On $L^p$ Estimates for Quasi-Elliptic Boundary Problems.

Quasi-linear elliptic boundary value problems

A Quasi-Monte Carlo Method for Elliptic Boundary Value Problems

In this paper we present and analyze a quasi-Monte Carlo method for solving elliptic boundary value problems. Our method transforms the given partial differential equation into an integral equation by employing a well known local integral representation. The kernel in this integral equation representation can be used as a transition density function to define a Markov process used in estimating...

Regularity estimates for elliptic boundary value problems in Besov spaces

We consider the Dirichlet problem for Poisson’s equation on a nonconvex plane polygonal domain Ω. New regularity estimates for its solution in terms of Besov and Sobolev norms of fractional order are proved. The analysis is based on new interpolation results and multilevel representations of norms on Sobolev and Besov spaces. The results can be extended to a large class of elliptic boundary val...

Sharp Boundary Estimates for Elliptic Operators

L 2 boundary decay properties of the heat kernel and spectral density. We deduce bounds on the rate of convergence of the eigenvalues when the region is slightly reduced in size. It is remarkable that several of the bounds do not involve the space dimension. AMS subject classifications: 35P99, 35P20, 47A75, 47B25 keywords: boundary decay, Laplacian, Hardy inequality, eigenfunctions, heat kernel...

Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains

We consider the model Dirichlet problem for Poisson’s equation on a plane polygonal convex domain W with data f in a space smoother than L2. The regularity and the critical case of the problem depend on the measure of the maximum angle of the domain. Interpolation theory and multilevel theory are used to obtain estimates for the critical case. As a consequence, sharp error estimates for the cor...