A nonvariational Neumann problem for the Helmholtz equation

Abstract

We consider a bounded open subset $\hat{I}\text{c◯}$ of ${\mathbb{R}}^n$ of class $C^{1,\hat{I}\pm }$ for some $\hat{I}\pm \in ]0,1[$ and we solve the Neumann problem for the Helmholtz equation both in $\hat{I}\text{c◯}$ and in the exterior of $\hat{I}\text{c◯}$. We look for solutions in the space for $\hat{I}\pm$-Hölder continuous functions that may not have a classical normal derivative at the boundary points of $\hat{I}\text{c◯}$ and that may have an infinite Dirichlet integral around the boundary of $\hat{I}\text{c◯}$. Namely for solutions that do not belong to the classical variational setting.

arXiv admin note: substantial text overlap with arXiv:2504.12349, arXiv:2504.11487; text overlap with arXiv:2405.01818