The Perron solution for vector-valued equations

TL;DR

This paper extends the classical Perron solution concept from scalar to vector-valued harmonic functions, providing methods for construction and applications to vector-valued elliptic and parabolic boundary value problems.

Contribution

It generalizes the Perron solution to vector-valued functions and explores multiple classical construction methods for these solutions.

Findings

Existence and uniqueness of vector-valued Perron solutions

Development of classical construction methods for vector-valued cases

Application to elliptic and parabolic boundary value problems

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. We also apply our results to solve elliptic and parabolic boundary value problems of vector-valued functions.