Semilinear elliptic systems with measure data

TL;DR

This paper investigates the existence and uniqueness of solutions to semilinear elliptic systems with measure data, employing probabilistic Dirichlet forms theory and providing a stochastic representation of solutions.

Contribution

It introduces a novel approach using probabilistic Dirichlet forms to establish solution existence and uniqueness for elliptic systems with measure data.

Findings

Unique solutions exist in generalized Sobolev spaces.

Solutions can be represented stochastically.

The method applies to nonregular domains.

Abstract

We study the Dirichlet problem for systems of the form -\Delta u^k=f^k(x,u)+\mu^k, x\in\Omega, k=1,...,n, where \Omega\subset R^d$ is an open (possibly nonregular) bounded set, \mu^1,...,\mu^n are bounded diffuse measures on \Omega, f=(f^1,...,f^n) satisfies some mild integrability condition and the so-called angle condition. Using the methods of probabilistic Dirichlet forms theory we show that the system has a unique solution in the generalized Sobolev space i.e. space of functions having fine gradient. We provide also a stochastic representation of the solution.