Antoine Henrot, Gérard A. Philippin, Abdesselam Safoui
In this paper we establish isoperimetric inequalities for the product of some moments of inertia. As an application, we obtain an isoperimetric inequality for the product of the $N$ first nonzero eigenvalues of the Stekloff problem in $\mathbb{R}^N$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromov's proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt t…
In this paper, we prove some isoperimetric inequalities and give a sharp bound for the positive solution of sublinear elliptic equations.
Several inequalities for the isoperimetric ratio for plane curves are derived. In particular, we obtain interpolation inequalities between the deviation of curvature and the isoperimetric ratio. As applications, we study the large-time behavior of some geometric flows of closed plane curves without a convexity assumption.
In this paper we study two different weighted isoperimetric inequalities. In the first part of the paper we prove a sharp stability result for the isoperimetric inequality with a log-convex weight. In the second part we analize the behavior of a negative power weight for the perimeter thus providing a complete picture of the isoperimetric problem in this context.
In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalises an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.
In this paper, we prove a Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities.
Isoperimetric inequalities for the fractional composite membrane problem
本文研究分数阶复合膜问题第一特征值的等周不等式，建立了Faber-Krahn不等式的类比，并探讨两域交集上的情形，为经典不等式提供新视角。
In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption.
We solve a class of weighted isoperimetric problems of the form $$ \min\left\{\int\_{\partial E}w e^V\,dx:\int\_E e^V\,dx={\rm constant}\right\} $$ where $w$ and $V$ are suitable functions on $\R^d$. As a consequence, we prove a comparison result for the solutions of degenerate elliptic equations.
We present the proof of several inequalities using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, we give a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric …
In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space $\H^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.
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.
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.
This paper deals with weighted isoperimetric inequalities relative to cones of $\mathbb{R}^{N}$. We study the structure of measures that admit as isoperimetric sets the intersection of a cone with balls centered at the vertex of the cone. For instance, in case that the cone is the half-space $\mathbb{R}_{+}^{N}={x \in \mathbb{R}^{N} : x_{N}>0}$ and the measure is factorized, we prove that this phenomenon occurs if and only if the measure has the form $dμ=ax_{N}^{k}\exp(c|x|^{2})dx $, for some $a>0$, $k,c\geq 0$. Our results are then used to obtain isoperimetric estimates for Neumann eigenvalue…
We consider a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the volume and the area element carry two different weights of the type $|x|^lx_N^α$. We solve them in a special case while a more detailed study is contained in \cite{ABCMP2}. Our results imply a weighted Polya-Szëgo principle and a priori estimates for weak solutions to a class of boundary value problems for degenerate elliptic equations
In this paper, we establish some sharp inequalities between the volume and the integral of the $k$-th mean curvature for $k+1$-convex domains in the Euclidean space. The results generalize the classical Alexandrov-Fenchel inequalities for convex domains. Our proof utilizes the method of optimal transportation.
In this paper we consider a non-local problem for a Laplace operator in a multidimensional bounded symmetric domain. The investigated problem is an analogue of the classical periodic boundary value problems in the case of non-rectangular domain. We prove self-adjointness of the problem and show a method of constructing eigenfunctions. We obtain an analogue of the Rayleigh type inequality and some spectral inequalities for the first eigenvalue of the nonlocal problem.
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 discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of $\mathbb R^2$ endowed with an Aharonov-Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the Neumann and the Steklov problems is positive. We establish isoperimetric inequalitites for the lowest eigenvalue in the spirit of the classical inequalities of Szegö-Weinberger, Brock and Weinstock, the model domain being a disk with the pole at its center. We consider more generally domains in the plane endowed with a rotationally invariant metric, which incl…
In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in $\mathbb{H}^3$. More precisely, there exists a torus, which must develop a singularity under MCF before the volume it encloses decreases to zero. There also exists a topological sphere in the shape of a dumbbell, which must develop a singularity in the flow before its area shrinks to zero. Simultaneously, by using the flow, we proved an isoperimetric inequality for some domains in $\mathbb{H}^3$.
Evolving smooth, compact hypersurfaces in R^{n+1} with normal speed equal to a positive power k of the mean curvature improves a certain 'isoperimetric difference' for k >= n-1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n <= 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.
The goal of this note is to give the unified approach to the solutions of a class of isoperimetric problems by relating them to the exterior differential systems studied by R.~Bryant and P.~Griffiths. In this note we list several classical by now isopereimetric inequalities which can be proved in a unified way. This unified approach reduces them to the so-called exterior differential systems studied by Robert Bryant and Phillip Griffiths. To the best of our knowledge, this is the first article where this connection is used. After reviewing a list of classical inequalities (log-Sobolev inequali…
The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free boundary, and the other is an integral inequality (due to Duff [9]) for any nonnegative function under Schwarz equimeasurable rearrangement. These ingredients also allow us to establish some Moser-Trudinger type inequalities, and obtain some estimates on the principal frequency of a membrane with partial free boundary, which extend early results of Nehari [15] and …
By using optimal mass transport theory we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of $n$-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of S. Brendle (2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp const…
We consider the monomial weight $|x_1|^{A_1}...|x_n|^{A_n}$ in $\mathbb R^n$, where $A_i\geq0$ is a real number for each $i=1,...,n$, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure $dx$ replaced by $|x_1|^{A_1}...|x_n|^{A_n}dx$, and they contain the best or critical exponent (which depends on $A_1$, ..., $A_n$). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When $A_i$ are nonnegative \textit{integers}, the…
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 prove integral formulas for closed hypersurfaces in C^{n+1}, which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the ``Soap Bubble Theorem'' for star-shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above.
Isoperimetric inequalities for Neumann eigenvalues on bounded domains in rank-1 symmetric spaces
本文证明了紧致与非紧致秩为1对称空间中有界区域上Neumann Laplacian低阶特征值的精确等周不等式，推广了Wang和Xia关于双曲空间的工作及Aithal和Santhanam的Szegö-Weinberger不等式。
First Eigenvalue and Torsional Rigidity: Isoperimetric Inequalities for the Fractional Laplacian
本文研究了分数阶拉普拉斯算子在等周不等式中的性质，证明了在具有相同扭刚度的集合中，球体是第一特征值最小的集合，并建立了相关反Holder不等式。
Isoperimetric inequalities for inner parallel curves
本文证明了光滑有界单连通区域中内平行曲线的加权等周不等式，表明固定周长下惯性矩在圆盘处最大，并将该结果推广到p≤2阶矩，同时刻画了p<3时圆盘为严格局部极大值、p>3时不等式失效的临界行为。
In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.
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…
The goal of this note is to have a systematic approach to generating isoperimetric inequalities from two concrete type of PDEs. We call these PDEs Bellman type because a totally analogous equations happen to rule many sharp estimates for singular integrals in harmonic analysis, and such estimates were obtained with the use of Hamilton--Jacobi--Bellman PDE. We show how classical inequalities of Brascamp--Lieb, Prekopa--Leindler, Ehrhard are particular case of this scheme, which allows us to augment the stock of such inequalities. We approach the isoperimetric inequalities as a maximum (minimum)…
Several isoperimetric inequalities of Dirichlet and Neumann eigenvalues of the Witten-Laplacian
本文利用重排技术与试验函数构造，在固定加权体积条件下建立空间形式中有界区域上 Witten-Laplacian 多个 Dirichlet 与 Neumann 特征值的等周不等式，推广了 Laplacian 的经典结果。
In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant $C(n)=Cn^{7}$ depending on the space dimension $n$ in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to $Cn^6$ for convex sets and to $Cn^5$ for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form $Cn^2,$ i.e., quadratic in $n.$ The tools are the Brenier's mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic…
We investigate the logarithmic and power-type convexity of the length of the level curves for $a$-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the $p$-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the $W^{2,2}$ regularity. The results are complemented by a number of estimates for the derivatives $L'$ and $L''$ of the length of the level curve function $L$, as well as by examples illustrating the presentation. Our work generalizes r…
We prove an inequality for submanifolds of Cartan-Hadamard manifolds, which relates the geometry of a submanifold to the measure of the geodesics in the ambient space which it intersects. For hypersurfaces, this gives an extension of Banchoff and Pohl's isoperimetric inequality to spaces of non-positive curvature. We also prove a modified version of Croke's isoperimetric inequality for hypersurfaces immersed in Cartan-Hadamard manifolds and a sharp, quantitative version of an isoperimetric inequality of Yau in spaces of negative curvature. We discuss the relationship between these results, and…
We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called …
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。
An Information-Theoretic Route to Isoperimetric Inequalities via Heat Flow and Entropy Dissipation
本文提出基于热流下熵耗散的等周不等式信息论方法，将扩散视作含噪信道推导互信息衰减与边界测度的关联，证明欧氏等周不等式并拓展至黎曼流形场景，搭建几何分析与信息论的联系。
Weighted fractional Poincaré inequalities via isoperimetric inequalities
本文利用分数阶等周不等式证明加权分数阶 Poincaré-Sobolev 不等式，改进 Bourgain-Brezis-Mironescu 估计与 Meyers-Ziemer 定理，并推广含 $A_p$ 权的结果。
Isoperimetric inequalities involving Steklov eigenvalues on surfaces
本文研究了带边界曲面上涉及 Steklov 特征值的等周不等式最优常数，在 Riemann 曲面拓扑和共形类层面证明了相关不等式，提供了实现最优常数的拓扑圆盘新实例，并证明了环面和 Möbius 带上第一共形 Steklov 特征值的刚性。
Isoperimetric and geometric inequalities in quantitative form: Stein's method approach
该研究通过引入带斜边界条件的椭圆型PDE解决边界项处理难题，将Stein方法几何化，得到Brock-Weinstock不等式、Steklov特征值约束下等周不等式及加权与未加权周长组合的稳定性结果，均基于α-Zolotarev距离表述。
Isoperimetric inequalities for the lowest magnetic Steklov eigenvalue
本文研究了平面域上磁斯托克罗夫特问题的最低特征值优化，证明圆盘在给定面积下最大化该特征值，并建立了外域下的等周不等式。
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。
Sloshing in containers with vertical walls: isoperimetric inequalities for the fundamental eigenvalue
本文针对垂直壁容器中的晃荡问题，在自由表面凸且具有两条对称轴、周长给定的条件下，证明当自由表面为正方形或等边三角形时无限深容器使基本特征值最大，并给出另一等周不等式作为Szegő经典结果的推论。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。