Two isoperimetric inequalities for the Sobolev constant

Details Two isoperimetric inequalities for the Sobolev constant Two isoperimetric inequalities for the Sobolev constant

2011-05-13

arxiv.org/abs/1105.2719v1

Mathematics

Analysis of PDEs

11 pages

Scientific paper

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.

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