A uniqueness theorem for nonvariational solutions of the Helmholtz equation

We consider a bounded open subset $\Omega$ of (Formula presented.) of class (Formula presented.) for some (Formula presented.), and we define a distributional outward unit normal derivative for α-Hölder continuous solutions of the Helmholtz equation in the exterior of $\Omega$ that may not have a classical outward unit normal derivative at the boundary points of $\Omega$and that may have an infinite Dirichlet integral around the boundary of $\Omega$. Namely for solutions that do not belong to the classical variational setting. Then we show a Schauder boundary regularity result for α-Hölder continuous functions that have the Laplace operator in a Schauder space of negative exponent and we prove a uniqueness theorem for α-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.