The Faber-Krahn inequality for the Hermite operator with Robin boundary condition

TL;DR

This paper establishes a Faber-Krahn inequality for the first eigenvalue of the Hermite operator with Robin boundary conditions, identifying the optimal domain as a half-space and analyzing the equality case.

Contribution

It extends Faber-Krahn inequalities to the Hermite operator with Robin boundary conditions, identifying the optimal domain and characterizing equality cases.

Findings

The optimal domain for the first eigenvalue is a half-space.

The inequality is proven for the Hermite operator with Robin boundary conditions.

The equality case in the inequality is characterized.

Abstract

In this paper we prove a Faber-Krahn type inequality for the first eigenvalue of the Hermite operator with Robin boundary condition. We prove that the optimal set is an half-space and we also address the equality case in such inequality.