Gerassimos Barbatis

A unified approach to improved ^{} Hardy inequalities with best constants

We present a unified approach to improved L p Hardy inequalities in R N . We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension 1 < k < N. In our main result, we add to the right hand side of the classical Hardy inequality a weighted L p norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted L q norms, q ¬= p.

Stability Estimates for Resolvents, Eigenvalues, and Eigenfunctions of Elliptic Operators on Variable Domains

We consider general second order uniformly elliptic operators sub- ject to homogeneous boundary conditions on open sets `(›) parametrized by Lipschitz homeomorphismsdeflned on a flxed reference domain ›. For two open sets `(›) and e `(›) we estimate the variation of resolvents, eigenvalues, and eigenfunctions via the Sobolev norm ke ` i `kW 1;p(›) for flnite values of p, under natural summability conditions on eigenfunctions and their gradi- ents. We prove that such conditions are satisfled for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigen- functions via the measure of the symmetric difierence of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.

SHARP HEAT-KERNEL ESTIMATES FOR HIGHER-ORDER OPERATORS WITH SINGULAR COEFFICIENTS

We obtain heat-kernel estimates for higher-order operators with measurable coefficients that can be singular or degenerate. Precise constants are given, which are sharp for small times. AMS 2000 Mathematics subject classification: Primary 35K30; 47D06. Secondary 35K25; 35K35

Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain

Bethe–Sommerfeld Conjecture for Pseudodifferential Perturbation

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Stability and Regularity of Higher Order Elliptic Operators with Measurable Coefficients

in

On the Hardy Constant of Some Non-convex Planar Domains

{\mathbb{R}}^N

On the approximation of Finsler metrics on Euclidean domains

. We consider deformations

Homogenization of random elliptic systems with an application to Maxwell's equations

\phi (\Omega)

Monotonicity, continuity and differentiability results for the L p Hardy constant

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