A uniqueness theorem for nonvariational solutions of the Helmholtz equation
M. Lanza de Cristoforis
TL;DR
This work advances the theory of the Helmholtz equation in exterior domains by developing a nonvariational framework that accommodates Hölder continuous solutions with potentially infinite Dirichlet energy near the boundary. It introduces a distributional outward normal derivative in this setting and proves a Schauder boundary regularity result for solutions whose Laplacian lies in a negative exponent Schauder space. The paper then establishes uniqueness results for exterior Dirichlet and impedance problems under the Sommerfeld radiation condition, without relying on classical variational energy finiteness. By combining a carefully constructed extension/trace theory with boundary integral–type operators, the results extend classical variational theory to nonvariational regimes relevant for exterior scattering problems.
Abstract
We consider a bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{1,α}$ for some $α\in]0,1[$, and we define a distributional outward unit normal derivative for $α$-Hölder continuous solutions of the Helmholtz equation in the exterior of $Ω$ that may not have a classical outward unit normal derivative at the boundary points of $Ω$ and that may have an infinite Dirichlet integral around the boundary of $Ω$. 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.