Boundary Value Problems with Measures for Elliptic Equations with Singular Potentials

TL;DR

This paper investigates boundary value problems involving Radon measures for elliptic equations with singular potentials, establishing conditions for solvability and analyzing boundary traces of solutions.

Contribution

It introduces specific capacity criteria and studies the reduced measure and boundary trace for solutions of elliptic equations with measures and singular potentials.

Findings

Provided sufficient conditions on boundary measures for problem solvability

Analyzed the boundary trace of positive solutions

Studied the reduced measure associated with the elliptic equation

Abstract

We study the boundary value problem with Radon measures for nonnegative solutions of $-\Delta u+Vu=0$ in a bounded smooth domain $\Gw$, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give sufficient conditions on a Radon measure $\gm$ on $\prt\Gw$ so that the problem can be solved. We study the reduced measure associated to this equation as well as the boundary trace of positive solutions.