Two isoperimetric inequalities for the Sobolev constant

Abstract

In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalises an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.

11 pages

References in corpus (2)

• Isoperimetric inequalities and variations on Schwarz's lemma
• Isoperimetric Inequalities and Sharp Estimate for Positive Solution of Sublinear Elliptic Equations

Cited by in corpus (5)

• Generic properties of free boundary problems in plasma physics
• On the rate of change of the best constant in the Sobolev inequality
• Self-commutators of Toeplitz operators and isoperimetric inequalities
• An isoperimetric inequality for extremal Sobolev functions
• A reverse Holder inequality for extremal Sobolev functions