The Perron solution for vector-valued equations

Abstract

Given a continuous function on the boundary of a bounded open set in $$\mathbb {R}^d$$ there exists a unique bounded harmonic function, called the Perron solution, taking the prescribed boundary values at least at all regular points (in the sense of Wiener) of the boundary. We extend this result to vector-valued functions and consider several methods of constructing the Perron solution which are classical in the real-valued case. Special emphasis is on the case where the codomain is a Banach lattice. In this case we investigate Perron’s classical construction via sub- and supersolutions. We also apply our results to solve elliptic and parabolic boundary value problems of vector-valued functions.