Two isoperimetric inequalities for the Sobolev constant

Tom Carroll; Jesse Ratzkin

doi:10.1007/s00033-012-0198-8

Two isoperimetric inequalities for the Sobolev constant

Tom Carroll
, Jesse Ratzkin

School of Mathematical Sciences

University of Cape Town

Research output: Contribution to journal › Article › peer-review

Abstract

In this note, we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first 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 generalizes an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.

Original language	English
Pages (from-to)	855-863
Number of pages	9
Journal	Zeitschrift fur Angewandte Mathematik und Physik
Volume	63
Issue number	5
DOIs	https://doi.org/10.1007/s00033-012-0198-8
Publication status	Published - Oct 2012

Keywords

Dirichlet eigenvalue
Schwarz Lemma
Sobolev constant
torsional rigidity

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10.1007/s00033-012-0198-8

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KW - Dirichlet eigenvalue

KW - Schwarz Lemma

KW - Sobolev constant

KW - torsional rigidity

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Two isoperimetric inequalities for the Sobolev constant

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Keywords

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