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

TL;DR

This paper proves that the first eigenfunction of the symmetric stable process in a bounded Lipschitz domain is superharmonic for specific values of alpha, extending known results from special cases to more general domains and parameters.

Contribution

It provides a new proof that the first eigenfunction is superharmonic for alpha=2/m in Lipschitz domains, generalizing previous results limited to balls and specific alpha values.

Findings

Eigenfunction is superharmonic for alpha=2/m in Lipschitz domains.

Extends superharmonicity results from balls to Lipschitz domains.

Provides a unified proof for specific alpha values.

Abstract

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