Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms

TL;DR

This paper studies semilinear elliptic equations involving measure data and quasi-regular Dirichlet forms, establishing existence, uniqueness, and regularity of solutions via probabilistic methods, with applications to various differential operators.

Contribution

It provides general results on solutions to such equations with measure data, extending to nonsymmetric and semi-Dirichlet forms, using backward stochastic differential equations.

Findings

Proved existence and uniqueness of solutions.

Established regularity properties of solutions.

Applied results to diverse operators including divergence form and Ornstein-Uhlenbeck.

Abstract

We are mainly concerned with equations of the form $-Lu=f(x,u)+\mu$, where $L$ is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, $f$ satisfies the monotonicity condition and mild integrability conditions, and $\mu$ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric divergence form operators, with gradient perturbations of some pseudodifferential operators and equations with Ornstein-Uhlenbeck type operators in Hilbert spaces. We also briefly discuss the existence and uniqueness of probabilistic solutions in the case where $L$ corresponds to a lower bounded semi-Dirichlet form.