Antonio De Rosa, Sławomir Kolasiński, Mario Santilli
Given an elliptic integrand of class $ \mathscr{C}^{3} $, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.
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We consider a class of nonlocal generalized perimeters which includes fractional perimeters and Riesz type potentials. We prove a general isoperimetric inequality for such functionals, and we discuss some applications. In particular we prove existence of an isoperimetric profile, under suitable assumptions on the interaction kernel.
We prove a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under convexity constraints. We show existence of minimizers, and we describe the shape of minimizers in certain parameter regimes.
Recently Frank and Seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from Sobolev seminorms of fractional order. This isoperimetric inequality is improved here in a quantitative form.
We prove a quantitative isoperimetric inequality for the Gaussian fractional perimeter using extension techniques. Though the exponent of the Fraenkel asymmetry is not sharp, the constant appearing in the inequality does not depend on the dimension but only on the Gaussian volume of the set and on the fractional parameter.
The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.
We consider a scale invariant functional involving the anisotropic $p-$momentum, the anisotropic perimeter and the volume. We show that the Wulff shape, associated with the Finsler norm $F$ considered and centered at the origin, is the unique minimizer of the anisotropic functional taken into consideration among all bounded convex sets.
We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter.
Stability of the ball in isoperimetric inequalities between two fractional perimeters
研究在s-周长和t-周长之间等周不等式中，球作为近球集的局部极小值问题，通过将F函数表示为参数函数u的泛函，并证明其关于Sobolev范数的稳定性结果。
We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms, for a quite wide range of perimeter functionals. Such generalized clusters are a natural "relaxed'' version of a cluster and can be thought of as ``albums'' with possibly infinite pages, having a minimal cluster drawn on each page, the total perimeter and the vector of masses being calculated by summation over all pages, the total perimeter being m…
It is a well known fact that in $\mathbb{R}^n$ a subset of minimal perimeter $L$ among all sets of a given volume is also a set of maximal volume among all sets of the same perimeter $L$. This is called the reciprocity principle for isoperimetric problems. The aim of this note is to prove this relation in the case where the class of admissible sets is restricted to the subsets of some subregion $G\subsetneq\mathbb{R}^n$. Furthermore, we give a characterization of those (unbounded) convex subsets of $\mathbb{R}^2$ in which the isoperimetric problem has a solution. The perimeter that we consider…
Isoperimetric conditions, lower semicontinuity, and existence results for perimeter functionals with measure data
本文在 $\mathbb{R}^n$ 上建立了带测度数据的周长泛函的下半连续性，推导了多种约束下极小值的存在性，并提出关键的小体积等周条件以捕捉抵消效应，为预定平均曲率问题奠定了变分基础。
Isoperimetric inequalities and spectral consequences in warped product manifolds
研究扭曲积流形上的中心等周不等式，推导几何条件并改进Cheeger不等式，得出极小正特征值的下界。
A spectral isoperimetric inequality on the n-sphere for the Robin-Laplacian with negative boundary parameter
本文研究带负边界参数的Robin-Laplacian在n维球面上的谱等周问题，证明固定周长小于半球周长时测地球最大化第一Robin特征值，并给出基于体积差的定量稳定性结果。
We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent $α$, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter $γ$. We show that for a wide class of density functions the energy admits a minimizer for any value of $γ$. Moreover these minimizers are bounded. For monomial densities of the form $|x|^p$ we prove that when $γ$ is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the…
On the reverse isoperimetric inequality in Gauss space
本文研究高斯空间中凸集的逆等周不等式，通过局部扰动分析证明光滑周长最大化集具有局部平坦边界，并给出更尖锐的周长界，特别对轴对称凸集和四边形退化情形进行了分析。
We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $t$-perimeter and the $s$-perimeter, with $s$ smaller than $t$. Exploiting the quantitative fractional isoperimetric inequality, we show that balls are the unique minimizers if the volume is sufficiently small, depending on $t-s$, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all $s,\,t$. When $s=0$ this problem reduces to the fractional isoper…
The aim of this paper is to prove the existence of minimizers for a variational problem involving the minimization under volume constraint of the sum of the perimeter and a non-local energy of Wasserstein type. This extends previous partial results to the full range of parameters. We also show that in the regime where the perimeter is dominant, the energy is uniquely minimized by balls.
Double-arc rearrangement and the symmetric isoperimetric problem in Gauss space
研究高斯空间中的双弧重排方法，用于分析对称集高斯周长的极小化子，为对称等周问题提供新的分析途径。
Quantitative isoperimetric inequalities in capillarity problems and cones in strong and barycentric forms
本文针对欧氏半空间中的经典毛细泛函和凸锥中的相对周长，建立了强形式与重心形式的尖锐定量等周不等式，证明等周亏量同时控制 Fraenkel 不对称性和法向量振荡不对称性，并显式给出计算中心。
A note on non-local Sobolev spaces and non-local perimeters
本文研究由积分核定义的非局部Sobolev空间，证明延拓结果、Sobolev与Poincaré不等式及非局部周长等周不等式，并讨论核非径向对称时的非局部等周问题。
We introduce a notion of fractional perimeter in an abstract Wiener space, and we show that halfspaces are the only volume-constrained minimisers.
A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$, such that, for every $Ω\subset\mathbb{R}^n$ open, bounded and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds \begin{equation*} λ_1(Ω)-λ_1(B)\geq c\left(P(Ω)-P(B) \right)^{2}, \end{equation*} where by $λ_1(\cdot)$ we denote the first Dirichlet eigenvalue of a set and by $P(\cdot)$ its perimeter. The hearth of the present pap…
In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular we prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either…
We prove a relative isoperimetric inequality in the plane, when the perimeter is defined with respect to a convex, positively homogeneous function of degree one, and characterize the minimizers.
We consider the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. We show that, in the small mass regime, if the Wulff shape of the anisotropic perimeter has certain symmetry properties, then it is the unique global minimizer of the total energy. In dimension two this applies to convex polygons which are reflection symmetric with respect to the bisectors of the angles. We further prove a rigidity result for the structure of (local) minimizers in two dimensions.
In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to two different densities. The case of a single density, or equivalently, when the two densities coincide, has been well studied in the last years; nonetheless, the problem with two different densities is an important generalisation, also in view of applications. We will prove the existence of isoperimetric sets in this context, extending the known results fo…
We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. This second term has a repulsive effect, and it is in competition with the perimeter. Because of that, in general the solution is not the half-space. We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the repulsive term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimet…
Characterization of sets of finite local and non local perimeter via non local heat equation
本文利用Γ-收敛刻画有限局部与非局部周长集合，并由此证明局部与非局部等周不等式。
We consider the magnetic Robin Laplacian with a negative boundary parameter. Among a certain class of domains, we prove that the disk maximizes the ground state energy under the fixed perimeter constraint provided that the magnetic field is of moderate strength. This class of domains includes, in particular, all domains that are contained upon translations in the disk of the same perimeter and all convex centrally symmetric domains.
We prove that finite perimeter subsets of $\mathbb{R}^{n+1}$ with small isoperimetric deficit have boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori integral curvature bounds. As an application, we answer a question raised by B. Colbois concerning the almost extremal hypersurfaces for Chavel's inequality.
We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the nonlocal term asymptotically localizes and contributes to the perimeter term to leading order. We establish existence of generalized minimizers for all values of the dipolar strength, mass and regularization cutoff and give conditions for existence of classical minimizers. For subcritical dipolar strengths we prove that the limiting functional is a renormalized per…
Barycentric stability of nonlocal perimeters: the convex case
本文研究了凸集的非局部周长稳定性，通过建立精确的非局部定量等周不等式，揭示了凸集在非局部意义下的稳定性特性。
An infinite cluster $\mathbf E$ in $\mathbb R^d$ is a sequence of disjoint measurable sets $E_k\subset \mathbb R^d$, $k\in \mathbb N$, called regions of the cluster. Given the volumes $a_k\ge 0$ of the regions $E_k$, a natural question is the existence of a cluster $\mathbf E$ which has finite and minimal perimeter $P(\mathbf E)$ among all clusters with regions having such volumes. We prove that such a cluster exists in the planar case $d=2$, for any choice of the areas $a_k$ with $\sum \sqrt a_k < \infty$. We also show the existence of a bounded minimizer with the property $P(\mathbf E)=\math…
On the Isoperimetric Profile of the Hypercube
本文证明了体积接近1/2的超立方体子集相对周长至少为1，解决了Brezis-Bruckstein猜想，并证明超立方体等周剖面不收敛于高斯等周剖面。
We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. W…
We study a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the densities of the weighted volume and weighted perimeter are given by two different non-radial functions of the type $|x|^k x_N^α$. Our results imply some sharp functional inequalities, like for instance, Caffarelli-Kohn-Nirenberg type inequalities.
We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.
In this note we present the solution of some isoperimetric problems in open convex cones of $\R^n$ in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. When the weight is constant, the result was established by Lions and Pacella in 1990.
We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal $s$-perimeter plus a non-local repulsive interaction term. In the particular case $s =1$ the $s$-perimeter coincides with the classical perimeter, and our results improve the ones of Knüpfer and Muratov concerning minimality of balls of small volume in isoperimetric problems with a competition b…
We study isoperimetric problems with respect to infinite measures on $R ^n$. In the case of the measure $μ$ defined by $dμ= e^{c|x|^2} dx$, $c\geq 0$, we prove that, among all sets with given $μ-$measure, the ball centered at the origin has the smallest (weighted) $μ-$perimeter. Our results are then applied to obtain Polya-Szego-type inequalities, Sobolev embeddings theorems and a comparison result for elliptic boundary value problems.
Quantitative isoperimetric inequalities for classical capillarity problems
本文研究欧几里得半空间中带常数边界权重的毛细管等周问题，利用ABP型技巧给出尖锐等周不等式的新证明，并通过定量ABP方法与选择型论证相结合，建立了关于最优截断球的Fraenkel不对称性和边界接触部分不对称性的两个定量估计。
Bubbling of almost critical points of anisotropic isoperimetric problems with degenerating ellipticity
研究各向异性表面能函数的近临界点在范数退化时的极限行为，证明其极限为有限个不相交的Wulff形状的并集。
The standard lens cluster in R^2 uniquely minimizes relative perimeter
本文研究平面三分区的等周问题，证明单位有限面积加两个无限面积区域的唯一解是圆弧透镜簇，其圆弧在轴上以120度角形成两个三重点，并证明该构型是周长的局部极小值。
On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian
本文研究在凸域上最小化拉普拉斯算子第k个特征值的问题，证明在直径约束下任意维数及二维周长约束下诺伊曼特征值最小化序列收敛于球，并扩展至混合边界情形。
Euler's elastica functional as a large mass limit of a two-dimensional non-local isoperimetric problem
研究二维非局部等周问题在大质量极限下的临界行为，证明能量下一阶Γ-极限为周长与欧拉弹性泛函的加权和，并得出极小元存在且收敛到圆盘或环的结论。
This note treats several problems for the fractional perimeter or $s$-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as $s \nearrow 1$ is proven. It is shown that their limit as $s \searrow -\infty$ can be expressed in terms of the volume.
The Anisotropic Gaussian Isoperimetric Inequality and Ehrhard Symmetrization
本文证明各向异性高斯等周不等式并刻画取等条件，同时通过构造反例说明Ehrhard对称化不满足递减性，进而提出含误差项的新不等式，为证明各向异性Ehrhard对称化的唯一性提供依据。
This paper is concerned with volume-constrained minimization problems derived from Gamow's liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. We consider a large class of potentials, given by general radial nonnegative kernels which are integrable on $\mathbb{R}^n$, such as Bessel potentials, and study the behavior of the problem for large masses (i.e., volumes). Contrarily to the small mass case, where the nonlocal term becomes negligible compared to the perimeter, here the nonlocal term explodes compared to it. However, …
A concentration-compactness principle for perturbed isoperimetric problems with general assumptions
本文在温和假设下建立了扰动等周问题广义极小元的存在性与密度估计，其假设涵盖经典、各向异性、分数阶周长等泛函，扰动项可包含势函数、平移不变核或含Wasserstein距离的非局部项。
In this paper we establish optimal isoperimetric inequalities for a nonlocal perimeter adapted to the fractional powers of a class of Kolmogorov-Fokker-Planck operators which are of interest in physics. These operators are very degenerate and do not possess a variational structure. The prototypical example was introduced by Kolmogorov in his 1938 paper on brownian motion and the theory of gases. Our work has been influenced by ideas of M. Ledoux in the local case.
We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form $\|x\|^β$ that provides the existence of a minimizer for any volume constraint, and we study the geometry of big minimizers. Then we provide a numerical method to adress this variational problem.
The anisotropic fractional $s$-perimeter with respect to a convex body $K$ in $\rn$ is shown to converge to the anisotropic perimeter with respect to the moment body of $K$ as $s\to 1^-$. A corresponding result is established for anisotropic fractional $s$-seminorms on $BV(\rn)$ (generalizing results of Bourgain, Brezis & Mironescu and Dávila). The minimizers of the anisotropic fractional $s$-isoperimetric inequality with respect to $K$ are shown to converge to the moment body of $K$. Anisotropic fractional Sobolev inequalities are established.
Existence and nonexistence results for a nonlocal isoperimetric problem on $\mathbb{H}^n$
本文研究了双曲空间中非局部等周问题，证明了小体积时测地球体为唯一极小解，并在特定指数范围内获得了大体积时的不存在性结果。
We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.
In this paper we prove that among all convex domains of the plane with two axis of symmetry, the maximizer of the first non trivial Neumann eigenvalue $μ_1$ with perimeter constraint is achieved by the square and the equilateral triangle. Part of the result follows from a new general bound on $μ_1$ involving the minimal width over the area. Our main result partially answers to a question addressed in 2009 by R. S. Laugesen, I. Polterovich, and B. A. Siudeja.
We solve a class of isoperimetric problems on $\mathbb{R}^2_+ :=\left\{ (x,y)\in \mathbb{R} ^2 : y>0 \right\}$ with respect to monomial weights. Let $α$ and $β$ be real numbers such that $0\le α<β+1$, $β\le 2 α$. We show that, among all smooth sets $Ω$ in $\mathbb{R} ^2_+$ with fixed weighted measure $\iint_{Ω} y^β dxdy$, the weighted perimeter $\int_{\partial Ω} y^α\, ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a lower bound for the first eigenvalue of a cla…
In tis paper we prove an isoperimetric inequality for the first twisted eigenvalue $λ_{1,γ}^T(Ω)$ of a weighted operator, defined as the minimum of the usual Rayleigh quotient when the trial functions belong to the weighted Sobolev space $H_0^1(Ω,dγ)$ and have weighted mean value equal to zero in $Ω$. We are interested in positive measures $dγ=γ(x) dx$ for which we are able to identify the isoperimetric sets, namely, the sets that minimize $λ_{1,γ}^T(Ω)$ among sets of given weighted measure. In the cases under consideration, the optimal sets are given by two identical and disjoint copies of th…
We study an isoperimetric problem the energy of which contains the perimeter of a set, Coulomb repulsion of the set with itself, and attraction of the set to a background nucleus as a point charge with charge $Z$. For the variational problem with constrained volume $V$, our main result is that the minimizer does not exist if $V - Z$ is larger than a constant multiple of $\max(Z^{2/3}, 1)$. The main technical ingredients of our proof are a uniform density lemma and electrostatic screening arguments.
Lattice tilings minimizing nonlocal perimeters
本文证明了在一般非局部周长泛函下，周期镶嵌的最小化存在性，通过将最优分割问题转化为等周问题并利用紧致性论证，同时探讨了平面六边形镶嵌的最优性。
On a convex set, we prove that the Poincaré-Sobolev constant for functions vanishing at the boundary can be bounded from above by the ratio between the perimeter and a suitable power of the $N-$dimensional measure. This generalizes an old result by Pólya. As a consequence, we obtain the sharp {\it Buser's inequality} (or reverse Cheeger inequality) for the $p-$Laplacian on convex sets. This is valid in every dimension and for every $1<p<+\infty$. We also highlight the appearing of a subtle phenomenon in shape optimization, as the integrability exponent varies.
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