The mixed problem for harmonic functions in polyhedra

TL;DR

This paper extends Brown's theorem on mixed boundary conditions for harmonic functions to more general polyhedral domains with complex boundary partitions, providing new insights into geometric conditions and their necessity.

Contribution

It introduces a broader boundary partition framework and considers non-graph manifold boundaries, expanding the applicability of existing harmonic function theories.

Findings

Extended Brown's theorem to more general polyhedral domains

Constructed examples demonstrating geometric hypothesis necessity

Analyzed implications of boundary partition generalizations

Abstract

R. M. Brown's theorem on mixed Dirichlet and Neumann boundary conditions is extended in two ways for the special case of polyhedral domains. A (1) more general partition of the boundary into Dirichlet and Neumann sets is used on (2) manifold boundaries that are not locally given as the graphs of functions. Examples are constructed to illustrate necessity and other implications of the geometric hypotheses.