A Faber-Krahn inequality for mixed local and nonlocal operators

TL;DR

This paper proves a Faber-Krahn inequality for a mixed local and nonlocal elliptic operator, showing that balls minimize the first eigenvalue for sets of fixed volume and providing stability results.

Contribution

It establishes a quantitative Faber-Krahn inequality for mixed local/nonlocal operators, including stability estimates for near-minimizers.

Findings

Balls minimize the first eigenvalue among sets of fixed volume.

A stability estimate quantifies how close a set is to a ball if it nearly minimizes the eigenvalue.

The inequality extends classical results to mixed local/nonlocal operators.

Abstract

We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum.