hence exists. ∎
Using the almost-monotonicity formula, we are now able to control the perimeter density ratios from below.
Proposition 4.9.
There exists a positive constant with the following property. If is a -almost-minimizer of in at scale , then for every , we have
| (4.64) |
for .
Proof.
Let . First consider the case . The limit of perimeter density rations at a point in the reduced boundary converge to as [Mag12, Corollary 15.8]. Note that and so for we can apply Theorem 4.7 with to obtain
| (4.65) |
By comparability to perimeter and (4.7), we have
| (4.66) |
Hence (4.64) holds for .
Now consider the general case (but perhaps not in ). Given , there is with by . It follows that
| (4.67) |
and so applying (4.64) at gives
| (4.68) |
Sending completes the proof. ∎
Let us recall a definition. For a set of locally finite perimeter of , the essential boundary of , denoted by , is the set of points with neither full nor zero volume density, that is,
| (4.69) |
Here denotes the points of volume density , that is,
| (4.70) |
In general, we always have . Federer’s theorem states that for sets of locally finite perimeter in .
A consequence of the volume density bounds (4.71) in the following proposition is that the topological boundary of an almost-minimizer cannot contain any points of zero or full volume density, that is, the essential boundary in equals the topological boundary in . This fact precludes the existence of sharp cusps in the topological boundary of as well as prevents two sheets of the topological boundary from touching tangentially. The perimeter bounds (4.72) show that the perimeter measure for is -Ahlfors regular up to scale .
Proposition 4.10 (Volume and perimeter bounds for almost-minimizers).
There exist positive constants , , and with the following property. with the following property. If is a -almost-minimizer of in at scale , then for every with ,
| (4.71) |
and
| (4.72) |
Moreover, the volume density bounds (4.71) imply and so Federer’s theorem gives
| (4.73) |
Proof.
The upper bound of (4.72) was proved in Lemma 4.4. For the lower bound of (4.72) take small enough so that where is the constant in Proposition 4.9.
Recall from the proof of Lemma 4.4 that for we have for a.e. . Then the inequality (4.30) becomes
| (4.74) |
So by the lower bound of (4.72) we have
| (4.75) |
Taking small enough so that and relabeling to gives for a.e. . Integrating on and modifying constants gives which is the lower bound of (4.71). Since is also a -almost-minimizer of , we can apply this lower bound of (4.71) to get which gives the upper bound of (4.71).
Hence, given any -almost-minimizer of in at scale , we can shrink by a fixed amount, depending only on the universal constants , and an upper bound for , so that at points the volume and perimeter bounds hold for all . Throughout the rest of this paper, we will work at this smaller scale and use the volume and perimeter bounds.
4.5. Compactness for the class of -energies
In addition to having fixed , , , , fix a positive constants and . Define the class of admissible matrix-valued functions by
| (4.78) |
Lemma 4.11.
is compact in the topology of uniform convergence on compact sets as a subspace of .
Proof.
Let be a sequence in . We apply Arzelà-Ascoli to noting that pointwise-boundedness follows from and equicontinuity follows from . Hence there is a subsequence and such that uniformly on compact sets. It follows that is symmetric and for any . For any ,
| (4.79) |
Sending , gives and . Thus and so is compact. ∎
Fix an open set and . Define the class of almost-minimizers of in for by
| (4.80) |
We will show that is compact by separately proving precompactness and closedness.
Proposition 4.12 (Precompactness of ).
Suppose that (that is, is a -almost-minimizer of in at scale for some and ), and that . For any open with , there exist as , a set of finite perimeter , and such that
| (4.81) |
Proof.
First we choose as such that . Let and with . Let be such that for . If , there is and so by upper density bound (4.72) since . By [Mag12, (16.10)], we have for that
| (4.82) |
and so . Since is open and compactly contained in , the balls with centers in that are contained in form a covering for . Hence we may cover by finitely many balls where satisfy with and for . Choose so that . Then
| (4.83) |
and so we may apply Theorem 3.1 to construct a set of finite perimeter and a further subsequence indices such that . Setting , we have that and . Finally, given and , we have
| (4.84) |
Since , this gives
| (4.85) |
So by density of in in the sup norm, we have . Finally, by Lemma 4.11 we may extract a further subsequence such that, up to relabeling, we also have uniformly on compact sets. ∎
Proposition 4.13 (Closedness of ).
Suppose that (that is, is a -almost-minimizer of in at scale for some and ), , is an open set with such that for a set of finite perimeter , and uniformly on compact sets for some . Then is a -almost-minimizer of in at scale . Moreover,
| (4.86) | |||
| (4.87) |
where we view and as Radon measures. In particular,
-
(i)
if , , and , then ;
-
(ii)
if , then there exists with such that .
Proof.
By the same argument as in the proof of Proposition 4.12 we can show . The weak convergence of (4.86) follows from as also shown in the proof of Proposition 4.12.
To show that is a -almost-minimizer of our strategy is as follows. Given a competitor for , we modify to construct competitors and apply the almost-minimality of with respect to . We then pass the minimality inequalities through limits to obtain the desired almost-minimality inequality for .
Suppose with and . For , set . Since and are Radon measures, we have that for a.e. ,
| (4.88) |
Note that because a Lebesgue measurable set is equivalent to its set of points of full density. By the coarea formula, , and , it follows that
| (4.89) |
as . Consequently, by Fatou’s lemma,
| (4.90) |
for a.e. . Since is compactly contained in , we may find finitely many balls with and satisfying (4.88) and (4.90) with , such that, setting , we have . Now consider the comparison sets defined by . Since , by (4.88) there holds
| (4.91) |
Additionally, since so that by (4.90) there holds
| (4.92) |
Observe that with . Since , there is such that for all . For now fix . By (4.91) we can apply Proposition 4.3 to obtain
| (4.93) |
Since is a competitor for the -almost-minimality of , we have
| (4.94) |
which simplifies to
| (4.95) |
Similar to (2.11) we have . Integrating yields
| (4.96) |
where we set . Taking the as gives
| (4.97) |
because by the uniform convergence on compact sets. Similarly,
| (4.98) |
Using the fact that for nonnegative sequences , we have
| (4.99) |
By and , this becomes
| (4.100) |
Noting since and using the lower semicontinuity of with respect to by Proposition 3.4, this implies
| (4.101) |
Now we combine our estimates, again using that , and obtain
| (4.102) | |||||
Since , we can add to obtain
| (4.103) |
as desired.
Next we prove the weak convergence of energy measures (4.87). Let and which are Radon measures on . It suffices to show the following claim.
Claim.
If is a Radon measure and is a subsequence such that , then .
Indeed, suppose the claim is true. By sequential compactness of Radon measures (which applies since ), for each subsequence of there exists a further subsequence that converges weakly to some Radon measure . By the claim and so converges weakly to . Since by the decomposition formula of the Gauss-Green measure for the intersection of two sets of locally finite perimeter (see (16.4) of [Mag12, Theorem 16.3]), this will complete the proof of (4.87).
Now we prove the above claim. Suppose for some Radon measure and subsequence . For convenience we will just write the indices as instead of .
Let us show on where denotes the Borel sets of . Let be an open bounded set and set for . Choose with . Note that (4.100) holds for any bounded set in place of by the same argument. So applying (4.100) with in conjunction with the lower semicontinuity of with respect to by Proposition 3.4 gives
| (4.104) |
By monotone convergence, taking gives . Since was an arbitrary open, bounded set, it follows that on .
Now let with . Define
| (4.105) |
for with
| (4.106) |
This holds for a.e. . Then with and for all larger than some . By the same argument as with above to prove (4.95), for such there holds
| (4.107) |
Since ,
| (4.108) |
Sending and using the lower semicontinuity of weak convergent Radon measures, we have by the same reasoning as for (4.5) that
| (4.109) |
The lower perimeter bound (4.72) and comparability to perimeter give . So by (4.5) and , which we know because on , it follows that
| (4.110) |
for a.e. . Sending gives for -a.e. . Since , we have that on , the Borel subsets of . This completes the proof of our claim.
We finish by showing and . For , suppose and for . Let with . Then for large enough . So by the weak convergence of the measures in , the lower perimeter bound of (4.72), and , we have
| (4.111) |
Hence . For , suppose and by way of contradiction that there does not exist a sequence with and . Then there is some and as such that and for every . It follows that
| (4.112) |
contradicting the fact that . ∎
5. The Excess and the Height Bound
The concept of the excess is a common key tool in the study of regularity for minimizers for many geometric variational problems. This quantity measures the average -oscillation of outward unit normal vector with respect to a fixed direction and will eventually allow us to control the average -oscillation of from its average. Our aim is to show decay estimates for the excess of almost-minimizers. For our variable coefficient surface energies and the change of variable, it will be useful to measure this oscillation over balls, ellipsoids, and cylinders.
5.1. Definition of the excess and basic properties
Given we decompose into by identifying with and with . With a slight abuse of notation, we write where and are the horizontal and vertical projections defined by
| (5.1) |
We define the open cylinder centered at of radius in the direction by
| (5.2) |
Note that balls and cylinders are comparable as we have
| (5.3) |
and we have by (4.7) that balls and the ellipsoids are comparable. Thus balls, ellipsoids, and cylinders can all be mutually contained in each other by shrinking or enlarging them by fixed scales.
Given a set of locally finite perimeter , a point , a radius and direction , we define the spherical excess by
| (5.4) |
the ellipsoidal excess by
| (5.5) |
and the cylindrical excess by
| (5.6) |
Since balls, ellipsoids, and cylinders are comparable, if we can control one of these types of excess, we can control all of them.
As mentioned in Section 4, it will often be convenient to prove estimates at points with the assumption . To do this, we make the change of variable under the transformation . The next proposition shows that the excess of the image set under this transformation is comparable to that of the original set.
Proposition 5.1 (Comparability of excess under change of variable ).
There exists a positive constant with the following property. If is a set of locally finite perimeter, for some , then for any and ,
| (5.7) |
where is defined by
| (5.8) |
Proof.
Without loss of generality assume and to simplify notation write , , and . Noting that is symmetric, the change of variable gives by Proposition A.2 that and
| (5.9) |
Note that , , and
| (5.10) |
For the first term, we have the estimate
| (5.11) |
since the maximum eigenvalue of is bounded by and its minimum eigenvalue is bounded by . For the second term, we have the estimate
| (5.12) |
as above. It follows that
| (5.13) |
Hence
| (5.14) |
or equivalently, . The upper bound for (5.7) follows by a symmetric argument. ∎
We now recall several known properties of the excess, referring readers to [Mag12, Chapter 22] for proofs of these facts.
Proposition 5.2 (Scaling of the excess).
Proposition 5.3 (Zero excess implies being a half-space).
If is a set of locally finite perimeter in , with , , , and , then if and only if is equivalent to the set .
Proposition 5.4 (Vanishing of the excess at the reduced boundary).
If is a set of locally finite perimeter in and , then
| (5.16) |
Proposition 5.5 (Excess at different scales).
If is a set of locally finite perimeter in , , , , then
| (5.17) |
Proposition 5.6 (Excess and changes of direction).
For every , there exists a constant
with the following property. If is a -almost-minimizer of in at scale , then
| (5.18) |
whenever , , .
5.2. Small-Excess Position and the Height Bound
We now recall some standard lemmas we will need about the excess and almost-minimizers and recall the height bound. The first lemma states that if the excess of an almost-minimizer in a given cylinder is small enough, then in a smaller cylinder the topological boundary sits within a narrow strip.
Lemma 5.7 (Small-excess position).
Given and , there is a positive constant with the following property. If is a -almost-minimizer of in with , , , and
| (5.19) |
then
| (5.20) | |||
| (5.21) | |||
| (5.22) |
We define the open disk in centered at and of radius by
| (5.23) |
Thus we may write .
The second lemma states that if the a set of locally finite perimeter satisfies the separation property of Lemma 5.7, then the difference of measure of perimeter sitting above a set and defines a measure which we call the excess measure.
Lemma 5.8 (Excess measure).
If is a set of locally finite perimeter in , with , and such that, for some ,
| (5.24) | |||
| (5.25) | |||
| (5.26) |
then, setting for brevity , we have for every Borel set , function , and that
| (5.27) | |||
| (5.28) | |||
| (5.29) | |||
| (5.30) |
where . In fact, the set function
| (5.31) | ||||
| (5.32) |
defines a Radon measure on , concentrated on , called the excess measure of over .
We now state the main result we need from this section which is a strengthening of Lemma 5.7 to quantitatively control the height of an almost-minimizer in a cylinder by the excess on a larger cylinder.
Proposition 5.9 (Height bound).
Given , there exist positive constants and with the following property. If is a -almost-minimizer of in at scale with , , and
| (5.33) |
then
| (5.34) |
Throughout the course of the proof of our regularity result, we shall keep track of a number of specific constants for which certain estimates hold. The estimate (5.34) with the constants and from Lemma 5.9 are the first of these. Subsequent ’s will be chosen to be larger than previous ones, i.e. and subsequent ’s will be chosen to be smaller than the previous ones, i.e. . This way previous estimates will also hold under any smallness of the excess assumptions. We shall also choose so that the height of our topological boundary is at most of the cylinder.
6. The Lipschitz Approximation Theorem
The next step in our proof is to show that, given a small excess assumption of an almost-minimizer in a cylinder, a large portion of the topological boundary can be covered by the graph of a Lipschitz function in a smaller cylinder. Moreover, if we assume , this Lipschitz function is quantitatively ‘almost-harmonic’ at with an error controlled in terms of the excess and the scale. Given a direction which decomposes into , we denote the gradient in the first directions by , that is, .
Theorem 6.1 (Lipschitz approximation theorem).
There exist positive constants , , and with the following property. If is a -almost-minimizer of in at scale with , , and
| (6.1) |
then, setting
| (6.2) |
there is a Lipschitz function with satisfying
| (6.3) |
such that the translation of the graph of over contains , that is, , and covers a large portion of in the sense that
| (6.4) |
Moreover, is ‘almost harmonic’ in in the sense that
| (6.5) |
and if , then
| (6.6) |
Proof.
Without loss of generality we may assume and . We simplify notation by setting . Everything up to and including (6.5) follows from [BEG+19, Theorem A.3] by Proposition 4.10, for an chosen sufficiently small. We also choose small enough so that
| (6.7) |
where is the constant from Lemma 5.7 with . It follows that
| (6.8) |
Let . By considering , we may assume and reduce to proving
| (6.9) |
By the Fundamental Theorem of Calculus and the fact that on , we have . Let be a cutoff function such that
| (6.10) |
and define by . Then , , and . Hence
| (6.11) |
and so . Consider the family of maps defined by . Then and so . We have that where denotes the operator norm and denotes the Frobenius norm. It then follows by [Mag12, Lemma 17.4] that there are positive constants , such that
| (6.12) |
for . Since is bounded, we can choose so that is a diffeomorphism for . Letting , we also have by [Mag12, Lemma 17.4] that for . Choosing , we claim that for .
To see why this is the case, take . Then for some . By definition of , and . So since and . Hence .
By [Mag12, Proposition 17.1] we have that
| (6.13) |
Claim.
We can choose small enough so that
| (6.14) |
for all , where .
To prove the claim, observe that and so, since for small by Taylor’s theorem, shrinking as necessary, we have
| (6.15) |
whenever . Combining this with (6.12) gives
| (6.16) |
for . Integrating with respect to completes the proof of the claim.
By the claim, Proposition 4.10, and Lemma 4.5, it follows that
| (6.17) |
whenever . Choosing gives that
| (6.18) |
Now, for -a.e. , there is such that
| (6.19) |
By (6.8) and definition of , we have on a neighborhood and so and for . Hence for -a.e. , there holds
| (6.20) |
since . Thus
| (6.21) |
by (6.18) and since . Since , it follows that
| (6.22) |
where we used the fact that , , and . Again, using , , we have
| (6.23) |
By (6.4) and (6.5), it follows that
| (6.24) |
completing the proof. ∎
7. The Reverse Poincaré Inequality
In Section 5 we saw that a small excess controls the height of the topological boundary of an almost-minimizer. In this section we show that given a small excess assumption on a cylinder, the flatness of the topological boundary controls the excess on a smaller cylinder. Recall that the cylindrical flatness of a set of locally finite perimeter at a point , radius , in the direction is defined by
| (7.1) |
This quantity measures how far in an sense the boundary of is from the best approximating plane with normal .
Theorem 7.1 (Reverse Poincaré Inequality).
Given , there is a positive constant
with the following property. If is a -almost-minimizer in with , , , and
| (7.2) |
where is the constant from Lemma 5.7, then
| (7.3) |
To prove this we modify the proofs presented in [Mag12, Chapter 24]. First we need several lemmas. Given and the decomposition of into , we define the narrow cylinders
| (7.4) |
for and .
Lemma 7.2 (Cone-like competitors [Mag12, Lemma 24.8]).
If and is an open set with smooth boundary in such that
| (7.5) |
then, for every and , there exists with such that for every , there exists an open set of locally finite perimeter in , satisfying,
| (7.6) |
| (7.7) |
| (7.8) |
| (7.9) |
Proof.
Lemma 7.3 (Weak reverse Poincaré inequality).
If is a -almost-minimizer of in , , at scale , such that
| (7.10) |
| (7.11) |
and if and are such that
| (7.12) |
then, for every ,
| (7.13) |
where .
Proof.
Properties (7.10) and (7.11) imply by the divergence theorem that
| (7.14) |
defines a Radon measure on concentrated on as in Lemma 5.8. By [Mag12, Theorem 13.8], given there exists a sequence of open sets with smooth boundary such that
| (7.15) |
where denotes the -neighborhood of . The coarea formula and Fatou’s lemma give
| (7.16) |
So for a.e. , there holds
| (7.17) |
Provided that is large enough, and imply by (7.10) and (7.11) that
| (7.18) |
| (7.19) |
Given and we can apply Lemma 7.2 to each for and to find the sets with such that for each there exists an open set satisfying (7.6), (7.7), (7.8), and (7.2). For each , we have the containment and so
| (7.20) |
It follows that there exists a subsequence as and such that
| (7.21) |
By Lemma 7.2 there exist a sequence of open sets of locally finite perimeter in such that
| (7.22) |
| (7.23) |
and
| (7.24) |
Now consider the comparison sets
| (7.25) |
Since and
| (7.26) |
we have that
| (7.27) |
By Proposition 4.5 we have
| (7.28) |
for some . Hence
| (7.29) |
It follows that
| (7.30) |
Taking the limit as , using the weak convergence of (7.15) since , and , we have
| (7.31) |
Hence
| (7.32) |
By (7.14), and so this inequality also holds for provided we take . Hence it holds for all . Minimizing the right hand side over all yields
| (7.33) |
as desired. ∎
Proof of Theorem 7.1.
By the scaling given in Proposition 4.2, we have that is a -almost-minimizer of in at scale with , , , and . Thus to prove (7.3), we may assume and show
| (7.34) |
By Proposition 5.7 and Proposition 5.8, it follows that
| (7.35) |
| (7.36) |
and
| (7.37) |
Hence and so it suffices to show that for every ,
| (7.38) |
If , then and so
| (7.39) |
and we are done provided we take . Thus we are left with the case . Set
| (7.40) |
which defines a Radon measure on , concentrated on . We apply Lemma 7.3 in every cylinder with and such that
| (7.41) |
to find that
| (7.42) |
An approximation argument, setting
| (7.43) |
for brevity, implies by (7.42) that
| (7.44) |
whenever since . We now use a covering argument to complete the proof. Let
| (7.45) |
and notice since for every , we have
| (7.46) |
Given , cover by finite many balls with centers . This can be done with a bounded number of balls depending only on the dimension , that is, . So by the subadditivity of the measure , (7.44), and the definition of , we have
| (7.47) |
where we used that . Hence for some . By Cauchy-Schwarz, we have . Combining these gives . Thus for some . Recalling the definitions of and , we see that this completes the proof of (7.38). ∎
8. Tilt-Excess Decay
We showed in Section 6 that almost-minimizers can be approximated by ‘almost-harmonic’ Lipschitz functions at points with small excess. Now we approximate these Lipschitz with harmonic functions which allow us to find new directions for which the excess experiences quadratic decay.
First we recall a couple lemmas about harmonic functions. These are just the rescaled versions of [Mag12, Lemma 25.1, Lemma 25.2]. Note that
| (8.1) |
denotes the integral average.
Lemma 8.1.
There is a positive constant with the following property. If is harmonic in and is defined by , then
| (8.2) |
for every . In particular,
| (8.3) |
Lemma 8.2 (Harmonic approximation).
For every there exists with the following property. If is such that
| (8.4) |
then there exists a harmonic function on such that
| (8.5) |
We now prove the excess improvement by tilting. This states that if the excess is small enough in a given direction, then there is a nearby direction in which the excess at a definite smaller scale sees quadratic decay with the error term seeing th power decay. Note in the theorem below, the fraction comes from the rough bound where the comes from the Lipschitz approximation theorem, the comes from small excess assumption in the reverse Poincaré inequality, and the comes from containing one cylinder inside of another cylinder that is tilted in a different direction.
Theorem 8.3 (Excess improvement by tilting).
Given , there exist positive constants and with the following property. If is a -almost-minimizer of in , , , and with
| (8.6) |
then there exists such that
| (8.7) |
Proof.
Assuming without loss of generality that and , it suffices to prove that given , there exist positive constants and with the following property. If is a -almost-minimizer of in , , , and with
| (8.8) |
then there exists such that
| (8.9) |
We set for brevity.
We shall select a number of criteria for to satisfy which together give the desired result. We place a around each of these choices to make it easy for the reader to check that all of these choices are consistent.
Choose to satisfy
| (8.10) |
where is from the Lipschitz approximation theorem. Then and thus there is a Lipschitz function such that such that
| (8.11) |
| (8.12) |
| (8.13) |
| (8.14) |
where is the constant from the Lipschitz approximation theorem, , and is the graph of . Choose to also satisfy
| (8.15) |
Then and so setting
| (8.16) |
we have
| (8.17) |
By Lemma 8.2, for every there is such that if
| (8.18) |
then there is which is harmonic in such that
| (8.19) |
Setting , we have that is harmonic in and
| (8.20) |
Since , setting for , we see by Lemma 8.1 that
| (8.21) |
By (8.20) and ,
| (8.22) |
and so
| (8.23) |
Noting , we have that
| (8.24) |
We apply the above with and choose to also satisfy
| (8.25) |
with as in (8.18). Then and so
| (8.26) |
Now set
| (8.27) |
and let’s estimate . Since , we have that
| (8.28) |
This last integral we split in terms of and .
For , observe that
| (8.30) |
By the height bound, we have
| (8.31) |
Also, . Since is harmonic,
| (8.32) |
By (8.20) and from (8.11), it follows that
| (8.33) |
Since , we have
| (8.34) |
Choose to also satisfy
| (8.35) |
Then
| (8.36) |
which gives
| (8.37) |
Combining these estimates we have
| (8.38) |
Next, we show that provided is suitably small, then
| (8.39) |
By Proposition 5.5 and Proposition 5.6, we have
| (8.40) |
where Additionally,
| (8.41) |
where the last several inequalities follow as above. Hence
| (8.42) |
for some . We choose to also satisfy
| (8.43) |
so that
| (8.44) |
since . The reverse Poincaré inequality, Theorem 7.1, implies that
| (8.45) |
as desired. ∎
9. Regularity of Almost-Minimizers
We are almost in the position to prove our main regularity result. All we first need is to prove the following lemma which allows us to remove the assumption and obtain an excess-decay estimate which we will iterate in the proof of our regularity theorem.
Lemma 9.1.
For each , there exist positive constants , , and with the following property. Let be a -almost-minimizer of in with and , and set
| (9.1) |
If and
| (9.2) |
then there exists such that
| (9.3) | ||||
| (9.4) |
Proof.
We will eventually make choices for positive constants and show that (9.3) holds if we set
| (9.5) |
where is the constant from Proposition 8.3 applied with .
Since and , we have
| (9.6) |
Consequently, we only need to show the existence of such that
| (9.7) |
If , then by Proposition 5.5 we have
| (9.8) |
and so we can take . Otherwise, . We will proceed by applying Proposition 8.3, but we need to use the change of variable since we are not assuming that equals the . This enables us to work with the set which is an almost-minimizer of with . Let denote the image of under this change of variable, that is,
| (9.9) |
First note that
| (9.10) |
Then is an almost-minimizer of in by Proposition 4.1 since
| (9.11) |
It also follows by (9.10) and Proposition 5.7 that
| (9.12) |
Hence by our assumption
| (9.13) |
where at this step we make our choice for . Thus Proposition 8.3 applies to with radius and so there is such that
| (9.14) |
Let denote the preimage of under the change of variable , that is,
| (9.15) |
Note that
| (9.16) |
So by definition of and Proposition 5.7, we have that
| (9.17) |
Combining this with (9.14) yields
| (9.18) |
Using this we now make our choice for by setting
| (9.19) |
The condition allows us to apply Proposition 8.3 as above and since , (9.18) implies
| (9.20) |
completing the proof of (9.3).
Now we prove our main theorem. Before we start, let’s briefly describe the structure of the argument. In the Lipschitz approximation theorem, Theorem 6.1, we saw that given a small excess assumption, there is a Lipschitz function such that, setting
| (9.23) |
the translated graph graph of over contains . We proceed by iterating (9.3) at points to obtain a sequence of unit vectors for which certain decay estimates of the excess hold, namely (9.35) and (9.36). Using this, we show that and that converges to . Moreover, our iteration gives estimates for Hölder continuity of . Lastly, we show in fact equals , that is, equals the graph of . Hölder estimates for follow from the ones for .
Theorem 9.2 (-regularity of almost-minimizers of ).
There exist positive constants
and with the following property. If is a -almost-minimizer of in with and such that
| (9.24) |
then there exists a Lipschitz function with satisfying
| (9.25) |
such that
| (9.26) | ||||
| (9.27) |
and with
| (9.28) | ||||
| (9.29) |
for every and .
Proof.
Without loss of generality we may assume . Let , , and denote the constants from Lemma 9.1 with the choice which hence depend only on , and . As mentioned before, we will choose and apply the Lipschitz approximation theorem, Theorem 6.1. This gives that there is a Lipschitz function with satisfying
| (9.30) |
and such that, setting
| (9.31) |
the translated graph graph of over contains , that is, . As in Lemma 9.1, we define
| (9.32) |
Let . Then and so
| (9.33) |
where . For this constant , choose to also satisfy
| (9.34) |
so that .
Claim.
There exists a sequence and with such that for every ,
| (9.35) | ||||
| (9.36) |
for some constant .
Proof of claim.
Since , we may apply Lemma 9.1 to find such that
| (9.37) | ||||
| (9.38) |
In particular, since ,
| (9.39) |
Proceeding inductively we find a sequence such that
| (9.40) | ||||
| (9.41) |
for . Stringing together the inequalities of (9.40) gives (9.35) and stringing together the inequalities of (9.41) gives
| (9.42) |
for . Given , it follows that
| (9.43) |
and so
| (9.44) |
where since depends only on these constants too. Hence is Cauchy and so there is such that as . Sending in (9.44) gives (9.36) and this first claim is proved.
Claim.
There is a constant such that
| (9.45) | ||||
| (9.46) |
Proof of claim.
Given , there is such that
| (9.47) |
Integrating with respect to over , and using the the perimeter bound (4.72) and (9.36), it follows that
| (9.48) |
which is (9.45). Now, take . In the case where , it follows that
| (9.49) |
Otherwise, and so integrating with respect to over , using (9.45) with and (9.36) with gives
| (9.50) |
Hence (9.46) holds. This completes the proof of our second claim.
Suppose . Then and so there is some such that
| (9.51) |
Integrating with respect to over and using the perimeter bounds (4.72) gives
| (9.52) |
Hence by (9.35) and the definition of we have,
| (9.53) |
By this, (9.36), (9.33), and , it follows that
| (9.54) |
and so
| (9.55) |
We now prove and so that (9.55) becomes (9.29), proving the Hölder continuity of the outer normal to .
By (9.45), . So by perimeter bounds (4.72), we
| (9.56) |
Expanding implies
| (9.57) |
Since and
| (9.58) |
this implies
| (9.59) |
Since , this by definition means with and hence (9.29) holds.
Combining (9.46) with (9.33) gives
| (9.60) |
Lastly, for this constant , we choose to also satisfy
| (9.61) |
where is the constant from the Lipschitz approximation theorem. It follows for that
| (9.62) |
and so . By the Lipschitz graph criterion, [Mag12, Theorem 23.1], the graph of the Lipschitz function coincides with in . Moreover,
| (9.63) |
for all . Since , for , it follows that
| (9.64) | |||
| (9.65) |
Theorem 9.3 (Regularity of the reduced boundary and characterization of the singular set).
If is an open set in , , and is a -almost-minimizer of in , then is a -hypersurface that is relatively open in , and it is -equivalent to . Hence the singular set of in ,
| (9.66) |
is closed. Moreover, is characterized in terms of the excess as follows:
| (9.67) |
where is the positive constant from Theorem 9.2.
Proof.
The regularity and relative openness of follows from Theorem 9.2 and the -equivalence follows from Proposition 4.10. Consequently, is closed. Hence all we need to show is (9.67). Consider the set defined by
| (9.68) |
We show .
If , then there is , , with such that
| (9.70) |
By Theorem 9.2, coincides with the graph of a -function and so . Hence . ∎
Now that we have established regularity of almost-minimizers at points in the reduced boundary, we wish to study the singular set which we do in the next section. However, before we move on to that, we prove the convergence of the outer unit normal vectors along sequences of almost-minimizers and points in the reduced boundaries. The contrapositive of this will be a useful tool in showing that the blow-ups at a singular point must converge to a singular point.
We first need the following lemma regarding almost upper semicontinuity of the excess. Recall from Section 4 the class of uniformly elliptic, Hölder continuous matrices with respect to the given universal constants and the class of -almost-minimizers of with .
Lemma 9.4 (Almost upper semicontinuity of the excess).
Suppose that is a sequence of -almost-minimizers of in at scale , , is an open set with such that for a set of finite perimeter, and uniformly on compact sets for some . Furthermore suppose and with , , and
| (9.71) |
then
| (9.72) |
for some positive constant .
Proof.
By Proposition 4.13, is a -almost-minimizer of in at scale , satisfying
| (9.73) | |||
| (9.74) |
We write for to simplify notation and claim
| (9.75) |
for some .
To show this, note
| (9.76) |
Noting , we bound the first term of (9.76) by
| (9.77) |
Similarly, for the second term of (9.76), we have
| (9.78) |
Since , by Proposition 4.13 there is a sequence such that . Given , we have for large . So
| (9.79) |
by the upper perimeter bound (4.72) for . Hence . We also have . So by (9.76), (9), and (9), we have
| (9.80) |
Note because and so by (9.74) and . This and the uniform convergence of on complete the proof of our claim.
Theorem 9.5 (Convergence of outer unit normals).
If and are -almost-minimizers of and , respectively, in the open set at scale , and
| (9.83) |
then for large enough. Moreover,
| (9.84) |
Proof.
Considering the translated sets , note that
| (9.85) |
, and uniformly on compact sets. Hence by replacing with and with , and with for some sufficiently small , we may assume that for every .
By applying the change of variable on , and , , we may assume without loss of generality that . Choose an open set with and . Lemma 9.4 with implies there is a constant for which
| (9.86) |
holds for every such that and . Since , by Proposition 5.4 there is and with , , , and
| (9.87) |
where is the constant from Theorem 9.2. Then for large and so by Theorem 9.3, for large . Moreover, by Theorem 9.2 there exist Lipschitz functions with such that
| (9.88) | ||||
| (9.89) |
and such that for ,
| (9.90) |
where . Then
| (9.91) |
It follows by integration by parts and the density of in that
| (9.92) |
for every . By (9.90), is equicontinuous and it is bounded by . Thus by Arzelà-Ascoli it is compact under uniform convergence. By (9.92), is the only possible limit point of . Hence uniformly on . Consequently, as , it follows that
| (9.93) |
as desired. ∎
10. Analysis of the Singular Set
In this final section, we turn to the portion of Theorem 1.1 which addresses the size of singular set.
Theorem 10.1 (Dimensional estimates of singular sets of -almost-minimizers).
If is a -almost-minimizer of in the open set at scale , then the following hold true:
-
(i)
if , then is empty;
-
(ii)
if , then has no accumulation points in ;
-
(iii)
if , then for every .
This result is known to be sharp in the case of perimeter minimizers in the sense that Simons’ cone,
| (10.1) |
is a perimeter minimizer in with singular set , and for , is perimeter minimizer in that gives . Since our surface energies include perimeter when , our theorem is also sharp.
We use blow-up analysis and a standard Federer dimension reduction argument to prove Theorem 10.1. The next theorem shows the convergence of the singular set along sequences of almost-minimizers. Recall again from Section 4 the class of uniformly elliptic, Hölder continuous matrices with respect to the given universal constants and the class of -almost-minimizers of with .
Theorem 10.2 (Closure and local uniform convergence of singularities).
If and are -almost-minimizers of and , respectively, in the open set at scale , and
| (10.2) |
then . Moreover, given and compact, then
| (10.3) |
for all large where denotes the neighborhood of a set.
Proof.
It must be that since would contradict Theorem 9.5. We prove (10.3) by contradiction. Indeed, assume there exist , compact, as , and such that . By compactness of and reducing to a further subsequence, we may assume for some . By Proposition 4.13 , we have . By the first part of this theorem, we have . This implies for large , a contradiction. ∎
10.1. Existence of blow-up limits
We now prove the existence of blow-up limits along subsequences and their convergence to singular minimizing cones. We say that is a cone with vertex if it is invariant under blow-ups at , that is, if
| (10.4) |
for all . If is a cone which is a (global) minimizer of in and , we say that is a singular minimizing cone of . A singular minimizing cone of perimeter we simply refer to as a singular minimizing cone.
Note that if is a singular minimizing cone, then , for otherwise the blow-ups would converge to a half-space, implying that is a half-space, in contradiction with .
Theorem 10.3 (Existence of blow-up limits at singular points).
If is a -almost-minimizer of in the open set at scale , , and , then, setting
| (10.5) |
there exist as and a set of locally finite perimeter in such that
| (10.6) |
and is singular minimizing cone of in with vertex .
Proof.
The change of variable allows us to assume without loss of generality that since the convergence properties and cones are preserved under this affine transformation. For each , is eventually contained in for large . Note that is a -almost-minimizer of in at scale by Proposition 4.2 and . Hence and for some positive constants and . Once , we have is a -almost-minimizer of . Thus we may apply the compactness of Proposition 4.12 and Proposition 4.13 to obtain as , a set of locally finite perimeter , and a uniformly elliptic, Hölder continuous matrix such that and uniformly on compact sets and is a minimizer of in at scale . Note that . Hence . By Proposition 4.13 and a diagonalization argument, we obtain a subsequence such that up to relabeling
| (10.7) |
By Theorem 10.2 we have because . All that remains is to show that is a cone with vertex . Choose one of the a.e. for which we have . By (10.1), Proposition 4.2, and Corollary 4.8, it follows that
| (10.8) |
since as . Hence
| (10.9) |
The monotonicity formula for perimeter minimizers [Mag12, Theorem 28.9] gives
| (10.10) |
So (10.9) implies
| (10.11) |
Hence
| (10.12) |
for a.e. , and thus
| (10.13) |
This implies for -a.e. . Thus is a cone with vertex by [Mag12, Proposition 28.8]. ∎
10.2. Dimension reduction argument
The next few results we recall from the standard argument for the characterization of the singular set for perimeter minimizers which we use for our adapted proof. We refer readers to [Mag12, Chapter 28] for proofs.
Theorem 10.4.
If and there exists a singular minimizing cone with , then .
Theorem 10.5 (Dimension reduction theorem).
If is a singular minimizing cone in , , , and if , then, up to extracting a subsequence and up to rotation, the blow-ups locally converge to a cylinder , where is a singular minimizing cone in .
Lemma 10.6 (Half-lines of singular points).
If is a singular minimizing cone in , , and , then and .
Lemma 10.7 (Cylinders of locally finite perimeter).
-
(i)
If is a set of locally finite perimeter in , then is of locally finite perimeter in , with
(10.14) Moreover, if is a perimeter minimizing in , then is a perimeter minimizer in .
-
(ii)
If is a set of locally finite perimeter in such that
(10.15) then there exists a set of locally finite perimeter in such that is equivalent to . If, moreover, is a perimeter minimizer in , then is a perimeter minimizer in .
Lemma 10.8.
-
(i)
If is a Borel set such that , , then
(10.16) -
(ii)
If is an -almost-minimizer of in the open set at scale , and , then, setting
(10.17) we have
(10.18) for every compact set .
-
(iii)
If , , and , then .
Proof.
and are proved in [Mag12, Lemma 28.14] and we now adapt the proof of his version of to the case of -almost-minimizers.
Let be a finite covering by open sets of the compact set . Then there exists such that . Eventually and , and so by Proposition 4.2 is a -almost-minimizer of in . Then by Theorem 10.2,
| (10.19) |
for large . Hence by definition of it follows that
| (10.20) |
Taking the infimum over all such coverings proves the result. ∎
Lastly we recall a technical result [Mat95, Theorem 8.16] before we jump into the proof of Theorem 10.1.
Lemma 10.9.
If and , then there exists with .
Proof of Theorem 10.1.
Let be a -almost-minimizer of in with . By way of contradiction, suppose there exists . As usual we may assume without loss of generality that by the change of variable . By Theorem 10.3 there exists a singular minimizing cone in , but this contradicts Simons’ theorem on the nonexistence of singular minimizing cones in dimensions , see [Mag12, Theorem 28.1 (i)].
Let be a -almost-minimizer of in . By way of contradiction, suppose is an accumulation point of . Then there is a sequence such that . Again we may assume without loss of generality that . Set and consider the blow-ups . By Theorem 10.3 there is a subsequence, which upon relabeling as , converges locally to a singular minimizing cone in . Let . Then and so by compactness there exists and a further subsequence so that, up to relabeling, we have . Note that as . So by Theorem 10.2 we have . Since , we have and so by Theorem 10.5 there exists a singular minimizing cone in , contradicting .
Let be a -almost-minimizer of in and suppose with . Then there exists . By the change of variable , we may assume without loss of generality that . By [Mat95, Theorem 8.16] and Lemma 10.8 , there exists such that
| (10.21) |
for all . This is equivalently rewritten in terms of the blow-ups as
| (10.22) |
for all . Eventually when is sufficiently and thus
| (10.23) |
By Theorem 10.3 there exists a subsequence, which up relabeling as , converges locally to a singular minimizing cone in . By Lemma 10.8 we have
| (10.24) |
We may apply the above argument with and in place of and . By Theorem 10.5 there exists a singular minimizing cone
| (10.25) |
By Lemma 10.8 we must have . If we now assume that and , repeating this argument times gives the existence of a singular minimizing cone in with , in contradiction to . Thus we conclude that . ∎
A. Appendix A
In this appendix we provide a proof for the change of variable formula for sets of locally finite perimeter with bounded, continuous integrands depending on both and . This is a generalization of [Mag12, Proposition 17.1] and [KMS19, Theorem A.1].
Proposition A.1 (Change of variable for sets of locally finite perimeter).
Suppose is a diffeomorphism of and denote . If is a set of locally finite perimeter in , then is a set of locally finite perimeter in such that
| (A.1) |
If is a bounded and continuous function and is an open, bounded set satisfying , and and are bounded, then the change of variable gives
| (A.2) |
Note that is the tangential Jacobian of with respect to .
Proof.
The fact that is a set of locally finite perimeter is shown in [Mag12, Proposition 17.1] and (A.1) is proved in [KMS19, Theorem A.1]. Hence we only need to show (A.2).
Let where denotes the standard mollifier and let . Then in and in as shown in the proof of [Mag12, Proposition 17.1]. Note that and so . By [Mag12, Remark 8.3], the change of variable gives
| (A.3) |
We shall show that this equation converges to (A.2) as . To do this, we shall apply the version of Reshetynak’s continuity theorem provided in [Spe11, Theorem 1.3]. Under the hypotheses that is bounded and continuous and is open, this states that
| (A.4) |
whenever , are finite -valued measures satisfying
| (A.5) |
where denotes the completion, with respect to the sup norm, of the compactly supported continuous functions from to .
Starting with the right-hand side of (A), first note that since is continuous. By [Mag12, Theorem 12.20], and where denotes Lebesgue measure. Since is bounded and is continuous, is bounded. Since is bounded, . Hence and
| (A.6) |
for all , since as is compact. Thus , are finite -valued measures which satisfy (A.5) (where we take discrete sequences of ). Hence for each , applying [Spe11, Theorem 1.3] to the bounded, continuous function gives
| (A.7) |
that is, . In particular, it follows from the fact that . Hence (A.5) holds for , and so we can apply [Spe11, Theorem 1.3] to which is bounded continuous since and are. We obtain
| (A.8) |
This is the convergence of the right-hand side of (A) to the right-hand side of (A.2).
Now moving on to the left-hand side of (A), note that for all ,
| (A.9) |
since in . So by density of in , we have . The change of variable ([Mag12, Remark 8.3]) gives
| (A.10) |
where the last equality is by [Mag12, Proposition 17.1]. Hence . Thus since and
| (A.11) |
for all , since as is compact. Hence satisfies (A.5) so by [Spe11, Theorem 1.3], we have
| (A.12) |
Hence left hand side of (A) converges to the left hand side of (A.2) and we are done. ∎
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