Abstract
We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the different behaviors of the energy at different scales allow the preservation of details and irregularities of the image in the process of removing white noises, thus improving the quality of the image without losing relevant features. In this paper, we provide a series of results concerning existence, rigidity and classification of minimizers, compactness results, isoperimetric inequalities, Poincaré–Wirtinger inequalities and density estimates. Furthermore, we provide the construction of planelike minimizers for this generalized perimeter under a small and periodic volume perturbation.
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1 Introduction
The main goal of this paper is to study a class of variational problems which interpolate between the classical perimeter and the volume functionals. Generalized energies of this type have been analyzed in [6, 9, 10, 16], and also in anisotropic contexts and in view of discretizations methods in [7, 8]. In terms of applications, nonlocal functionals interpolating between perimeter and volume are often used in image processing to keep fine details and irregularities of the image while denoising additive white noises, see e.g. [2, 12].
These objects are modeled by an energy which, at large scales, resembles the perimeter using an approximation based on the Minkowski content, but on small scales they present predominant volume contributions, giving rise to a sort of nonlocal behavior, which may produce severe losses of regularity and compactness properties. We will call r-perimeters these nonlocal perimeter functionals.
We recall that in recent years a lot of attention has been devoted to the analysis of other type of nonlocal perimeter functionals, starting from the seminal work of Caffarelli et al. [5], where it was initiated the study of the Plateau problem for such kind of nonlocal perimeters. A regularity theory for minimizers of such perimeters has been developed in analogy to the regularity theory of classical minimal surfaces, and also the geometric and variational relation with the classical perimeter has been investigated. For a general overview on the subject we refer to [29] and references therein.
In this paper, we develop a preliminary study of the main properties of the r-perimeters. First of all we analyze the main features of sets with finite r-perimeter, in particular compactness properties, to get existence for the Dirichlet problem, and then isoperimetric inequalities. The global isoperimetric inequality is a direct consequence of the Brunn–Minkowski inequality, whereas its local version is valid at the appropriate scale.
We show some rigidity results for minimizers of the r-perimeter, in dimension 2, and we presents some properties of minimizers. In particular we consider their density properties, pointing out an interesting phenomenon not appearing in the classical case. Indeed in the density estimates two scales of growth appear: if the initial density is below a given threshold depending on r, then there is an exponential density growth, then, over the threshold, the growth reduces to the usual one, that is the radius to the power n.
An important feature of these results is that they always need to capture the “local” behavior of the minimizers, which can be rather different than the “global” one, due to nonlocal effects at small scales. In addition, these problems are not scale invariant and they do not possess any associated extended problem of local type, therefore many classical techniques related to scaled iterations and monotonicity formulas are not easily applicable in our setting. In particular, we show with a concrete example (see Theorem 1.19) that compactness and regularity properties can fail, at a small scale, for minimizers of the r-perimeter with the addition of a sufficiently large volume term.
Finally the last section is devoted to the construction of plane-like minimizers for the r-perimeters in a periodic medium. A classical problem in different fields, including geometry, dynamical systems and partial differential equations, consists in the determination of objects that are embedded into a periodic medium and present bounded oscillations with respect to a reference hyperplane. These objects are somehow the natural extension of “flat” objects such as hyperplanes and linear functions and have the important property that, for these solutions, the forcing term produced by the lack of homogeneity of the medium “averages out” at a large scale. We refer to [20, 22] for the first results of this type on geodesics, to [23] for the introduction of this setting in the case of elliptic integrands, to [1, 3, 11] for the case of hypersurfaces with prescribed mean curvature, to [26, 28] for the case of partial differential equations and to [4, 14] for problems related to statistical mechanics. The planelike structures are also useful to construct pinning effects and localized bump solutions, see e.g. [24, 25]. See also [15] for planelike constructions related to nonlocal problems of fractional type and [13] for a general review.
We also address the problem of existence of planelike minimizers for energy functionals in which the r-perimeter in (1.2) is modulated by a volume term which is periodic and with a sufficiently small size. This is a setting not comprised in the existing literature, since, as far as we know, the only nonlocal cases taken into account are the ones arising from fractional minimal surfaces or related to the Ising model.
In the rest of this section, we formalize the mathematical setting in which we work and we present our main results. The nonlocal perimeter functional based on Minkowski content will be introduced in Sect. 1.1. Some rigidity properties of minimizers will be also discussed.
In Sect. 1.2 we present some compactness results at large scales and some \(\Gamma \)-convergence results for this nonlocal perimeter. Then, in Sect. 1.3 we discuss the Dirichlet problem and in Sect. 1.4 we present some rigidity results.
In Sect. 1.5 we introduce global and relative isoperimetric inequalities. Furthermore, we provide density estimates for the nonlocal perimeter, which in turn show that the compactness and regularity properties of the nonlocal perimeter minimizers may be deeply influenced by oscillations at small scales.
Finally, the planelike minimizers for the nonlocal perimeter are discussed in Sect. 1.6.
A detailed organization of the paper is then presented at the end of the Introduction, in Sect. 1.7.
1.1 The nonlocal perimeter and the corresponding Dirichlet energy
We start with some preliminary definitions. Given \(r>0\) and \(E\subseteq \mathbb {R}^n\), we let
We shall identify a set \(E\subseteq \mathbb {R}^n\) with its points of density one and \(\partial E\) with the topological boundary of the set of points of density one.
Given \(r>0\) and a domain \(\Omega \subseteq \mathbb {R}^n\), for any measurable set \(E\subseteq \mathbb {R}^n\), we use the notation in (1.1) and we consider the functional
As customary, we denoted here by \({\mathscr {L}}^n\) the n-dimensional Lebesgue measure. When \(\Omega =\mathbb {R}^n\), we simply write \(\mathrm {Per}_r(E):=\mathrm {Per}_r(E,\mathbb {R}^n)\). Note that our definition agrees with that in [7, 8], since we are identifying a set with its points of density one, therefore
We observe (see e.g. [9]) that \(\mathrm {Per}_r\) is weak lower semicontinuous in \(L^1_\mathrm{{loc}}\) and that, for every A, B measurable sets,
The definition of \(\mathrm {Per}_r\) is inspired by the classical Minkowski content (which would be recovered in the limit, see e.g. [7, 8, 18]). In particular, for sets with compact and \((n-1)\)-rectifiable boundaries, the functional in (1.2) may be seen as a nonlocal approximation of the classical perimeter functional, in the sense that
Hence, in some sense, \(\mathrm {Per}_r\) recovers a perimeter functional for small r and a volume energy for large r.
We observe that recently a great attention has been devoted to the fractional perimeters introduced in [5], which also interpolate the classical perimeter with an area type functional (see e.g. [17] for a review on such topic). The functional in (1.2) is however very different in spirit from that in [5], since the lack of scaling invariance does not allow a classical regularity theory and causes severe lack of compactness at small scales (as we will discuss in details in the sequel).
More generally, given a domain \(\Omega \subseteq \mathbb {R}^n\) and a function \(g\in L^1_\mathrm{loc}(\mathbb {R}^n)\), we define the energy
The functional in (1.2) is related via a coarea formula to a Dirichlet energy which takes into account the local oscillation of a function, which is described as follows. Let \(\Omega \subseteq \mathbb {R}^n\) be a domain and \(u\in L^1_\mathrm{loc}(\Omega \oplus B_r)\). Then for any \(x\in \Omega \) we consider the oscillation of u in \(B_r(x)\), given by
In this paper, in the \(\sup \) and \(\inf \) notation, we mean the “essential supremum and infimum” of the function (i.e., sets of null measure are neglected). It can be checked by using the definition that a triangular inequality holds: for all \(v,u\in L^1_\mathrm{loc}(\Omega \oplus B_r)\)
We have the following generalized coarea formula [see formulas (4.3) and (5.7) in [10] for similar formulas in very related contexts]:
Lemma 1.1
It holds that
In the setting of (1.2) and (1.3), we introduce the definition of local minimizer and Class A minimizer. We are interested in existence, compactness and regularity properties of such minimizers. Moreover we will also provide construction of planelike minimizers for such energies in periodic media.
Definition 1.2
(Local minimizer and Class A minimizer) A set E is a minimizer for \({\mathrm {Per}_r}\) (resp. for \({\mathscr {F}}_{r,g}\)) in a bounded domain \(\Omega \) if for any measurable set \(F\subseteq \mathbb {R}^n\) with \(F{\setminus }(\Omega \ominus B_r)=E{\setminus }(\Omega \ominus B_r)\) it holds that
E is a Class A minimizer if it is a minimizer in this sense in any ball of \(\mathbb {R}^n\).
We observe that if E is a Class A minimizer, then
Indeed,
Note that in Definition 1.2 we allow competitors only away from the boundary of the domain, in a way compatible with the natural scale of the problem. Actually, this is the appropriate notion of minimizer, since the following result shows that the problem trivializes if competitors are allowed to produce modifications up to the boundary of the domain.
Proposition 1.3
Let \(E\subseteq \mathbb {R}^n\) be such that for every ball \(B\subseteq \mathbb {R}^n\), and for any measurable set \(F\subseteq \mathbb {R}^n\) with \(F{\setminus } B=E{\setminus } B\) it holds that
Then either \(E=\varnothing \) or \(E=\mathbb {R}^n\).
1.2 \(\Gamma \)-Convergence results and compactness properties
We start with some convergence results on \(\mathrm {Per}_{r_k}\) as \(r_k\rightarrow r\). We focus also on compactness properties of sets with bounded energy.
Theorem 1.4
Let \(\Omega \) be either an open and bounded subset of \(\mathbb {R}^n\), or equal to \(\mathbb {R}^n\). Let also \(r_k\rightarrow r\in (0,+\infty )\). Then the following holds.
-
(1)
\(\mathrm {Per}_{r_k}(E, \Omega ) \rightarrow \mathrm {Per}_{r}(E, \Omega )\) and \(\mathscr {F}_{r_k,g}(E, \Omega ) \rightarrow \mathscr {F}_{r,g}(E, \Omega )\) for all \(E\subseteq \mathbb {R}^n\).
-
(2)
\(\mathrm {Per}_{r_k}(\cdot , \Omega )\) (resp. \(\mathscr {F}_{r_k,g}(\cdot , \Omega )\)) \(\Gamma \)-converges in \(L^1_\mathrm{{loc}}(\Omega )\) to \(\mathrm {Per}_{r}(\cdot , \Omega )\) (resp. to \(\mathscr {F}_{r,g}(\cdot , \Omega )\)).
-
(3)
Let \(E_k\subseteq \mathbb {R}^n\) be such that
$$\begin{aligned}\sup _{k\in \mathbb {N}} \mathrm {Per}_{r_k}(E_k,\Omega )<+\infty .\end{aligned}$$Assume that, up to subsequences, \(\chi _{E_k}\rightharpoonup u\), in \(L^1_\mathrm{{loc}}\big (\Omega )\) with \(u:\mathbb {R}^n\rightarrow [0,1]\). Let \(\Sigma := \{x\in \Omega : u(x)\in (0,1)\}\) and
$$\begin{aligned} \ell :=\liminf _{r_k\rightarrow r} \mathrm {Per}_{r_k}(E_k,\Omega ).\end{aligned}$$(1.5)Then the following holds true:
$$\begin{aligned} {\mathscr {L}}^n(\Sigma \oplus B_r\cap \Omega )\leqslant 2\ell r , \qquad \chi _{E_k} \rightarrow u \;{ \text{ in } }\;L^1_\mathrm{{loc}}(\Omega {\setminus } \Sigma \big ), \qquad \int _\Omega \mathop {\hbox {osc}}\limits _{B_r(x)}u\, dx\leqslant 2r \ell .\quad \end{aligned}$$(1.6)
Observe that we cannot expect a stronger compactness result, due to the following observation.
Remark 1.5
Families of sets \(E_k\) for which \(\mathrm {Per}_r(E_k,\Omega )\leqslant 1\) are not necessarily compact in \(L^1(\Omega )\) (and, more generally, it is not necessarily true that \(\chi _{E_k}\) converges pointwise up to a subsequence).
Remark 1.6
When \(\Omega \) is unbounded and \(r_k\searrow r>0\), some pathological counterexamples to the claim in (1) of Theorem 1.4 may arise. For instance, one may have that
We recall also the following result, which is proved in [8, Theorem 3.1 and Remark 3.4].
Theorem 1.7
Let \(\Omega \subseteq \mathbb {R}^n\) be open and bounded, and \(r\rightarrow 0\). Then the following holds.
-
(1)
\(\mathrm {Per}_{r}(\cdot , \Omega )\) \(\Gamma \)-converges in \(L^1_\mathrm{{loc}}(\Omega )\) to \(\mathrm {Per}(\cdot , \Omega )\), where \(\mathrm {Per}\) is the standard perimeter.
-
(2)
Let \(E_r\subseteq \mathbb {R}^n\) be such that
$$\begin{aligned} \sup _{r} \mathrm {Per}_{r}(E_r,\Omega )<+\infty .\end{aligned}$$Then there exists \(E\subseteq \mathbb {R}^n\) such that
$$\begin{aligned} \chi _{E_r}\rightarrow \chi _E \quad { \text{ in } } L^1_\mathrm{{loc}}(\Omega )\quad {\text{ up } \text{ to } \text{ a } \text{ subsequence }}\end{aligned}$$and
$$\begin{aligned} \mathrm {Per}(E,\Omega )\leqslant \liminf _{r\rightarrow 0} \mathrm {Per}_r(E_r,\Omega ).\end{aligned}$$
An analogous result holds for the functional \(\mathscr {F}_{r,g}\).
Dealing with compactness issues, it is interesting to point out that sequences of minimizers (differently than sequences of sets with bounded r-perimeter) always provide a limit which is also a minimizer, on a smaller set. From the technical point of view, such limit is obtained from the support of the weak limit, once sets of zero measure are neglected. The precise statement goes as follows:
Theorem 1.8
Let \(\Omega \) be a bounded open set and \(E_k\) be a sequence of local minimizers of \({\mathscr {F}}_{r,g}\) in \(\Omega \) such that \(\chi _{E_k}\rightharpoonup u\), with \(u:\mathbb {R}^n\rightarrow [0,1]\), in \(L^1(\Omega )\).
Let E be such that \(\{ u=1\}\subseteq E\subseteq \Omega {\setminus } \{u=0\}\), and \(\Sigma := \{x\in \Omega : u(x)\in (0,1)\}\). Then E is a local minimizer of \({\mathscr {F}}_{r,g}\) in \(\Omega \ominus B_r\) and \(g(x)=0\) for a.e. \(x\in \Sigma \).
Moreover, if \({\mathscr {L}}^n(\{ g=0\})=0\) then \(\chi _{E_k}\rightarrow \chi _E\).
1.3 The Dirichlet problem
We now consider the Dirichlet problem for the functional \(\mathrm {Per}_r\).
Theorem 1.9
Let \(E_o\subseteq \mathbb {R}^n\) and \(\Omega \) a bounded open set. Fix \(\Omega '\Subset \Omega \). Then, there exists \(E\subseteq \mathbb {R}^n\) such that \(E{\setminus }\Omega '= E_o{\setminus } \Omega '\), and
for any \(F\subseteq \mathbb {R}^n\) for which \(F{\setminus }\Omega ' =E_o{\setminus } \Omega '\).
The same holds for the functional \(\mathscr {F}_{r,g}\).
1.4 Class A minimizers
In this subsection, we present some rigidity results for the nonlocal functionals introduced in (1.2).
Next result shows that half-spaces are always Class A minimizers for \(\mathrm {Per}_r\).
Proposition 1.10
Let \(\omega \in \mathbb {R}^n\) and \(E=\{x\ | x\cdot \omega <0\}\). Then E is a Class A minimizer for \(\mathrm {Per}_r\).
In addition, we give the complete characterization of Class A minimizers in dimension 1, according to the following result:
Theorem 1.11
If E is a Class A minimizer for \({\mathrm {Per}_r}\) and \(n=1\), then E is either \(\varnothing \) or \(\mathbb {R}\) or a halfline of the type either \((a, +\infty )\) or \((-\infty , a)\), for some \(a\in \mathbb {R}\).
It would be interesting to study the Bernstein problem for \(\mathrm {Per}_r\). In particular, in analogy with the classical perimeter, one could expect that the the Class A minimizers are only \(\varnothing \) or \(\mathbb {R}^n\) or half-spaces, at least in small dimension.
1.5 Isoperimetric inequalities and density estimates
We now discuss the isoperimetric properties of the functional \(\mathrm {Per}_r\). To this end, we first point out that balls are isoperimetric for the functional in (1.2), as a consequence of the Brunn–Minkowski Inequality. Namely, we have that:
Lemma 1.12
-
(i)
For any \(R>0\) and any measurable set \(E\subseteq \mathbb {R}^n\) such that \({\mathscr {L}}^n(E)= {\mathscr {L}}^n(B_R)\) it holds that
$$\begin{aligned} \mathrm {Per}_r(E)\geqslant \mathrm {Per}_r(B_R).\end{aligned}$$(1.8) -
(ii)
Viceversa, if \({\mathscr {L}}^n(E)= {\mathscr {L}}^n(B_R)\) and
$$\begin{aligned} \mathrm {Per}_r(E)=\mathrm {Per}_r(B_R),\end{aligned}$$then \(E=B_{ R}(p){\setminus }{\mathscr {N}}\), for some set \({\mathscr {N}}\) of null measure and some \(p\in \mathbb {R}^n\).
We present now a version of the relative isoperimetric inequality for \(\mathrm {Per}_r\) in an appropriate scale:
Theorem 1.13
Let assume there exists \(\lambda \geqslant 1\) such that
There exists \(C>0\), possibly depending on n, such that for all \(E\subseteq \mathbb {R}^n\) with
there holds
For the proof of this result we will need the following technical lemma (which can be seen as a working version of the compactness result in Theorem 1.7).
Lemma 1.14
Let \(\Omega \subseteq \mathbb {R}^n\) be open and bounded. Consider a sequence of sets \(\Omega _k \supseteq \Omega \), such that
Let
and \(E_k\subseteq \mathbb {R}^n\) such that
Then, there exist \({\widehat{E}}_k\subseteq \mathbb {R}^n\), \(E\subseteq \mathbb {R}^N\) and a constant \(C>0\) only depending on n such that
Remark 1.15
We stress that (1.11) holds with a constant which depends on \(\lambda \), that is, on the ratio \(\frac{r}{R}\), if \(r>R\). Namely, if \(r>R\), (1.11) may fail to be true for C just depending on n.
As a simple consequence of Theorem 1.13, we also provide the following nonlocal Poincaré–Wirtinger inequality:
Theorem 1.16
There exists \(C>0\), only depending on n, such that the following statement holds. Let \(\lambda \geqslant 1\), with \(\lambda R\geqslant r>0\), and \(u\in L^\infty (B_r)\cap L^1(B_R)\). Let
Then,
Remark 1.17
When \(r>R\), the estimate (1.19) does not necessarily hold true with a constant independent of \(\lambda \).
We address now the density properties of the minimizers of \(\mathrm {Per}_r\). Differently than the classical cases, the density properties of the minimizers may depend on the initial density for small scales: nevertheless, we can obtain a density growth in larger balls, and the constants become uniform once a suitable density threshold is reached. More precisely, our result is the following:
Theorem 1.18
Let \(\Omega \subseteq \mathbb {R}^n\), \(r>0\) and E be a minimizer for \(\mathrm {Per}_r\). Let \(R_o>0\). Suppose that \(B_{R_o}\subseteq \Omega \) and
Let also \(k\in \mathbb {N}\) be such that \(B_{R_o+2kr} \subseteq \Omega \). Then,
for a suitable \(c_{\star }>0\), possibly depending on n, r and \(\omega _o\).
Moreover,
Also,
In addition, if
for some \(\overline{C}>0\), then
for some \({\widetilde{c}}>0\), depending on n and \(\overline{C}\).
It is interesting to point out that Theorem 1.18 detects two scales of growth (and this fact is different from the case of the classical minimal surfaces, as well as of the nonlocal minimal surfaces in [5], where there is only one type of growth, given by the dimension of the space). Indeed, in our framework, if the initial density is below the threshold prescribed by \(r^n\) [as stated in (1.24)], then there is an exponential density growth [as stated in (1.25)], till the density reaches the quantity \(r^n\). Then, once a density of order \(r^n\) is reached, the growth reduces to the usual one, that is the radius to the power n [as stated in (1.21)]. In such case of polynomial growth away from an initial \(r^n\), the constant become uniform [as stated in (1.23), being the onedimensional case special, in view of (1.22)].
We think that it would be interesting to establish whether or not the growth in (1.25) is optimal or if sharper estimates may be obtained independently on the initial density.
Finally it is interesting to remark that compactness and regularity properties related to \(\mathrm {Per}_r\) can be problematic, or even fail, at a small scale, also for minimizers. To make a concrete example, we consider \(K>0\) and the function \(g(x):=-K \chi _{B_r{\setminus } B_{r/2}}\). We let
Then, minimizers are not necessarily smooth and sequences of minimizers are not necessarily compact. Indeed, we have:
Theorem 1.19
There exists \(C>0\), only depending on n, for which the following statement holds true.
Suppose that
Then, there exists \(E_*\subseteq \mathbb {R}^n\) satisfying
for any bounded set \(E\subseteq \mathbb {R}^n\), and such that \(\partial E_*\) is not locally a continuous graph (and, in fact, can be “arbitrarily bad” inside \(B_{r/2}\)).
Moreover, there exists a sequence \(E_k\subseteq \mathbb {R}^n\) satisfying
for any bounded set \(E\subseteq \mathbb {R}^n\), and such that \(\chi _{E_k}\) is not precompact in \(L^1(B_r)\).
Given the negative result in Theorem 1.19, we think that it is an interesting problem to develop a regularity theory for minimizers of \(\mathrm {Per}_r\) and of functionals such as \({\mathscr {F}_{r,g}}\).
1.6 Planelike minimizers in periodic media
In the spirit of [3], we recall the following definition:
Definition 1.20
We say that a set \(E\subseteq \mathbb {R}^n\) is planelike if, up to an appropriate change of coordinates, there exists \(K>0\) such that
To state our result, we recall some notation. We say that a direction \(\omega \in S^{n-1}\) is rational if the orthogonal space has maximal dimension over the integers, i.e.
Given a rational direction \(\omega \in S^{n-1}\), we say that a set E is \(\omega \)-periodic if, for any \(k\in \mathbb {Z}^n\) with \(\omega \cdot k=0\), we have that \(E+k=E\). Similarly, a function \(u:\mathbb {R}^n\rightarrow \mathbb {R}\) is said to be \(\omega \)-periodic if, for any \(k\in \mathbb {Z}^n\) with \(\omega \cdot k=0\), it holds that \(u(x+k)=u(x)\) for any \(x\in \mathbb {R}^n\).
Then, we state the following:
Theorem 1.21
There exist \(\eta \in (0,1)\) and \(M>1\), only depending on n, such that the following result holds true. Let \(r\in (0,1)\), \(g:\mathbb {R}^n\rightarrow \mathbb {R}\) be \(\mathbb {Z}^n\)-periodic, with zero average in \([0,1]^n\) and such that \(\Vert g\Vert _{L^\infty (\mathbb {R}^n)}\leqslant \eta \).
Let \(\omega \in S^{n-1}\). Then, there exists \(E^*_\omega \) which is a Class A minimizer for \( {\mathscr {F}}_{r,g}\), such that
Moreover, if \(\omega \) is rational, then \(E^*_\omega \) is \(\omega \)-periodic.
1.7 Organization of the paper
The rest of the paper is devoted to the proofs of our main results. Section 2 contains the proof of Proposition 1.3.
The \(\Gamma \)-convergence results and the compactness properties for the functional \(\mathscr {F}_{r,g}\), together with the proof of Theorem 1.4, Remarks 1.5 and 1.6, and Theorem 1.8, are presented in Sect. 3.
The proof of Theorems 1.9 is contained in Sect. 4.
The characterizations of Class A minimizers in Proposition 1.10 and in Theorem 1.11 are dealt with in Sect. 5.
We address the isoperimetric inequalities in Sect. 6, which contains the proof of Lemmas 1.12, 1.14, Theorem 1.13, Remark 1.15, Theorem 1.16 and Remark 1.17.
The regularity and density estimates, with the proof of Theorems 1.18 and 1.19, are discussed in Sect. 7.
Finally, in Sect. 8, we deal with the construction of the planelike minimizers in periodic media and we prove Theorem 1.21.
2 Basic properties of minimizers of \({\mathrm {Per}_r}\): Proof of Proposition 1.3
We provide the proof of Proposition 1.3, which justifies our definitions of local and Class A minimizers, given in Definition 1.2.
Proof of Proposition 1.3
First of all, we claim that there exists a universal \(\varepsilon >0\) such that
A picture can easily convince the reader about this simple geometric fact.
From now, we fix \(\varepsilon \) as in (2.1) and, without loss of generality, we take \(\varepsilon \in \left( 0,\frac{1}{2}\right] \). As a matter of fact, scaling (2.1), we see that
and therefore
Now we take E to be a Class A minimizer for \(\mathrm {Per}_r\) and we claim that
The proof of (2.3) is by contradiction: if not, any ball of radius \(r/\varepsilon \) contains both points of E and of its complement, and so it contains at least one point of \(\partial E\).
Let now \(M\geqslant 10\) to be taken suitably large in the sequel and \(R:=Mr/\varepsilon \). We consider N disjoint balls of radius \(2r/\varepsilon \) contained in the ball \(B_R\), and we observe that we can take \(N\geqslant \frac{c\,R^n}{(r/\varepsilon )^n} =c M^n\), for some universal \(c>0\). Let us call \(B_{2r/\varepsilon }(p_1),\dots ,B_{2r/\varepsilon }(p_N)\) such balls. We know that each ball \(B_{r/\varepsilon }(p_j)\) contains a point \(q_j\in \partial E\) and so \((\partial E)\oplus B_r\) contains at least the balls \(B_r(q_j)\) which are disjoint and contained in \(B_R\).
Consequently,
for some \({\bar{c}}>0\).
Now we consider \(F:=E\cup B_{R-r}\). Notice that \(\partial F\subseteq \mathbb {R}^n{\setminus } B_{R-r}\) and thus \((\partial F)\oplus B_r\subseteq \mathbb {R}^n{\setminus } B_{R-2r}\). This and the minimality of E give that
From this and (2.4) a contradiction easily follows by taking M appropriately large (possibly also in dependence of the fixed \(\varepsilon \)). This completes the proof of (2.3).
Now, from (2.3), we can suppose that E contains a ball of radius \({r/\varepsilon }\) and we prove that \(E=\mathbb {R}^n\) (if instead \(\mathbb {R}^n{\setminus } E\) contains a ball of radius \(B_{r/\varepsilon }\), a similar argument would prove that E is void).
Sliding the ball till it touches the boundary of E, we find a ball of radius \(r/\varepsilon \) which lies in E and whose boundary contains a point of \(\partial E\). Therefore, up to a rigid motion, we can suppose that \(0\in \partial E\) and \(B_{r/\varepsilon }(re_n/\varepsilon )\subseteq E\). We define
Notice that
and so
This, (2.2) and the minimality of E give that
On the other hand, since \(0\in \partial E\), we have that \( (\partial E)\oplus B_r\supseteq B_r\) and therefore
This is in contradiction with (2.5). The proof of Proposition 1.3 is thus complete. \(\square \)
3 \(\Gamma \)-Convergence results and compactness properties for the functional \(\mathscr {F}_{r,g}\): Proof of Theorem 1.4, Remarks 1.5 and 1.6, and Theorem 1.8
We start with a preliminary result on the convergence of characteristic functions.
Lemma 3.1
Let \(\Omega \) be an open subset of \(\mathbb {R}^n\) and let \(E_k\) be a sequence of sets such that \(\chi _{E_k}\) converges to u weakly in \(L^1_\mathrm{loc}(\Omega )\). Then, letting \(\Sigma :=\{x\in \Omega :\,u(x)\in (0,1)\}\), there holds
In particular, if u is a characteristic function, then \(\chi _{E_k}\rightarrow u\) in \(L^1_\mathrm{loc}(\Omega )\).
Proof
Without loss of generality we can assume that \(\Omega \) is bounded.
Let \(u_k:=\chi _{E_k}\). Since \(0\leqslant u_k\leqslant 1\), we have that
which proves (3.1). \(\square \)
Remark 3.2
Note that a consequence of the previous lemma is the following fact: if \(F_k\) is a sequence of sets such that \(\chi _{F_k}\rightarrow \chi _F\) in \(L^1_\mathrm{loc}(\Omega )\), for some open set \(\Omega \) and for some \(F\subset \mathbb {R}^n\) then \(\chi _{\lambda _k F_k}\rightarrow \chi _F\) in \(L^1_\mathrm{loc}(\Omega )\) for all \(\lambda _k\rightarrow 1\). Indeed it is sufficient to prove that \(\chi _{\lambda _kF_k}\) converges to \(\chi _F\) weakly in \(L^1_\mathrm{loc}(\Omega )\) and then apply the previous lemma.
We now provide the proof of the convergence result for \(\mathrm {Per}_r\).
Proof of Theorem 1.4
First of all, we prove the claim in (1). For this, we observe that, for every \(r>0\), it holds that
This can be obtained e.g. as a consequence of the estimate proved in [21, Theorem 2]: for all closed sets A and all \(r>0\), it holds that \(\mathscr {H}^{n-1} (\partial (A\oplus B_r))\leqslant \frac{C}{r} \mathscr {L}^{n}\big ((A\oplus B_r){\setminus } A\big )\), where \(C>0\) is a dimensional constant. Using this with \(A:=A_m=(\partial E)\cap B_m\), for any fixed \(m\in \mathbb {N}\), we find that \( \mathscr {H}^{n-1} (\partial (A_m\oplus B_r))<+\infty \), and therefore
We conclude observing that
Using (3.2), we see that \(\chi _{(\partial E)\oplus B_{r_k}}\rightarrow \chi _{ (\partial E)\oplus B_r}\) a.e. in \(\Omega \).
Hence, if \(\Omega \) is bounded, or if \(\Omega =\mathbb {R}^n\) and \(\partial E\) is bounded, the assertion in (1) follows from the Dominated Convergence Theorem.
To complete the proof of (1), we have only to consider the case in which \(\Omega =\mathbb {R}^n\) and \(\partial E\) is unbounded. In this case, we can take a sequence \(p_j\in \partial E\), with \(|p_j|>2r+2+|p_{j-1}|\). In this way \(B_\rho (p_j)\cap B_\rho (p_i)\) is void when \(j\ne i\) and \(\rho \in (0,r+1)\), which gives that \({\mathscr {L}}^n\big ( (\partial E)\oplus B_\rho \big )=+\infty \). This says that, in this case,
and so (1) holds true.
Now, we prove the claim in (2). By (1), we immediately deduce that
We are left to prove that, if \(E_{k}\rightarrow E\) in \(L^1_\mathrm{{loc}}(\Omega )\), then
To see this, we observe that, if we set
then
and, recalling Remark 3.2,
Notice that, again by Remark 3.2 applied with \(F_k:= {\widetilde{\Omega }}_k\) and \(F:=\Omega \), we have that
Therefore, by the lower semicontinuity of the functional \(\mathrm {Per}_r\), from (3.6), (3.7) and (3.8) we conclude that
This completes the proof of (2).
To prove (3), we define \({\widetilde{E}}_{r_k}\) as in (3.5). Then
Let \(u_k:=\chi _{{\widetilde{E}}_{r_k}}\). Then, up to subsequences, \(u_k\) converges to u weakly-\(\star \) in \(L^\infty _\mathrm{{loc}}(\Omega )\) and then also \(u_k\) converges to u weakly in \(L^1_\mathrm{{loc}}(\Omega )\), with \(u:\mathbb {R}^n\rightarrow [0,1]\). So, by the lower semicontinuity of the functional \(\mathscr {E}_{1,\Omega }\), we have that
which proves the third inequality in (1.6).
So, by Lemma 3.1, we have
Also, (3.9) gives that, for all \(x\in \Sigma \oplus B_r\), it holds that
Indeed, if this is not true, then either \(B_r(x)\subseteq E_k\) or \(B_r(x)\subseteq \mathbb {R}^n{\setminus } E_k\) for infinitely many k’s, and so either \(u_k=1\) or \(u_k=0\) a.e. in \(B_r(x)\) for infinitely many k. This would imply that either \(u=1\) or \(u=0\) a.e. in \(B_r(x)\), in contradiction with the fact that \(\Sigma \cap B_{r}(x)\ne \varnothing \), and so (3.10) is proved.
Using (3.10), we get that \(\mathop {\hbox {osc}}\limits _{B_r(x)} u_k\rightarrow 1\) for \(x\in \Sigma \oplus B_r\), therefore
which completes the proof of (1.6).
Then, the proof of Theorem 1.4 is complete. \(\square \)
We now exhibit the lack of compactness that was claimed in Remark 1.5.
Proof of Remark 1.5
Let us take, for example, \(r:=1\) and \(\Omega :=(-3,3)\times (0,1)^{n-1}\subseteq \mathbb {R}^{n}\). Let also, for any \(k\geqslant 1\),
If \(\chi _{E_k}\) converged pointwise, it would also converge in \(L^1(\Omega )\), due to the Dominated Convergence Theorem. But this is not the case, since the norm in \(L^1(\Omega )\) of \(\chi _{E_k}-\chi _{E_{k+m}}\) is always bounded from below independently on k and m. \(\square \)
Now we present the pathological counterexample to Theorem 1.4, as stated in Remark 1.6.
Proof of Remark 1.6
We take \(n\geqslant 2\), \(r>0\), \(r_k:=r+\frac{1}{k}\), \(\Omega :=\{x_n>0\}\) and \(E:=\{x_n < -r\}\). In this way, for any \(\rho \geqslant r\),
Therefore
These considerations prove (1.7). \(\square \)
Now we show that sequences of minimizers for the functional \({\mathscr {F}}_{r,g}\) produce a limit minimizer.
Proof of Theorem 1.8
First of all we consider the case in which \(E=\{u=1\}\), and we show that it is a local minimizer in \(\Omega \) and that \(g=0\) a.e. on \(\Sigma \).
Observe that for all \( x\in A:=\left( \Sigma \cup \partial E\right) \oplus B_r\) it holds that
The proof of this fact is the same as the proof of (3.10) above (in the proof of Theorem 1.4).
Fix \(\Omega '\subseteq \Omega \ominus B_r\) and let
Observe that
with
Note that
We define \(\Sigma _k:= (E_k\Delta \{g<0\})\cap \Sigma \cap \Omega '\). So (3.12) reads
We also let
Notice that for all \(x\in D:=(C\oplus B_r)\cap \Omega '\) there holds
To check this, we argue by contradiction and we suppose that, for instance, \(B_r(x)\subseteq E_k\) for k large enough. Then, \(u_k=1\), and so \(u=1\) a.e. in \(B_r(x)\), i.e. \(B_r(x)\cap \Omega '\subseteq E\cap \Omega '\). Recalling that \(B_r(x){\setminus } \Omega '\subseteq E_k{\setminus } \Omega '\), this implies that \(B_r(x)\subseteq E_k^\star \). Accordingly, we have that \(B_r(x)\cap C=\varnothing \), and so \(x\not \in D\), against our assumption. This proves (3.14).
Properties (3.11) and (3.14) imply that
for some \(O_k\subseteq \mathbb {R}^n\) with
By definition of \(E_k^\star \), there holds
Therefore, from (3.15) and (3.17) it follows that
and then
Therefore, by minimality of \(E_k\) we deduce that
This implies that \(u=0\) on \(\Sigma \cap \{g> 0\}\) and \(u=1\) on \(\Sigma \cap \{g< 0\}\), which, recalling the definition of \(\Sigma \), implies that \({\mathscr {L}}^n(\Sigma \cap \{g> 0\})=0={\mathscr {L}}^n(\Sigma \cap \{g<0\})\), so \(g=0\) almost everywhere on \(\Sigma \). If \({\mathscr {L}}^n(\{ g=0\})=0\), we deduce that \({\mathscr {L}}^n(\Sigma )=0\), and then we conclude the strong convergence of \(\chi _{E_k}\) to \(\chi _E\).
Moreover, since \(\Sigma _k\subseteq \Sigma \), we conclude that
Let now F be such that \(F\Delta E\subseteq \Omega '\ominus B_r\). We define
By construction
Recalling (7.5) we then get
where the last inequality follows by the minimality of \(E_k\). Now we send \(k\rightarrow +\infty \) and we obtain that
thanks to (3.16). This concludes the proof of the local minimality of \(E=\{u=1\}\).
Now, let E be any set such that \(\{u=1\}\subseteq E\subseteq \Omega {\setminus } \{u=0\}\). Then we can define \(E_k^\star = ((E\cup (\Sigma \cap \{g<0\})\cap \Omega ') \cup (E_k{\setminus } \Omega ')\) and repeat the same argument as above (recalling that \(g=0\) almost everywhere on \(\Sigma \)) to get that (7.5) holds. The proof of Theorem 1.8 is thus complete. \(\square \)
4 The Dirichlet problem: Proof of Theorem 1.9
Proof of Theorem 1.9
Let \(E_k\) be a minimizing sequence. Then, up to subsequences, \(\chi _{E_k}\rightharpoonup u\) in \(L^1_\mathrm{{loc}}(\Omega )\), with \(u:\mathbb {R}^n\rightarrow [0,1]\). By the lower semicontinuity in \(L^1\) of the functional \(v\rightarrow \int _{\Omega } \mathop {\hbox {osc}}\limits _{B_r(x)} v\, dx\) proved in [8] and the coarea formula, we get
Notice that
for a suitable \(s_\Omega \in (0,1)\). So, we define \(E:=\{u>s_\Omega \}\). Since \(\chi _{E_k}\) does not depend on k outside \(\Omega '\), we have that \(E=E_k\) outside \(\Omega '\) and thus it is an admissible competitor. Then, (4.1) says that E is a minimizer for \(\mathrm {Per}_r\), and this proves Theorem 1.9. \(\square \)
5 Class A minimizers: Proof of Proposition 1.10 and of Theorem 1.11
Now we prove the results about Class A minimizers. We start showing that half-spaces are Class A minimizers for \(\mathrm {Per}_r\) in every dimension.
Proof of Proposition 1.10
In this proof, we write \(x=(x', x_n)\in \mathbb {R}^n\). Up to translations and rotations, we can assume that \(E=\{x\in \mathbb {R}^n { \text{ s.t. } } x_n<0\}\). We fix \(B_R\) with \(R>r\), and we consider \(F\subseteq \mathbb {R}^N\) such that \(F\Delta E\Subset B_R\). Let \(C_R\) be the cylinder \(\{x'\in \mathbb {R}^{n-1}{ \text{ s.t. } } |x'| \leqslant R\} \times [-R, R]\), and observe that \(\mathrm {Per}_r(E, C_R)= n\omega _n R^{n-1}\).
For any fixed \(x'\in \mathbb {R}^{n-1}\), let also \(\ell _{x'}=\{(x',x_n)\in \mathbb {R}^{n} { \text{ s.t. } } x_n\in \mathbb {R}\}\). We compute
where we used the observation that \(\mathscr {H}^{1} ((\partial F\oplus B_r )\cap \ell _{x'}) \geqslant 2r\), for every \(x'\). This proves Proposition 1.10. \(\square \)
Now we characterize the Class A minimizers of the nonlocal perimeter functional in dimension 1
Proof of Theorem 1.11
Suppose that \(E\subseteq \mathbb {R}\) is a Class A minimizer for \(\mathrm {Per}_r\). Assume also that \(E\ne \varnothing \) and \(E\ne \mathbb {R}\). Observe that this implies that \(E\not \subseteq (a,b)\) and \(\mathbb {R}^n {\setminus } E\not \subseteq (a,b)\) for every \(-\infty<a<b<+\infty \). Indeed, if \(E\subseteq (a,b)\) with \( -\infty<a<b<+\infty \), then the empty set would be an admissible competitor for E in \((a-r,b+r)\) and this would contradict the minimality of E. Similarly for \(\mathbb {R}^n{\setminus } E\).
To conclude, it is sufficient to show that E is connected:
We prove (5.1) by contradiction.
Assume it is not true, then there exists a point \(\beta \in (\partial E)\cap (p,q)\). We define \(F:=E\cup (p,q)\) and we observe that F and E coincide outside (p, q). Also,
We also observe that
We claim that
Indeed, if \(\zeta \in (\partial F){\setminus } (p,q)\) then either \(\zeta \in (\partial F){\setminus }[p,q]\), or \(\zeta \in \{p,q\}\). If \(\zeta \in (\partial F){\setminus }[p,q]\), then, by (5.3), we have that \(\zeta \in (\partial E){\setminus }[p,q]\subseteq (\partial E){\setminus }(p,q)\), and we are done.
Hence, we can focus on the case in which, for instance, \(\zeta =p\). Since F contains (p, q), the fact that \(\zeta \in \partial F\) implies that there exists \(\zeta _k\in \mathbb {R}^n{\setminus } F\) with \(\zeta _k\leqslant \zeta =p\). Then, by the definition of F, we see that \(\xi _k\in \mathbb {R}^n{\setminus } E\). On the other hand, we know that \(\xi =p\in E\) [recall (5.1)]. These observations imply that \(\zeta =p\in \partial E\). This proves (5.4) also in this case.
From (5.2) and (5.4) we get that
This implies that \(\mathrm {Per}_r(E,(p-r,q+r))>\mathrm {Per}_r(F,(p-r,q+r))\), which is against minimality, and so the proof of (5.1) is completed. \(\square \)
6 Isoperimetric inequalities: Proof of Lemmas 1.12, 1.14, Theorem 1.13, Remark 1.15, Theorem 1.16 and Remark 1.17
Now, we deal with the isoperimetric problems.
Proof of Lemma 1.12
First of all, we prove (i). To this end, we remark that, without loss of generality, we can suppose that \(\partial E\) is bounded (if not, there would exist a sequence \(x_j\in \partial E\) such that \(|x_j|\geqslant j\) and \(|x_{j+1}-x_j|\geqslant 2r+1\), and thus \(\partial E\oplus B_r\) would contain the disjoint balls \(B_r(x_j)\), thus yielding that \(\mathrm {Per}_r(E)=+\infty \)).
In addition, we notice that \((\partial B_R)\oplus B_r=B_{R+r}{\setminus } B_{(R-r)^+}\) and therefore
By the Brunn–Minkowski Inequality (see e.g. [27] or Theorem 4.1 in [19]) we have that
As a consequence, we get
We observe that if \(R<r\), then \(B_{R-r}=\emptyset \) and \(E\ominus B_r=\emptyset \). Therefore (6.2) implies (i).
On the other hand, if \(R\geqslant r\), let us take \({\widetilde{R}}\in [0,R]\) such that
Also, recalling that \(\big ( E\ominus B_r\big )\oplus B_r\subseteq E\), we have that
Accordingly, applying again the Brunn–Minkowski Inequality we get that
which implies that \({\widetilde{R}}\leqslant R-r\).
From this, we obtain that if \(R\geqslant r\)
Putting together (6.2) and (6.3) if \(R\geqslant r\) we obtain
thus proving (i).
Now, we prove (ii). For this, we observe that if equality holds, then all the previous equalities hold true with equal sign. In particular, formula (6.1) would give that
Hence (see e.g. page 363 in [19]), since equality holds in the Brunn–Minkowski inequality if and only if the two sets are homothetic convex bodies (up to removing sets of measure zero), we have that \(E=B_{\lambda R}(p){\setminus }{\mathscr {N}}\), for some set \({\mathscr {N}}\) of null measure, some \(p\in \mathbb {R}^n\) and some \(\lambda >0\). Since
we obtain that \(\lambda =1\), which establishes (ii). \(\square \)
Having settled the global isoperimetric problem, we now deal with the proof of the relative isoperimetric inequality. First of all we give the proof of the technical lemma.
Proof of Lemma 1.14
We consider a partition of \(\mathbb {R}^n\) into adjacent cubes of side \(\frac{r_k}{4\sqrt{n}}\) (hence, the diameter of each cube is \(\frac{r_k}{4}\)). These cubes will be denoted by \(\{ Q_j\}_{j\in \mathbb {N}}\). For any \(k\in \mathbb {N}\), we set
Let also
Notice that (1.15) is obvious in this setting. We now prove (1.16). For this, we say that \(Q_j\) is a k-boundary cube if \(j\in I_k\) and there exists a cube \(Q_i\) that is adjacent to \(Q_j\) with \(i\not \in I_k\). We let \(\beta _k\) be the number of k-boundary cubes which intersect \(\Omega _k\).
We remark that
for some \(C>0\). We also claim that
up to renaming \(C>0\). To this end, let \(Q_j\) be a k-boundary cube and \(Q_i\) be its adjacent cube with \(j\in I_k\) and \(i\not \in I_k\). Thus, by (6.4), there exists \(p_{j,k}\in Q_j\cap E_k\) and \(p_{i,k}\in Q_i{\setminus } E_k\). Consequently, we find a point \(p^\star _k\in \partial E_k\) which lies at distance at most \(r_k/4\) from \(Q_j\). Therefore
In addition, if \(Q_j\) intersects \(\Omega _k\), it follows from (1.12) that (for large k)
for some \(C>0\). Hence, if \(Q_j^\star \) denotes the dilation of \(Q_j\) by a factor 2 with respect to its center, we have that \(Q^\star _j\supseteq Q_j\oplus B_{\frac{r_k}{100}}\) and
This and (6.7) give that
Our goal is now to sum up (6.8) for all the indices j for which \(Q_j\) is a boundary cube that intersects \(\Omega _k\). Notice that the family \(\{ Q_j^\star \}_{j\in \mathbb {N}}\) is overlapping (differently from the original nonoverlapping family \(\{Q_j\}_{j\in \mathbb {N}}\)), but the number of overlappings is finite, say bounded by some \(C^\star >0\). Hence, since (6.8) is valid for any k-boundary cube \(Q_j\) which intersect \(\Omega _k\), summing up (6.8) over the indices j gives that
and thus
that establishes (6.6), up to renaming constants.
From (6.5) and (6.6) it follows that (1.16) holds true, as desired.
In addition, from (1.14) and (1.16), we obtain a uniform bound for \(\mathrm {Per}( {\widehat{E}}_k,\, \Omega _k )\) and thus on \( \mathrm {Per}( {\widehat{E}}_k,\, \Omega )\), so by compactness, up to a subsequence we have that there exists \(E\subseteq \mathbb {R}^n\) for which
Now we prove (1.17). For this, let
Notice that
To check this, let \(x\in ({\widehat{E}}_k {\setminus } E_k)\cap \Omega _k\). Then, there exists \(j\in I_k\) such that \(x\in Q_j\). Notice that \(x\in Q_j{\setminus } E_k\) and \(x\in Q_j\cap \Omega _k\), which means that \(j\in J_k\), and so \(x\in H_k\), thus proving (6.10).
Now we prove that
Indeed, if \(j\in J_k\), then also \(j\in I_k\), therefore \(Q_j\cap E_k\ne \varnothing \) and also \(Q_j{\setminus } E_k\ne \varnothing \). Hence there exists \(x_{j,k}\in Q_j\cap (\partial E_k)\). Notice that
Also, \(Q_j\cap \Omega _k\ne \varnothing \). Consequently, making use of (1.12) and (6.12), we see that
up to renaming \(C>0\). Since this is valid for any \(j\in J_k\) and there is a finite number of overlaps between different \(Q_j\oplus B_{\frac{r_k}{100}}\), we conclude that
up to renaming \(C>0\) that implies (6.11).
Now, in view of (6.10) and (6.11), we find that
This implies (1.17). Finally, (1.13), (1.14) and (1.17) give that
and this, combined with (6.9), implies (1.18), as desired. \(\square \)
With this, we can now complete the proof of Theorem 1.13.
Proof of Theorem 1.13
First of all we consider the case in which \(R<r\leqslant \lambda R\) for \(\lambda >1\). By assumption we know that \({\mathscr {L}}^n (E\cap B_{R})\leqslant \frac{1}{2} {\mathscr {L}}^n (B_{R})=\frac{1}{2} \omega _n R^n\). So either \({\mathscr {L}}^n (E\cap B_{R})=0\) and there is nothing to prove, or \({\mathscr {L}}^n (E\cap B_{R})\not =0\). In this case \(\partial E\cap B_R\not =\emptyset \), and so \(\mathrm {Per}_r(E, B_R)\geqslant \frac{1}{2r} \frac{{\mathscr {L}}^n (B_{R})}{3}= \frac{\omega _n R^n}{6r} \). Summarizing we get
We consider now the case \(\lambda =1\), so \(r\leqslant R\), and we argue by contradiction. If (1.11) were not true, recalling also (1.9) and (1.10), we would infer that there exist sequences
and \(E_k\subseteq \mathbb {R}^n\) such that
We define \(\lambda _k:=\big ( {\mathscr {L}}^n(E_k\cap B_{R_k})\big )^{-\frac{1}{n}}\), \({\widetilde{E}}_k:= \lambda _k E_k\), \({\widetilde{r}}_k:= \lambda _k r_k\) and \({\widetilde{R}}_k=\lambda _k R_k\). With this scaling, we have that
Moreover,
Therefore (6.14) becomes
Thanks to the first inequality in (6.16), setting
we have that \(R_o\in (0,+\infty ]\) and
Here, the obvious notation \(B_{{\widetilde{R}}_o}=\mathbb {R}^n\) if \(R_o=+\infty \) has been used.
Now we claim that
For this, we observe that \(\widetilde{R}_k\geqslant {\widetilde{r}}_k\), thanks to (6.13).
In addition,
thanks to (6.15) and (6.16). Therefore both \({\widetilde{E}}_k\cap B_{{\widetilde{R}}_k}\) and \(B_{{\widetilde{R}}_k}{\setminus }{\widetilde{E}}_k\) are nonvoid, and so there exists \(p_k\in (\partial {\widetilde{E}}_k)\cap B_{{\widetilde{R}}_k}\). Accordingly,
for some \(c>0\). From this and (6.16) we deduce that
which proves (6.18), as desired.
In light of (6.18), we can now exploit Lemma 1.14 [with \(\Omega _k:=B_{{\widetilde{R}}_k}\) and \(\Omega :=\cap _k B_{{\widetilde{R}}_k}\), which is nontrivial thanks to (6.17)]. In particular, from (1.15) and (1.16), we know that there exists \({\widehat{E}}_k\subseteq \mathbb {R}^n\) such that
and
Therefore, recalling (6.16),
Moreover, using (1.17),
Using (6.15) and (6.21), we see that
This and (6.17) imply that
So, we can assume that, for large k,
hence we can apply the classical relative isoperimetric inequality and find that
Consequently, recalling (6.19) and (6.20),
From this, sending \(k\rightarrow +\infty \) and recalling (6.15), we obtain a contradiction that proves Theorem 1.13. \(\square \)
Now we check that (1.11) cannot hold with a constant independent of \(\lambda \).
Proof of Remark 1.15
As an example, let \(n=2\), \(R=100\) and \(E:=B_1\). Notice that (1.10) is satisfied, but (1.11) cannot be true for arbitrarily large r for some constant C independent of the rate \(\frac{r}{R}\). Indeed, we have that \(\partial E\subseteq B_{100}\), hence
As a consequence, if r is sufficiently large,
thus violating (1.11). \(\square \)
Now, we can provide the easy proof of the Poincaré–Wirtinger inequality in Theorem 1.16:
Proof of Theorem 1.16
Up to a vertical translation, we may and do suppose that
Moreover,
Thus, possibly exchanging u with \(-\,u\), we may and do suppose that
Let also \(u^+:=\max \{u,0\}\). Then, using (6.23), we see that
Hence, integrating with respect to the distribution function (see e.g. Theorem 5.51 in [30]), we have that
In addition, from (6.24), for any \(s\geqslant 0\) we have that
Consequently, we can exploit our relative isoperimetric inequality in Theorem 1.13 with \(E:=\{u^+>s\}\) and conclude that, for any \(s\geqslant 0\),
for some \(C>0\). Multiplying this estimate by \( \Big ( {\mathscr {L}}^n (\{u^+>s\}\cap B_R)\Big )^{\frac{1}{n}}\), we obtain that, for any \(s\geqslant 0\),
up to renaming \(C>0\). Accordingly,
thanks to the coarea formula in (1.4). Hence, recalling (6.25), we conclude that
which is the desired result, up to renaming constants. \(\square \)
Now we check that Theorem 1.16 does not hold in general when \(r>R\) with a constant independent of the rate \(\frac{r}{R}\):
Proof of Remark 1.17
Let \(R=1\) and
Notice that u has zero average and its oscillation is always bounded by 2. Therefore, if r is large enough,
which violates (1.19). \(\square \)
7 Regularity issues and density estimates: Proof of Theorems 1.18 and 1.19
In this section we prove the nonlocal density estimates in Theorem 1.18:
Proof of Theorem 1.18
We set \(f(R):={\mathscr {L}}^n(E\cap B_R)\). We notice that if \(R-r\geqslant r\) and \(f(R-r)\leqslant \frac{{\mathscr {L}}^n(B_R) }{2}\), then we can apply the relative isoperimetric inequality in Theorem 1.13 and obtain that
Furthermore,
Observe that
Consequently
and therefore
Assume also that \(B_{R+r}\subseteq \Omega \). Then, the minimality of E in \(B_{R+r}\) and (7.2) give that
This and (7.1) give that, if \(B_{R+2r}\subseteq \Omega \), \(R\geqslant 2r\) and \(f(R-r)\leqslant \frac{{\mathscr {L}}^n(B_R) }{2}\), then
That is, if \(R\geqslant r\) and \(f(R)\leqslant \frac{{\mathscr {L}}^n(B_R) }{2}\),
Now we define, for \(k\in \mathbb {N}\), the sequence \(x_k:= f(R_o+2kr)\), and we claim that, if \(B_{R_o+2kr}\subseteq \Omega \), then
where
being \(\omega _o\) as in (1.20) and C as in (7.3). The proof of (7.4) is by induction. First of all, from (1.20) we have that
and so (7.4) holds true when \(k=0\). Now we suppose that it holds true for \(k-1\), namely
Thus, from (7.3),
Now, by a first order Taylor expansion, we see that, for any \(\tau \in [0,1]\),
and therefore
As a consequence,
thanks to (7.5). This and (7.6) give that \(x_k \geqslant (\omega _o^{\frac{1}{n}}+ 2 c_{\star } k r)^{n}\), which completes the inductive proof of (7.4).
From (7.4) and (7.5), we obtain (1.21), (1.22) and (1.23).
Now, we prove (1.25). To this end, we take k as in (1.24) and we observe that
Hence, for any \(j\in \{1,\dots ,k\}\),
thus we deduce from (7.3) that
Iterating, we thus obtain
that establishes (1.25). This completes the proof of Theorem 1.18. \(\square \)
Now we address the compactness and lack of regularity issues exemplified in Theorem 1.19:
Proof of Theorem 1.19
We start with some preliminary observations. First of all, if we denote by \(\{e_1,\dots ,e_n\}\) the Euclidean basis of \(\mathbb {R}^n\), it is clear that
Moreover, there exists a constant \(c_\star >0\), only depending on n, such that, for any \(x\in \overline{B_{3/2}}\) it holds that
To prove (7.8), we argue for a contradiction: if not, there exists a sequence of points \(x_k\in \overline{B_{3/2}}\) such that
Up to a subsequence, we may assume that \(x_k\rightarrow {\bar{x}}\) as \(k\rightarrow +\infty \), for some \({\bar{x}}\in \overline{B_{3/2}}\), and, passing to the limit (7.9), we obtain that
Up to a rotation, we can assume that \({\bar{x}}\) is parallel to \(e_1\), namely \({\bar{x}}=\lambda e_1\), for some \(\lambda \in \left[ 0,\frac{3}{2}\right] \). We define
Notice that, by (7.7), we have that
In addition, we note that \(|{\bar{x}}-\lambda _\star e_1|= |\lambda -\lambda _\star |\leqslant 1/2\). Consequently, if \(p\in B_{1/8}(\lambda _\star e_1)\), we have that \( |{\bar{x}}-p|\leqslant |{\bar{x}}-\lambda _\star e_1|+ |\lambda _\star e_1-p|\leqslant \frac{1}{2}+\frac{1}{8}<1\), which gives that \(B_{1/8}(\lambda _\star e_1)\subseteq B_1({\bar{x}})\).
Therefore, we conclude that \( B_{1/8}( \lambda _\star \,e_1)\cap (B_1{\setminus } B_{1/2})\subseteq B_{1}( {\bar{x}})\cap (B_1{\setminus } B_{1/2})\). From this and (7.11), we obtain that \({\mathscr {L}}^n\big ( B_{1}( {\bar{x}})\cap (B_1{\setminus } B_{1/2})\big )>0\), and this is in contradiction with (7.10). The proof of (7.8) is thus completed.
We also notice that, by scaling (7.8), it holds that, for any \(x\in \overline{B_{3r/2}}\),
Now we claim that there exists \(\delta _\star >0\), only depending on n, such that
To prove this, let
Our aim is to show that
since this would imply that \(x\in \{0\}\cup \big ((\partial H)\oplus B_r\big )\), thus establishing (7.13).
Also, since (7.15) is obvious when \(x=0\), we can assume that
Notice that, from (7.14), we know that there exists \(y\in B_r(x)\cap \big (\partial (H\cup B_{r/2})\big )\). Consequently, we can find \(\xi _k\in (H\cup B_{r/2})\) and \(\eta _k\in (\mathbb {R}^n{\setminus } H)\cap (\mathbb {R}^n{\setminus } B_{r/2})\) with the property that \(\xi _k\rightarrow y\) and \(\eta _k\rightarrow y\) as \(k\rightarrow +\infty \).
We observe that \(\eta _k\in \mathbb {R}^n{\setminus } H\): hence, if \(\xi _k\in H\), it follows that \(y\in \partial H\) and so (7.15) holds true. Therefore, we can restrict ourselves to the case in which \(\xi _k\in (B_{r/2}{\setminus } H)\). In particular
and so \(y\in \partial B_{r/2}\).
Consequently, we see that \(|x|\leqslant |y|+|x-y|\leqslant \frac{r}{2}+r=\frac{3r}{2}\), and so we are in the position of exploiting (7.12). Accordingly, we have that
In addition, from the hypothesis of (7.13), we find that
This says that
as long as we choose \(\delta _\star \) appropriately small. Thus, recalling (7.17), we find that
which gives that \( {\mathscr {L}}^n\big ( B_r(x)\cap (B_r{\setminus } B_{r/2})\cap H\big )\geqslant \frac{c_\star }{2}\,r^n\). In particular, we have that
So, we claim that
To prove (7.19), we suppose the contrary, namely that \(B_r(x)\cap (\partial H)=\varnothing \). Then, from (7.18) we have that \(B_r(x)\subseteq H\). In particular, recalling (7.16), we have that, if \(p_j:=x+\left( r-\frac{1}{j}\right) \frac{x}{|x|}\), then
for large j. Accordingly, we obtain that \( p_j\in B_r(x)\subseteq H\subseteq B_r\), where one assumption in (7.13) has been used for the latter inclusion.
So, we have found that
This is a contradiction with (7.16), and so we have proved (7.19).
Then, since (7.19) implies (7.15), we have thus completed the proof of (7.13).
Now, we deal with the core of the proof of Theorem 1.19. For this, we observe that
Now we claim that
for any bounded set \(E\subseteq \mathbb {R}^n\). To this end, we distinguish two cases, namely
being \(\delta _\star \) the constant in (7.13).
When (7.22) holds true, we have that
Accordingly, from (7.20), we have that
provided that K is large enough, as prescribed by (1.26). This proves (7.21) when (7.22) holds true, hence we can now focus on the case in which (7.23) is satisfied.
Thanks to (7.23), we can exploit (7.13) with
In this way, setting
we have that \( \{0\}\cup \big ((\partial H)\oplus B_r \big )\supseteq (\partial G)\oplus B_r\). In particular, we have that
We also point out that \( {\mathscr {L}}^n\big ( G\cap (B_{r}{\setminus } B_{r/2})\big ) ={\mathscr {L}}^n\big ( H\cap (B_{r}{\setminus } B_{r/2})\big )\), thanks to (7.25). Hence, exploiting (7.26), we find that
In addition, we claim that
To check this, we recall [see formulas (2.4)–(2.5) in [9]] that
Let now \(R\geqslant r\) be such that \({\mathscr {L}}^n(E\cup B_r)= {\mathscr {L}}^n(B_R)\). Then, from the isoperimetric inequality in (1.8), we see that
Hence, we insert this inequality into (7.29) and we obtain (7.28), as desired.
We also notice that, by (7.24), we have that \( {\mathscr {L}}^n\big (H\cap (B_r{\setminus } B_{r/2})\big )= {\mathscr {L}}^n\big (E\cap (B_r{\setminus } B_{r/2})\big )\), and so
thanks to (7.28).
Let also \(\rho \geqslant 0\) be such that
We point out that, by (7.24) and (7.25),
and so
Also, making use of the isoperimetric inequality in (1.8), we see that
Furthermore,
Hence, by (7.34) and (7.35), we have that
We notice that, for any \(t\in \left[ \frac{r}{2},\,r\right] \),
as long as K is large enough, as prescribed in (1.26). Therefore, recalling (7.20) and (7.33), we have that
Hence, we insert this information into (7.36), and we conclude that \({\mathscr {F}}_K(G)\geqslant {\mathscr {F}}_K (B_r{\setminus } B_{r/2})\). From this, (7.30) and (7.27), we conclude that
which completes the proof of (7.21).
Now, for any (arbitrarily ugly) set \(U\subseteq B_{r/2}\), we set \(E_U:= (B_r{\setminus } B_{r/2})\cup U\). We notice that \( (\partial E_U)\oplus B_r = B_{2r}= (B_r{\setminus } B_{r/2})\oplus B_r\), and also \( {\mathscr {L}}^n\big (E_U\cap (B_r{\setminus } B_{r/2})\big )={\mathscr {L}}^n (B_r{\setminus } B_{r/2})\), and therefore
Hence, from (7.21), we have that \(E_U\) is also a minimizer for \({\mathscr {F}}_K\), from which the claims in Theorem 1.19 plainly follow. \(\square \)
8 Planelike minimizers in periodic media: Proof of Theorem 1.21
In this section we establish the existence of planelike minimizers for periodic volume perturbations of \(\mathrm {Per}_r\).
Proof of Theorem 1.21
The proof is given in two steps: in the first one, we fix a rational slope \(\omega \) and we provide the construction of a planelike minimizer \(E^*_\omega \) which is also \(\omega \)- periodic. Then, in the second step, we consider irrational slopes by means of an approximation procedure.
Step 1: construction of planelike minimizers with rational slope. The idea of the proof is to perform an argument based on a constrained minimal minimizer procedure, as in [3]. A major difference with [3] here is that optimal density estimates at small scales do not hold, hence the width of the strip may depend, in principle, on r. Indeed, roughly speaking, here one needs an initial density to improve the density in the large, and so, to let the density reach a uniform threshold, a large number (in dependence of r) of fundamental cubes may be needed, and this has a rather strong consequence on the energy estimates when r is small.
Hence, the proof of this step will be performed in two parts: first, we obtain an initial bound on the width of the strip that depends on r, and then we improve this bound up to a uniform scale. This method will combine the minimal minimizer argument in [3] with an ad-hoc procedure of finely selecting appropriating cubes and performing a cut at a suitable level. These estimates will be based on a fine analysis of cubes, to detect local densities and energy contributions.
The details of the proof go like this. We consider a “fundamental domain” for the \(\omega \)-periodicity, i.e. we take \(K_1,\dots ,K_{n-1}\in \mathbb {Z}^n\) which are linearly independent and such that \(\omega \cdot K_j=0\) for any \(j\in \{1,\dots ,n-1\}\), and we set
Notice that the existence of \(K_1,\dots ,K_{n-1}\) is a consequence of the rationality of \(\omega \) in (1.27).
Given \(M\geqslant 2\), we also consider the parallelepipedon
We consider the functional
We now introduce the set of periodic constrained minimizers for this functional. Namely we define \({\mathscr {C}}_{\omega ,M}\) the family of sets \(E\subseteq \mathbb {R}^n\) which are \(\omega \)-periodic and such that
Let also \(L_\omega :=\{\omega \cdot x\leqslant 0\}\). Then
for some \(C>0\).
We also consider the family of finite overlapping dilated cubes
We define \({\mathscr {Q}}_M\) the family of cubes \(Q\in {\mathscr {Q}}\) which intersect \(\{\omega \cdot x=\pm M\}\). The fact that g has zero average in each \(Q\in {\mathscr {Q}}\) implies that
up to renaming \(C>0\), and therefore, in view of (8.1), it holds that
This says that there exists at least one set in \({\mathscr {C}}_{\omega ,M}\) with finite energy, hence we can proceed to the minimization of the functional. The existence of the minimum in this case follows along the lines of Theorem 1.9.
So we define \({\mathscr {M}}_{\omega ,M}\) as the family of sets \(E\in {\mathscr {C}}_{\omega ,M}\) such that
Following a classical idea of [3], we now define the minimal minimizer as
We remark that
for any E, \(F\subseteq \mathbb {R}^n\) and any domain \(\Omega \), and thus
By (8.4), we have that \(E^*_{\omega ,M}\in {\mathscr {M}}_{\omega ,M}\), that is the minimal minimizer is indeed a minimizer. Moreover, \(E^*_{\omega ,M}\) satisfies the inclusion properties (for a proof of this we refer to [3, Lemma 6.5])
Consequently, since \(E^*_{\omega ,M}\) is the smallest possible minimizers,
We now divide the cubes in \({\mathscr {Q}}\) according to their “color”, i.e. their density properties with respect to the set \(E^*_{\omega ,M}\) (pictorially, we think that the set \(E^*_{\omega ,M}\) is “black” and its complement is “white”).
Namely, we consider the “family of black cubes” given by
and the “family of white cubes”
We also take into account the “family of grey cubes”
We also subdivide the grey cubes into the ones which are “foggy black” and the ones which are “foggy white”: the first family contains cubes with a sufficient density of \(E^*_{\omega ,M}\), while the second family contains cubes with a sufficient density of the complement of \(E^*_{\omega ,M}\), being the notion of “sufficient density” the one compatible with uniform scales in the density estimates of Theorem 1.18. That is, we define
Notice that, since \(r\in (0,1)\),
On the other hand, in general, we have that \({\mathscr {Q}}_\mathrm{{{f.Bl}}} \cap {\mathscr {Q}}_\mathrm{{{f.Wh}}}\ne \varnothing \), since there might be cubes with sufficiently high density of both \(E^*_{\omega ,M}\) and its complement (these cubes are, in some sense, “multicolored” inside). So, we define
Notice that the cubes in \({\mathscr {Q}}_\mathrm{{{f.Bl}}}{\setminus } {\mathscr {Q}}_\mathrm{{{Mu}}}\) have a sufficiently high density of \(E^*_{\omega ,M}\) and a rather low density of its complement, so they “look almost black”. For this reason, we set
Similarly, we define the family of almost white cubes as
We are now going to show that the strip is divided into five ordered “color layers”: on the bottom stay all the black cubes, then the almost black ones, then cubes of multicolor type, then almost white cubes and finally white cubes on the top (rigorous statements below). We also estimate carefully the width of these layers.
To this end, we observe that the “color density” of the cubes is monotone with respect to \(\omega \), in the sense that the color of an upper translation is more pale than the color of a lower translation. The precise statement goes as follows: we claim that, for any \(k\in \mathbb {Z}^n\) with \(\omega \cdot k\geqslant 0\), we have that
To check this, we exploit (8.5) to see that \( E^*_{\omega ,M}-k\subseteq E^*_{\omega ,M}\subseteq E^*_{\omega ,M}+k\) and therefore
Accordingly
thus proving (8.6).
As a consequence of (8.6), we have that, for any \(k\in \mathbb {Z}^n\) with \(\omega \cdot k\geqslant 0\),
and similar statements hold in the case \(\omega \cdot k\leqslant 0\).
We now point out that, if \(Q\in {\mathscr {Q}}_\mathrm{{{Gr}}}\), then \(Q\cap (\partial E^*_{\omega ,M})\ne \varnothing \) and therefore
for some \(c>0\). We also observe that, for any \(E\subseteq \mathbb {R}^n\),
To check this, let us first suppose that \({\mathscr {L}}^n(E\cap Q)\leqslant {\mathscr {L}}^n(Q{\setminus } E)\). Then,
which gives (8.9) in this case. Conversely, if \({\mathscr {L}}^n(E\cap Q)> {\mathscr {L}}^n(Q{\setminus } E)\) we use that g has zero average and we write
thus completing the proof of (8.9).
In view of (8.8) and (8.9), we know that
provided that \(\eta \) is small enough.
On the other hand, if \(Q\in {\mathscr {Q}}_\mathrm{{{Mu}}}\), we are in the uniform density setting of (1.23) and, consequently, by (1.21) we can write that
up to renaming \(c>0\), where \(Q'\) is the dilation of Q with respect to its center by a factor 2 (we stress that the condition \(Q\in {\mathscr {Q}}_\mathrm{{{Mu}}}\) has been used here to guarantee an initial estimate on the density, which makes the constants in Theorem 1.18 uniform).
Then, from (8.11) and the relative isoperimetric inequality in Theorem 1.13, up to renaming \(c>0\), we have that if \(Q\in {\mathscr {Q}}_\mathrm{{{Mu}}}\), then
Therefore
provided that \(\eta \) is small enough.
Now we denote by \(J_\mathrm{{{Mu}}}\), \(J_\mathrm{{{a.Bl}}}\) and \(J_\mathrm{{{a.Wh}}}\) the number of cubes in \({\mathscr {Q}}_\mathrm{{{Mu}}}\), \({\mathscr {Q}}_\mathrm{{{a.Bl}}}\) and \({\mathscr {Q}}_\mathrm{{{a.Wh}}}\), respectively. Then, up to renaming constants, we deduce from (8.10) and (8.12) that
Comparing with (8.2) and using minimality, we thus obtain that
up to renaming \(C>0\). Hence, in view of the layer structure described in (8.7), we have that
while
We observe that the bound in (8.13) is already satisfactory, but the one in (8.14) needs to be improved if we want to arrive at a strip of uniform width (independent of r). That is, we are now in a situation in which “almost white” or “almost black” cubes may have a long tail in the strip when r is small, and we want to rule out this possibility.
For this, we need a careful procedure of cutting cubes in \({\mathscr {Q}}_\mathrm{{{a.Wh}}}\). The idea is that once we have a cube which is “almost white” we can color the region above it in full white, gaining energy.
The goal is thus to replace (8.14) with
up to renaming C (the situation of formulas (8.13) and (8.15) is graphically depicted in Fig. 1). So, if we can bound the width in (8.14) with a uniform bound, we are done; otherwise suppose that, for instance, \({\mathscr {Q}}_\mathrm{{{a.Wh}}}\) occupies a strip of width \(W_r\geqslant 2n\), possibly depending on r (from (8.14), we only know that \(W_r\leqslant C/r^{n-1}\)), say \(\{C_o\leqslant \omega \cdot x\leqslant C_o+W_r\}\) (notice that the position of the lower boundary of this strip is uniformly bounded, thanks to (8.13), so we denoted it by \(C_o\) for the sake of clarity).
The idea is now to replace \(E^*_{\omega ,M}\) with \(E^*_{\omega ,M}\cap \{\omega \cdot x\leqslant C_o+\sqrt{n}\}\). To compute the effect of this cut, let us consider that, at levels \(\{\omega \cdot x\in [ C_o,C_o+n]\}\), we may have created additional r-perimeter adding portions of \(\{\omega \cdot x=C_o+\sqrt{n}\}\) to the boundary of the set. Since this portion is flat, the cut procedure has produced an energy increasing for the r-perimeter of size at most \(C\,{\mathscr {H}}^{n-1}(F_\omega )\,r^{n-1}\). As for the bulk energy produced by g, in each cube Q in \(\{\omega \cdot x=C_o+\sqrt{n}\}\), we have produced an energy increasing of at most
thanks to (8.9) and to the fact that \(Q\in {\mathscr {Q}}_\mathrm{{{a.Wh}}}\). That is, the total bulk energy increased at levels \(\{\omega \cdot x\in [ C_o,C_o+n]\}\) is bounded by \(C\,\eta \, {\mathscr {H}}^{n-1}(F_\omega )\,r^n\). Summarizing, the modifications of the cubes in \({\mathscr {Q}}_\mathrm{{{a.Wh}}}\) at levels \(\{\omega \cdot x\in [ C_o,C_o+n]\}\) produce an energy increasing bounded by
To prove that the total energy has in fact decreased, we now check that the cut procedure produces a considerable gain at the other levels \(\{\omega \cdot x\in [ C_o+n,C_o+W_r]\}\). For this, notice that
In each of these cubes, the cut has produced an energy gain, due to the r-perimeter, and possibly an energy loss due to the bulk energy of g. From (8.8), we know that the energy gain in each of these cubes is at least \(c r^{n-1}\), up to renaming \(c>0\). On the other hand, from (8.9) and the fact that the cube belongs to \({\mathscr {Q}}_\mathrm{{{a.Wh}}}\), we deduce an upper bound of the bulk energy loss in each cube of the form \(C\eta r^n\), for some \(C>0\). Hence, the variation of energy in each of these cubes is of the form \(-c r^{n-1}+C\eta r^n\) (which is negative for small \(\eta \)).
Summarizing, and recalling (8.17), we have that the cut procedure has produced in \(\{\omega \cdot x\in [ C_o+n,C_o+W_r]\}\) an energy variation bounded from above by
up to renaming c and C. From this and (8.16), up to renaming constants line after line, we obtain that the variation of the energy produced by the cut is in total bounded from above by
Since, by the minimal property of \(E^*_{\omega ,M}\), this energy variation has to be positive, we conclude that
and thus, for small \(\eta \), we obtain that \(W_r\) is bounded uniformly, by a constant independent of r.
This proves (8.15) for the families of cubes in \({\mathscr {Q}}_\mathrm{{{a.Wh}}}\) (the cases of the cubes in \({\mathscr {Q}}_\mathrm{{{a.Bl}}}\) is similar).
From (8.15), one can exploit the methods in [3], namely find that there exists a uniform \(M_0>0\) such that if \(M\geqslant M_0\), then \( E^*_{\omega ,M}=E^*_{\omega ,M_0}\) and then, checking that the minimal minimizer is stable with respect to multiples of the period, establish that it is a Class A minimizer, thus completing the proof of Theorem 1.21 for rational slopes \(\omega \).
Step 2: planelike minimizers with irrational slopes. Since the quantity M is a universal constant, independent of n, in order to construct minimizers with an irrational slope \(\omega \in S^{n-1}\) we approximate \(\omega \) with rational frequencies \(\omega _k\), which produce planelike minimizers \(E^*_{\omega _k}\) and then pass to the limit in k, using Theorem 1.8, which applies in particular to the Class A planelike minimizers. \(\square \)
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Communicated by L. Ambrosio.
This work has been supported by the Andrew Sisson Fund 2017 and the Australian Research Council Discovery Project N.E.W. Nonlocal Equations at Work.
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Cesaroni, A., Dipierro, S., Novaga, M. et al. Minimizers for nonlocal perimeters of Minkowski type. Calc. Var. 57, 64 (2018). https://doi.org/10.1007/s00526-018-1335-9
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DOI: https://doi.org/10.1007/s00526-018-1335-9