Stability of isoperimetric type inequalities for some Monge–Ampère functionals

We investigate the stability of some inequalities of isoperimetric type related to Monge–Ampère functionals. In particular, firstly we prove the stability of a reverse Faber–Krahn inequality for the Monge–Ampère eigenvalue and its generalization. Then we give a stability result for the Brunn–Minkowski inequality and for a consequent Urysohn’s type inequality for the so-called \(n\) -torsional rigidity, a natural extension of the usual torsional rigidity.     

Isoperimetric Inequalities for a Wedge-Like Membrane

For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for the fundamental eigenvalue which in some cases improves the classical isoperimetric inequality of Faber–Krahn. In this work, we introduce “relative torsional rigidity” for this type of membrane and prove new isoperimetric inequalities in the spirit of Saint-Venant, Pólya–Szeg?, Payne, Payne–Rayner, Chiti, and Talenti, which link the eigenvalue problem with the boundary value problem in a fundamental way.     

Brunn-Minkowski type inequalities for conformal and Euclidean moments of domains

We prove Brunn-Minkowski type inequalities for three new functionals which are power moments for conformal and Euclidean characteristics of domains.     

Symmetry results for overdetermined problems on convex domains via Brunn–Minkowski inequalities

We consider some well-posed Dirichlet problems for elliptic equations set on the interior or the exterior of a convex domain (examples include the torsional rigidity, the first Dirichlet eigenvalue, and the electrostatic capacity), and we add an overdetermined Neumann condition which involves the Gauss curvature of the boundary. By using concavity inequalities of Brunn–Minkowski type satisfied by the corresponding variational energies, we prove that the existence of a solution implies the symmetry of the domain. This provides some new characterizations of spheres, in models going from solid mechanics to electrostatics.     

Interpolating between torsional rigidity and principal frequency

A one-parameter family of variational problems is examined that interpolates between torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. The associated partial differential equation is derived, which is shown to have positive solutions in many cases. Results are obtained regarding extremal domains and regarding variations of the domain or the parameter.     

Lp Brunn-Minkowski型仿射均质积分不等式

经典的仿射均质积分不等式是Brunn-Minkowski理论中一个关键不等式.建立了L_p Brunn-Minkowski型仿射均质积分不等式,定义了L_p Brunn-Minkowski型仿射混合均质积分且推广得到了L_p Brunn-Minkowski型仿射混合均质积分不等式.     

Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds

This paper studies eigenvalues of the drifting Laplacian on compact Riemannian manifolds with boundary (possibly empty) and provides a general inequality for them. Using the general inequality, we obtain universal inequalities for eigenvalues of the drifting Laplacian of Payne-Pólya-Weinberger-Yang type for manifolds supporting some special functions. We also obtain a lower bound for the first eigenvalue of the square of the drifting Laplacian on compact manifolds with boundary under some condition on the Bakry-Ricci curvature.     

Eigenvalue problems on Riemannian manifolds with a modified Ricci tensor

In this paper, we study the eigenvalue problems on a Riemannian manifold with a modified Ricci tensor. We obtain some sharp lower bound estimates for the first eigenvalue of Laplacian. We also prove some rigidity theorems for the Riemannian manifold with some suitable conditions.     

On the characterization of the compact embedding of Sobolev spaces

For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of p-capacity zero, we characterize the compactness of the embedding ${W^{1,p}({\mathbb R}^N)\cap L^p ({\mathbb R}^N,\mu)\hookrightarrow L^q({\mathbb R}^N)}$ in terms of the qualitative behavior of some characteristic PDE. This question is related to the well posedness of a class of geometric inequalities involving the torsional rigidity and the spectrum of the Dirichlet Laplacian introduced by Pólya and Szeg? (Isoperimetric inequalities in mathematical physics, Annals of mathematics studies, 1951). In particular, we prove that finite torsional rigidity of an arbitrary domain (possibly with infinite measure), implies the compactness of the resolvent of the Laplacian.     

质心体与射影体的极

建立了关于Blaschke与调和Blaschke线性组合的射影体极的几个精度不同的Brunn-Minkowski型不等式,给出了调和Blaschke线性组合的质心体极的Brunn- Minkowski不等式和类似结果．     

Sharp Stability of Some Spectral Inequalities

In this work we review two classical isoperimetric inequalities involving eigenvalues of the Laplacian, both with Dirichlet and Neumann boundary conditions. The first one is classically attributed to Krahn and P. Szego and asserts that among sets of given measure, the disjoint union of two balls with the same radius minimizes the second eigenvalue of the Dirichlet–Laplacian, while the second one is due to G. Szegő and Weinberger and deals with the maximization of the first non-trivial eigenvalue of the Neumann–Laplacian. New stability estimates are provided for both of them.     

对偶Brunn-Minkowski型不等式

本文研究了对偶Brunn-Minkowski不等式问题.利用对偶混合体和Blaschke径向和的性质,建立了对偶混合体均质积分的Brunn-Minkowski不等式的隔离形式,推广了对偶Brunn-Minkowski理论的几个不等式.     

Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non-zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.     

UNIVERSAL INEQUALITIES FOR A HORIZONTAL LAPLACIAN VERSION OF THE CLAMPED PLATE PROBLEM ON CARNOT GROUP

In this paper, we investigate a horizontal Laplacian version of the clamped plate problem on Carnot groups and obtain some universal inequalities. Furthermore, for the lower order eigenvalues of this eigenvalue problem on carnot groups, we also give some universal inequalities.     

Inequalities for dual quermassintegrals of mixed intersection bodies

In this paper, we first introduce a new concept ofdual quermassintegral sum function of two star bodies and establish Minkowski’s type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov-Fenchel inequality and the Brunn-Minkowski inequality for mixed intersection bodies and some related results. Our results present, for intersection bodies, all dual inequalities for Lutwak’s mixed prosection bodies inequalities.     

Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators

In this paper we prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the operator , that is the p‐Laplacian, and , namely the pseudo‐p‐Laplacian. Moreover we prove a stability result by means of a suitable isoperimetric deficit. Finally, we give a sharp lower bound for the anisotropic p‐torsional rigidity.     

On an inequality of Minkowski for mixed volumes

Using a recently established stability result regarding the Brunn-Minkowski theorem and simple facts about convex functions we find strengthened versions of known inequalities for the mixed volumes of convex bodies. These results improve previously known inequalities of this type.Supported by National Science Foundation Research Grant DMS 870189.     

INEQUALITIES FOR MIXED INTERSECTION BODIES

In this paper, some properties of mixed intersection bodies are given, and inequalities from the dual Brunn-Minkowski theory (such as the dual Minkowski inequality, the dual Aleksandrov-Fenchel inequalities and the dual Brunn-Minkowski inequalities) are established for mixed intersection bodies.     

Sharp estimates for the anisotropic torsional rigidity and the principal frequency

In this paper we generalize some classical estimates involving the torsional rigidity and the principal frequency of a convex domain to a class of functionals related to some anisotropic nonlinear operators.     

Volume Difference Inequalities for the Polars of Mixed Complex Projection Bodies

In this paper, based on the notion of mixed complex projection and generalized the recent works of other authors, we obtain some volume difference inequalities containing Brunn-Minkowski type inequality, Minkowski type inequality and Aleksandrov-Fenchel inequality for the polars of mixed complex projection bodies.