A nonvariational form of the acoustic single layer potential

Abstract

We consider a bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1[$ and the space $V^{-1,\alpha}(\partial\Omega)$ of (distributional) normal derivatives on the boundary of $\alpha$-H\"{o}lder continuous functions in $\Omega$ that have Laplace operator in the Schauder space with negative exponent $C^{-1,\alpha}(\overline{\Omega})$. Then we prove those properties of the acoustic single layer potential that are necessary to analyze the Neumann problem for the Helmholtz equation in $\Omega$ with boundary data in $V^{-1,\alpha}(\partial\Omega)$ and solutions in the space of $\alpha$-H\"{o}lder continuous functions in $\Omega$ that have Laplace operator in $C^{-1,\alpha}(\overline{\Omega})$, \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.