Representation theorems for nonvariational solutions of the Helmholtz equation

TL;DR

This work develops a comprehensive layer-potential framework for the Helmholtz equation in $Ω$ and its exterior, accommodating $\alpha$-Hölder nonvariational solutions ($m=0$) with potentially infinite Dirichlet energy. It establishes interior and exterior representation theorems expressing solutions via single-layer and double-layer potentials, possibly with distributional densities, and identifies finite-dimensional corrections tied to Dirichlet/Neumann spectra. It then provides nonvariational solvability results for interior/exterior Dirichlet and Neumann problems, together with a unifying representation formula that bijectively maps boundary data supplements to solution spaces, modulo spectral obstructions. The results extend classical variational theory to nonvariational contexts and multi-component exteriors, offering a rigorous basis for applications in singular perturbation problems and electromagnetism.

Abstract

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

Theorems & Definitions (60)

• Definition 2.2
• Proposition 2.4
• Lemma 2.8
• Definition 2.9
• Lemma 2.11
• Definition 2.14
• Theorem 2.16
• Lemma 2.19
• Lemma 2.21
• Lemma 2.23