Representation theorems for nonvariational solutions of the Helmholtz equation

We consider a possibly multiply connected bounded open subset $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ and in the exterior of $\mathrm{\Omega }$ 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 $\alpha$-Hölder continuous solutions which may not have a classical normal derivative at the boundary points of $\mathrm{\Omega }$ and that may have an infinite Dirichlet integral around the boundary of $\mathrm{\Omega }$, $\mathit{\text{i.e.}}$, case $m=0$. Namely for solutions that do not belong to the classical variational setting.