Reduced measures for semilinear elliptic equations involving Dirichlet operators

TL;DR

This paper studies solutions to semilinear elliptic equations involving Dirichlet operators, introducing a probabilistic solution concept, analyzing measure properties, and characterizing good measures for specific nonlinearities.

Contribution

It extends the theory of good and reduced measures to general Dirichlet operators and develops a probabilistic framework for solutions.

Findings

Introduces a probabilistic definition of solutions for (E).

Establishes basic properties and inequalities for solutions.

Characterizes good measures for nonlinearities of the form -u^p.

Abstract

We consider elliptic equations of the form (E) $-Au=f(x,u)+\mu$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $\mu$ is a Borel measure of finite potential. We introduce a probabilistic definition of a solution of (E), develop the theory of good and reduced measures introduced by H. Brezis, M. Marcus and A.C. Ponce in the case where $A=\Delta$ and show basic properties of solutions of (E). We also prove Kato's type inequality. Finally, we characterize the set of good measures in case $f(u)=-u^p$ for some $p>1$.