Quasi-regular Dirichlet forms and the obstacle problem for elliptic equations with measure data

TL;DR

This paper investigates the obstacle problem for semilinear elliptic equations with measure data using quasi-regular Dirichlet forms, establishing existence, uniqueness, regularity, and solution representation.

Contribution

It introduces a novel framework for solving obstacle problems with irregular barriers via quasi-regular Dirichlet forms, extending classical methods.

Findings

Existence and uniqueness of solutions are proven.

Solutions can be represented as envelopes of supersolutions.

Regularity results and Lewy-Stampacchia inequality are established.

Abstract

We consider the obstacle problem with irregular barriers for semilinear elliptic equation involving measure data and operator corresponding to a general quasi-regular Dirichlet form. We prove existence and uniqueness of a solution as well as its representation as an envelope of a supersolution to some related partial differential equation. We also prove regularity results for the solution and the Lewy-Stampacchia inequality.