An observation on eigenfunctions of the Laplacian
Agnid Banerjee, Nicola Garofalo · arXiv (Cornell University) · 2023
In his seminal 1943 paper F. Rellich proved that, in the complement of a cavity $Ω= \{x\in \mathbb R^n\mid |x|>R_0\}$, there exist no nontrivial solution $f$ of the Helmholtz equation $Δf = - λf$, when $λ>0$, such that $\int_Ω |f|^2 dx < \infty$. In this note we generalise this result by showing that if $\int_Ω |f|^p dx < \infty$ for some $0 \frac{2n}{n-1}$, eigenfunctions do exist in $Ω$.