Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form

TL;DR

This paper extends the concept of renormalized solutions to a broad class of semilinear elliptic and parabolic equations involving operators from general Dirichlet forms, establishing their equivalence with probabilistic solutions and proving existence and uniqueness.

Contribution

It introduces a generalized definition of renormalized solutions for equations with nonlocal operators and measure data, linking them to probabilistic representations and ensuring well-posedness.

Findings

Renormalized solutions coincide with probabilistic solutions under mild conditions.

Existence and uniqueness of solutions are established for a wide class of equations.

The framework applies to both local and nonlocal operators associated with Dirichlet forms.

Abstract

We generalize the notion of renormalized solution to semilinear elliptic and parabolic equations involving operator associated with general (possibly nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We show that under mild integrability assumption on the data a quasi-continuous function $u$ is a renormalized solution to an elliptic (or parabolic) equation in the sense of our definition iff $u$ is its probabilistic solution, i.e. $u$ can be represented by a suitable nonlinear Feynman-Kac formula. This implies in particular that for a broad class of local and nonlocal semilinear equations there exists a unique renormalized solution.