Renormalized solutions of semilinear elliptic equations with general measure data

TL;DR

This paper introduces a new definition of renormalized solutions for semilinear elliptic equations with general measure data, establishing their equivalence with existing solution concepts under mild conditions.

Contribution

It proposes a unified framework for renormalized solutions involving general Dirichlet forms and compares it with other solution notions in the literature.

Findings

All solution concepts coincide under mild integrability conditions.

The new definition applies to equations with nonlocal operators and nonsmooth measures.

The paper clarifies relationships between different solution frameworks.

Abstract

In the paper, we first propose a definition of renormalized solution of semilinear elliptic equation involving operator corresponding to a general (possibly nonlocal) symmetric regular Dirichlet form satisfying the so-called absolute continuity condition and general (possibly nonsmooth) measure data. Then we analyze the relationship between our definition and other concepts of solutions considered in the literature (probabilistic solutions, solution defined via the resolvent kernel of the underlying Dirichlet form, Stampacchia's definition by duality). We show that under mild integrability assumption on the data all these concepts coincide.