Representation theorems for nonvariational solutions of the Helmholtz equation

TL;DR

This paper develops representation theorems for nonvariational solutions of the Helmholtz equation in bounded domains, focusing on solutions with limited regularity and extending classical potential theory methods.

Contribution

It introduces integral representation results for solutions lacking classical normal derivatives, broadening the scope of Helmholtz equation analysis beyond traditional variational frameworks.

Findings

Representation theorems for nonvariational solutions using acoustic layer potentials.

Extension of potential theory to solutions with Hölder continuity and infinite boundary integrals.

Analysis applicable to multiply connected domains with limited boundary regularity.

Abstract

We consider a possibly multiply connected bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{\max\{1,m\},\alpha}$ for some $m\in {\mathbb{N}}$, $\alpha\in]0,1[$ and we plan to solve both the Dirichlet and the Neumann problem for the Helmholtz equation in $\Omega$ and in the exterior of $\Omega$ in terms of acoustic layer potentials. Then we turn to prove an integral representation theorem solutions of the Helmholtz equation in terms of an acoustic single layer potential. The main focus of the paper is on $\alpha$-H\"{o}lder continuous solutions which may not have a classical normal derivative at the boundary points of $\Omega$ and that may have an infinite Dirichlet integral around the boundary of $\Omega$\, \textit{i.e.}, case $m=0$. Namely for solutions that do not belong to the classical variational setting.