Dirichlet forms and semilinear elliptic equations with measure data

TL;DR

This paper introduces a probabilistic framework for solving semilinear elliptic equations with measure data using Dirichlet forms and backward stochastic differential equations, establishing existence, uniqueness, and regularity of solutions.

Contribution

It provides a novel probabilistic approach to semilinear elliptic equations with measure data, extending the theory to nonlocal operators and establishing key existence and regularity results.

Findings

Existence and uniqueness of solutions under monotonicity and integrability conditions.

Regularity results for solutions with smooth measures.

Applications demonstrating the framework's versatility.

Abstract

We propose a probabilistic definition of solutions of semilinear elliptic equations with (possibly nonlocal) operators associated with regular Dirichlet forms and with measure data. Using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right-hand side of the equation is monotone and satisfies mild integrability assumption, and the measure is smooth. We also study regularity of solutions under the assumption that the measure is smooth and has finite total variation. Some applications of our general results are given.