An observation on eigenfunctions of the Laplacian

Abstract

In his seminal 1943 paper F. Rellich proved that, in the complement of a cavity $\Omega =\{x\in {\mathbb{R}}^n\mid |x|>R_0\}$, there exist no nontrivial solution $f$ of the Helmholtz equation $\Delta f=-\lambda f$, when $\lambda >0$, such that ${\int }_{\Omega }|f{|}^{2}dx<\infty$. In this note we generalise this result by showing that if ${\int }_{\Omega }|f{|}^{p}dx<\infty$ for some $0<p\le \frac{2n}{n-1}$, then $f\equiv 0$. This result is sharp since for any $p>\frac{2n}{n-1}$, eigenfunctions do exist in $\Omega$.