We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in $R^n$. Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in $R^2$ in the small asymmetry regime.
We study the optimal constant in a Sobolev inequality for BV functions with zero mean value and vanishing outside a bounded open set. We are interested in finding the best possible embedding constant in terms of the measure of the domain alone. We set up an optimal shape problem and we completely characterize the behavior of optimal domains.
The relative isoperimetric inequality inside an open, convex cone $\mathcal C$ states that, at fixed volume, $B_r \cap \mathcal C$ minimizes the perimeter inside $\mathcal C$. Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov's proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal C$. Our proof follows the line of reasoning in \cite{Fi}, though several new ideas are needed in order to deal with the lack o…
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…
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.
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.
We prove a generalization of Reifenberg's isoperimetric inequality. The main result of this paper is used to establish existence of a minimizer for an anisotropically-weighted area functional among a collection of surfaces which satisfies a set of axioms, namely being closed under certain deformations and Hausdorff limits. This problem is known as the axiomatic Plateau problem.
We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics.
We prove an sharp anisotropic isoperimetric inequality for a domain outside an Euclidean ball in $\mathbb{R}^n$ for $n\geq 2$. The proof applies the ABP method to a Neumann boundary value problem.
We prove an isoperimetric inequality of the Rayleigh-Faber-Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue. More precisely, we show that the minimizer among sets of given volume is the union of two equal balls.
We obtain a generalized version of an inequality, first derived by C. Bandle in the analytic setting, for weak subsolutions of a singular Liouville-type equation. As an application we obtain a new proof of the Alexandrov isoperimetric inequality on singular abstract surfaces. Interestingly enough, motivated by this geometric problem, we obtain a seemingly new characterization of local metrics on Alexandrov's surfaces of bounded curvature. At least to our knowledge, the characterization of the equality case in the isoperimetric inequality in such a weak framework is new as well.
We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Lojasiewicz-Simon inequality. A main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.
We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in\re$, $Ω\subset\re^2$ then the inequality $$ \left(\frac{|Ω|}π\right)^{\frac{p+1}{2}}\leq\frac{1}{2π}\int_{\delomega}|x|^pdσ(x) $$ holds true under appropriate assumptions on $Ω$ and $p.$ This solves an open problem arising in the context of isoperimetric problems with density and poses some new ones (for instance generalizations to $\re^n$). We prove the equivalence with a Hardy-Sobolev inequality, giving the best constant, and generalize thereby the equivalence betwee…
We establish a partial generalization of a prior isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate to plates of nonzero Poisson's ratio.
We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.
We provide a full quantitative version of the Gaussian isoperimetric inequality. Our estimate is independent of the dimension, sharp on the decay rate with respect to the asymmetry and with optimal dependence on the mass.
For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.
This paper deals with various questions related to the isoperimetic problem for smooth positive measure $dμ= φ(x)dx$, with $x \in Ω\subset \mathbb{R}^N$. Firstly we find some necessary conditions on the density of the measure $ φ(x)$ that render the intersection of half spaces with $Ω$ a minimum in the isoperimetric problem. We then identify the unique isoperimetric set for a wide class of factorized finite measures. These results are finally used in order to get sharp inequalities in weighted Sobolev spaces and a comparison result for solutions to boundary value problems for degenerate ellipt…
We prove a quantitative reverse isoperimetric inequality for embedded surfaces with Willmore energy bounded away from $8π$. We use this result to analyze the negative $L^2$ gradient flow of the Willmore energy plus a positive multiple of the inclosed volume. We show that initial surfaces of Willmore energy less than $8π$ with positive inclosed volume converge to a round point in finite or infinite time.
In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application we obtain a global existence result for the surface diffusion flow, providing that an initial curve is $H^2$-close to a multiply covered circle and sufficiently rotationally symmetric.
We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hoelder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs.
We consider the monomial weight $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N}}$, where $a_{i}$ is a nonnegative real number for each $i\in\{1,\ldots,N\}$, and we establish the existence and nonexistence of isoperimetric inequalities with different monomial weights. We study positive minimizers of $\int_{\partialΩ}x^{A}\mathcal{H}^{N-1}(x)$ among all smooth bounded sets $Ω$ in $\mathbb{R}^{N}$ with fixed Lebesgue measure with monomial weight $\int_Ωx^{B}dx$.
Let D be a bounded domain in n-dimensional Euclidean space, where n>2, and let 1<p< (2n)/(n-2). We prove a reverse-Holder inequality for functions realizing equality in the Sobolev inequality, which finds a lower bound for their (p-1)-norm in terms of their p-norm. This inequality is sharp, and it is an equality if and only if the domain is a round ball. Our result generalizes a theorem of Payne and Rayner and our proof relies on integral rearrangements and an analysis of the ODE corresponding to the radial case.
The quantitative isoperimetric inequality: A calibration argument
Universität Leipzig, Fakultät für Mathematik und Informatik（莱比锡大学数学与计算机学院）；Universität Heidelberg, Institut für Mathematik & Interdisziplinäres Zentrum für Wissenschaftliches Rechnen（海德堡大学数学系与跨学科科学计算中心）
} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $Δ^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero eigenvalue $Υ_1(\Om)$ admits the isoperimetric inequality of Szegö-Weinberger type: $Υ_1(\Om)\le Υ_1(B_{\Om})$, where $B_{\Om}$ is a ball in ${\bf R}^n$ with the same volume of $\Om$. The isoperimetric inequality of Szegö-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators $Δ^{2m}$ ($m\ge1$) on $\Om$ is also exploited.
We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in $R^n$. Combining these results with a refinement of the selection principle introduced by the authors in arXiv:0911.0786, we describe a method suitable for the determination of the best constants in the quantitative isoperimetric inequality with higher order terms. Then, applying Bonnesen's annular symmetrization in a very elementary way, we show that, for $n=2$, the above-mentioned constants can be explicitly computed through a one-parameter family of convex sets known as ovals. This proves a f…
A well known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions. In dimension 2, the constant can be controlled by the integral of the positive part of the Gaussian curvature. In this paper, we showed that on simply connected conformal flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson $Q$- curvature. We achieve this by exploring the relationship between $A_p$ weights and integrals of the Q-curvature.
Let $c, k_1,..., k_N $ be non-negative numbers, and define a measure $μ$ in the wedge $W:= \{x\in \mathbb{R} ^N :\, x_i >0, i=1,...,N\} $ by $dμ= e^{c|x|^2} x_1 ^{k_1}...x_N ^{k_N} \, dx $. It is shown that among all measurable subsets of $W$ with fixed $μ$ -measure, the intersection of $W$ with a ball centered at the origin renders the weighted perimeter relative to $W$ a minimum.
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 …
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.
We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds, and develop some applications of this result to the isoperimetric problem in spaces of nonpositive curvature.
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…
We discuss various representations of planar $p$-harmonic systems of equations and their solutions. For coordinate functions of $p$-harmonic maps we analyze signs of their Hessians, the Gauss curvature of $p$-harmonic surfaces, the length of level curves as well as we discuss curves of steepest descent. The isoperimetric inequality for the level curves of coordinate functions of planar $p$-harmonic maps is proven. Our main techniques involve relations between quasiregular maps and planar PDEs. We generalize some results due to P. Lindqvist, G. Alessandrini, G. Talenti and P. Laurence.
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…
An isoperimetric inequality for Neumann eigenvalues with radial log-concave measures
College of informatics, Huazhong Agricultural University（华中农业大学信息学院）；School of Mathematical Sciences, Soochow University（苏州大学数学科学学院）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
In this note we investigate three-dimensional Schrödinger operators with $δ$-interactions supported on $C^2$-smooth cones, both finite and infinite. Our main results concern a Faber-Krahn-type inequality for the principal eigenvalue of these operators. The proofs rely on the Birman-Schwinger principle and on the fact that circles are unique minimizers for a class of energy functionals. The main novel idea consists in the way of constructing test functions for the Birman-Schwinger principle.
The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull $Ω^*$ of a set $Ω$ is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where $Ω$ has $\mathscr{C}^{1, α}$-boundary, the area of $\partial Ω^*$ is recovered as the limit of the $p$-capacities of $Ω$, as $p \to 1^+$. Finally, building on the existence of strictly …
In the present paper we introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a bounded simply-connected $C^2$-smooth open set. The considered perturbation is of lower order and corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if, and only if, the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of…
