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 $0<p\leq \frac{2n}{n-1}$, then $f\equiv 0$. This result is sharp since for any $p> \frac{2n}{n-1}$, eigenfunctions do exist in $\Omega$.