A Faber--Krahn inequality for indented and cut membranes

TL;DR

This paper extends the Faber--Krahn inequality to domains with cuts and indentations within reflex angles, demonstrating that certain geometric modifications influence the fundamental eigenvalue of the Dirichlet Laplacian.

Contribution

It introduces a Faber--Krahn type inequality for indented and cut membranes within reflex angles, expanding classical results to more complex domain geometries.

Findings

The inequality holds for domains with cuts in reflex angles.

Indented domains also satisfy the inequality.

The results generalize classical Faber--Krahn inequalities.

Abstract

In 1960, Payne and Weinberger proved that among all domains that lie within a wedge (an angle whose measure is less than or equal to $\pi$), and have a given value of a certain integral the circular sector has the lowest fundamental eigenvalue of the Dirichlet Laplacian. Here, it is shown that an analogue of this assertion is true for domains with a cut and for indented domains; that is, for those located in a reflex angle (its measure is between $\pi$ and $2 \pi$).