Two isoperimetric inequalities for the Sobolev constant |
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| Authors: | Tom Carroll Jesse Ratzkin |
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| Institution: | 1. Department of Mathematics, University College Cork, Cork, Ireland
2. Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Rondebosch, 7701, South Africa
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| 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. |
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