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Two isoperimetric inequalities for the Sobolev constant

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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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Authors and Affiliations

  1. Department of Mathematics, University College Cork, Cork, Ireland

    Tom Carroll

  2. Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Rondebosch, 7701, South Africa

    Jesse Ratzkin

Authors
  1. Tom Carroll
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  2. Jesse Ratzkin
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Correspondence to Tom Carroll.

Additional information

T.C. is partially supported by the ESF Network ‘Harmonic and Complex Analysis and Applications’ (HCAA). J.R. is partially supported by the University of Cape Town Research Committee.

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Carroll, T., Ratzkin, J. Two isoperimetric inequalities for the Sobolev constant. Z. Angew. Math. Phys. 63, 855–863 (2012). https://doi.org/10.1007/s00033-012-0198-8

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  • DOI: https://doi.org/10.1007/s00033-012-0198-8

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