A uniqueness theorem for nonvariational solutions of the Helmholtz equation

We consider a bounded open subset $\mathrm{\Omega }$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha }$ for some $\alpha \in ]0,1[$, and we define a distributional outward unit normal derivative for $\alpha$-Hölder continuous solutions of the Helmholtz equation in the exterior of $\mathrm{\Omega }$ that may not have a classical outward unit normal derivative at the boundary points of $\mathrm{\Omega }$ and that may have an infinite Dirichlet integral around the boundary of $\mathrm{\Omega }$. Namely for solutions that do not belong to the classical variational setting. Then we show a Schauder boundary regularity result for $\alpha$-Hölder continuous functions that have the Laplace operator in a Schauder space of negative exponent and we prove a uniqueness theorem for $\alpha$-Hölder continuous solutions of the exterior Dirichlet and impedance boundary value problems for the Helmholtz equation that satisfy the Sommerfeld radiation condition at infinity in the above mentioned nonvariational setting.