On the First Eigenfunction of the Symmetric Stable Process in a Bounded Lipschitz Domain

Abstract

We give a proof that the first eigenfunction of the $\alpha$-symmetric stable process on a bounded Lipschitz domain in ${\mathbb{R}}^d$, $d\ge 1$, is superharmonic for $\alpha =2/m$, where $m>2$ is an integer. This result was first proved for the ball by M. Kaßmann and L. Silvestre (personal communication) with different methods. For $\alpha =1$, the result was proved in \cite[Theorem 4.7]{BanKul}.