Sharp boundary estimates for elliptic operators

TL;DR

This paper establishes precise boundary decay estimates for eigenfunctions of elliptic operators and their derivatives, and analyzes how eigenvalues change with slight domain reductions, using Hardy inequalities.

Contribution

It provides sharp boundary decay bounds and eigenvalue variation estimates for elliptic operators, employing Hardy inequalities, which advances understanding of boundary behavior in elliptic PDEs.

Findings

Sharp L^2 boundary decay estimates for eigenfunctions

Bounds on eigenvalue changes under domain reduction

Use of Hardy inequality for boundary estimates

Abstract

We prove sharp L^2 boundary decay estimates for the eigenfunctions of certain second order elliptic operators acting in a bounded region, and of their first order space derivatives, using only the Hardy inequality. We then deduce bounds on the change of the eigenvalues when the region is reduced slightly in size, subject to DBCs.