Semilinear elliptic systems with measure data

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.