Semilinear elliptic systems with measure data
arXiv:1303.4930
Abstract
We study the Dirichlet problem for systems of the form -Δu^k=f^k(x,u)+μ^k, x\inΩ, k=1,...,n, where Ω\subset R^d$ is an open (possibly nonregular) bounded set, μ^1,...,μ^n are bounded diffuse measures on Ω, 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.