Isoperimetric inequalities for the fractional composite membrane problem

TL;DR

This paper extends classical isoperimetric inequalities to the fractional composite membrane problem, establishing new bounds for the first eigenvalue and exploring geometric properties in the fractional setting.

Contribution

It introduces fractional analogues of the Faber-Krahn inequality and investigates eigenvalue inequalities on intersecting domains, expanding understanding of fractional membrane problems.

Findings

Established a fractional Faber-Krahn inequality.

Derived isoperimetric inequalities for eigenvalues on intersecting domains.

Provided new insights into fractional geometric analysis.

Abstract

In this article, we investigate some isoperimetric-type inequalities related to the first eigenvalue of the fractional composite membrane problem. First, we establish an analogue of the renowned Faber-Krahn inequality for the fractional composite membrane problem. Next, we investigate an isoperimetric inequality for the first eigenvalue of the fractional composite membrane problem on the intersection of two domains-a problem that was first studied by Lieb [23] for the Laplacian. Similar results in the local case were previously obtained by Cupini-Vecchi [9] for the composite membrane problem. Our findings provide further insights into the fractional setting, offering a new perspective on these classical inequalities.