Isoperimetric inequalities and variations on Schwarz's lemma

Details Isoperimetric inequalities and variations on Schwarz's lemma Isoperimetric inequalities and variations on Schwarz's lemma

2010-06-11

arxiv.org/abs/1006.2310v1

Mathematics

Spectral Theory

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

Scientific paper

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.

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