Isoperimetric inequalities and variations on Schwarz's lemma

2010/06/11 by Tom Carroll, Carroll, Tom, Jesse Ratzkin +1 · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1006.2310

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.

Cited by

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

Related

• An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane
• Isoperimetric Bounds for Eigenvalues of the Wentzell-Laplace, the Laplacian and a biharmonic Steklov Problem
• Some inequalities related to isoperimetric inequalities with partial free boundary
• Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces
• Isoperimetric inequalities for the principal eigenvalue of a membrane and the energy of problems with Robin boundary conditions