A nonvariational form of the acoustic single layer potential

Abstract

We consider a bounded open subset $\hat{I}\text{c◯}$ of ${\mathbb{R}}^n$ of class $C^{1,\hat{I}\pm }$ for some $\hat{I}\pm \in ]0,1[$ and the space $V^{-1,\hat{I}\pm }(\partial \hat{I}\text{c◯})$ of (distributional) normal derivatives on the boundary of $\hat{I}\pm$-Hölder continuous functions in $\hat{I}\text{c◯}$ that have Laplace operator in the Schauder space with negative exponent $C^{-1,\hat{I}\pm }(\overline{\hat{I}}\text{c◯})$. Then we prove those properties of the acoustic single layer potential that are necessary to analyze the Neumann problem for the Helmholtz equation in $\hat{I}\text{c◯}$ with boundary data in $V^{-1,\hat{I}\pm }(\partial \hat{I}\text{c◯})$ and solutions in the space of $\hat{I}\pm$-Hölder continuous functions in $\hat{I}\text{c◯}$ that have Laplace operator in $C^{-1,\hat{I}\pm }(\overline{\hat{I}}\text{c◯})$, \textit{i.e.}, in a space of functions that may have infinite Dirichlet integral. Namely, a Neumann problem that does not belong to the classical variational setting.

arXiv admin note: substantial text overlap with arXiv:2504.11487; text overlap with arXiv:2408.17192