A Wiener estimate for relaxed Dirichlet problems in dimension $N\geq 2$

TL;DR

This paper establishes a Wiener energy estimate for relaxed Dirichlet problems involving elliptic operators with measure data, extending classical results to more general measure settings in higher dimensions.

Contribution

It introduces a Wiener energy estimate for relaxed Dirichlet problems with measure data, generalizing classical energy estimates to broader measure contexts in multiple dimensions.

Findings

Proves a Wiener energy estimate for relaxed Dirichlet problems.

Extends energy estimates to classical variational Dirichlet problems.

Applicable to elliptic operators with measure data in higher dimensions.

Abstract

We prove a Wiener energy estimate for relaxed Dirichlet problems $Lu + \mu u =\nu$ in $\Omega$, with $L$ an uniformly elliptic operator with bounded coefficients, $\mu$ a measure of ${\cal M}_0(\Omega)$, $\nu$ a Kato measure and $\Omega$ a bounded open set of ${\bf R}^N$, $N \geq 2$. Choosing a particular $\mu$, we obtain an energy estimate also for classical variational Dirichlet problems.