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

Tomasz Klimsiak,Andrzej Rozkosz

doi:10.1007/s00030-015-0350-1

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.

Highlights

The aim of this paper is to extend the notion of renormalized solution to encompass semilinear elliptic and parabolic equations involving measure data and operators associated with Dirichlet forms
In the first one we are concerned with elliptic equations of the form
In (1.1), L is the operator associated with a regular Dirichlet form (E, D(E)) on L2(E; m) and f : E × R → R is a measurable function

Summary

Introduction

The aim of this paper is to extend the notion of renormalized solution to encompass semilinear elliptic and parabolic equations involving measure data and operators associated with Dirichlet forms. Under the assumption that E is transient, we define renormalized solution of (1.1) as a quasi-continuous function u : E → R such that f (·, u) ∈ L1(E; m), Tku ∈ De(E) for k > 0 and there is a sequence {νk} of bounded smooth measures on E such that νk T V → 0 and. In case Lt are local, a definition of a renormalized solution of equations of the form (1.5) involving more general nonlinear local operators Lt of Leray– Lions type but with f not depending on u have been introduced in [12] In [15, 16] definitions of renormalized solutions to (1.5) with Leray–Lions type operators and f depending on u have been proposed (in [16] equations with general, not necessarily smooth measures are considered). In the paper we confine ourselves to equations with operators corresponding to regular forms, but our results can be generalized to quasi-regular forms (see remarks at the end of Sects. 3 and 4)

Preliminaries

Dirichlet forms

Time dependent Dirichlet forms

Markov processes and additive functionals

Elliptic equations

Parabolic equations