A boundary formula for reproducing kernel Hilbert spaces of real harmonic functions in Lipschitz domains

TL;DR

This paper introduces a novel Hilbert space framework for real harmonic functions in Lipschitz domains, providing boundary representations, reproducing kernels, and solutions for the Dirichlet problem with $L^2$ boundary data.

Contribution

It develops a new Hilbert space approach to characterize harmonic function spaces, including boundary representations and kernel constructions, in Lipschitz domains.

Findings

Boundary integral representation for weak Dirichlet problem solutions

Explicit reproducing kernels and orthonormal bases for harmonic spaces

Characterization of harmonic spaces via trace data and inner products

Abstract

This paper develops a new Hilbert space method to characterize a family of reproducing kernel Hilbert spaces of real harmonic functions in a bounded Lipschitz domain $\Omega \subset \mathbb R^d, d\geq 2$ involving some families of positive self-adjoint operators and making use of characterizations of their trace data and of a special inner product on $H^1(\Omega).$ We also establish boundary representation results for this family in terms of the $L^2-$ Bergman kernel. In particular, a boundary integral representation for the very weak solution of the Dirichlet problem for Laplace's equation with $L^2- $ boundary data is provided. Reproducing kernels and orthonormal bases for the harmonic spaces are also found.