Isoperimetric inequalities and variations on Schwarz's lemma

TL;DR

This paper extends Schwarz's lemma to the first eigenvalues of the Laplacian with Dirichlet boundary conditions, using isoperimetric inequalities for eigenfunctions and conformal metrics.

Contribution

It introduces a novel version of Schwarz's lemma for eigenvalues, leveraging isoperimetric inequalities for eigenfunctions and conformal metrics.

Findings

Proves a new Schwarz lemma for Laplacian eigenvalues

Reinterprets Payne and Rayner's eigenfunction inequality as a conformal metric inequality

Establishes isoperimetric inequalities for eigenfunctions in planar domains

Abstract

In this note we prove a version of the classical Schwarz lemma for the first eigenvalues of the Laplacian with Dirichlet boundary data. A key ingredient in our proof is an isoperimetric inequality for the first eigenfunction, due to Payne and Rayner, which we reinterpret as an isoperimetric inequality for a (singular) conformal metric on a bounded domain in the plane.