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, analyzing boundary traces, and addressing questions about the Poisson kernel's vanishing set.

Contribution

It introduces specific capacity conditions for measure solvability and explores the boundary trace and reduced measure for elliptic equations with singular potentials.

Findings

Provided sufficient conditions on Radon measures for problem solvability

Analyzed the boundary trace of positive solutions

Addressed the vanishing set of the Poisson kernel for certain potentials

Abstract

We study the boundary value problem with Radon measures for nonnegative solutions of $L_Vu:=-\Delta u+Vu=0$ 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. In the appendix A. Ancona solves a question raised by M. Marcus and L. V\'eron concerning the vanishing set of the Poisson kernel of $L_V$ for an important class of potentials $V$.