Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form

TL;DR

This paper studies the obstacle problem for semilinear evolution equations with measure data and semi-Dirichlet form operators, establishing existence, uniqueness, and extending theory for irregular barriers with applications to switching problems.

Contribution

It introduces a framework for solving obstacle problems with irregular barriers involving measure data and semi-Dirichlet operators, extending existing theories.

Findings

Proved existence and uniqueness of solutions under monotonicity and integrability conditions.

Extended the theory of precise versions of functions for irregular barriers.

Applied results to switching problems in evolution equations.

Abstract

In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of solutions under the assumption that the right-hand side of the equation is monotone and satisfies mild integrability conditions. To treat the case of irregular barriers, we extend the theory of precise versions of functions introduced by M. Pierre. We also give some applications to the so-called switching problem.