A uniqueness theorem for nonvariational solutions of the Helmholtz equation

Abstract

We consider a bounded open subset $\Omega$ 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 $\Omega$ 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 $\alpha$-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.

arXiv commentarXiv admin note: text overlap with arXiv:2405.01818, arXiv:2403.15057