Isoperimetric inequalities and variations on Schwarz's lemma

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.

reproves results of Laugesen and Morpurgo by different methods, 9 pages, 3 figures

Cited by in corpus (1)

• The Semi-linear Torsional Rigidity on a Complete Riemannian Two-Manifold