A bridge between Sobolev and Escobar inequalities and beyond

Francesco Maggi; Robin Neumayer

doi:10.1016/j.jfa.2017.06.013

A bridge between Sobolev and Escobar inequalities and beyond

Francesco Maggi^*, Robin Neumayer

^*Corresponding author for this work

Mathematics

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

2 Scopus citations

Abstract

The classical Sobolev and Escobar inequalities are embedded into the same one-parameter family of sharp trace-Sobolev inequalities on half-spaces. Equality cases are characterized for each inequality in this family by tweaking a well-known mass transportation argument and lead to a new comparison theorem for trace Sobolev inequalities. The case p=2 corresponds to a family of variational problems on conformally flat metrics which was previously settled by Carlen and Loss with their method of competing symmetries. In this case minimizers interpolate between conformally flat spherical and hyperbolic geometries, passing through the Euclidean geometry defined by the fundamental solution of the Laplacian.

Original language	English (US)
Pages (from-to)	2070-2106
Number of pages	37
Journal	Journal of Functional Analysis
Volume	273
Issue number	6
DOIs	https://doi.org/10.1016/j.jfa.2017.06.013
State	Published - Sep 15 2017

Funding

Acknowledgments. We thank Eric Carlen and Michael Loss for their advice concerning [5], and an anonymous referee for some valuable suggestions. RN supported by the NSF Graduate Research Fellowship under Grant DGE-1110007. FM supported by the NSF Grants DMS-1265910 and DMS-1361122.

Keywords

Escobar inequality
Mass transportation
Sobolev inequality
Yamabe problem

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2017.06.013

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abstract = "The classical Sobolev and Escobar inequalities are embedded into the same one-parameter family of sharp trace-Sobolev inequalities on half-spaces. Equality cases are characterized for each inequality in this family by tweaking a well-known mass transportation argument and lead to a new comparison theorem for trace Sobolev inequalities. The case p=2 corresponds to a family of variational problems on conformally flat metrics which was previously settled by Carlen and Loss with their method of competing symmetries. In this case minimizers interpolate between conformally flat spherical and hyperbolic geometries, passing through the Euclidean geometry defined by the fundamental solution of the Laplacian.",

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AB - The classical Sobolev and Escobar inequalities are embedded into the same one-parameter family of sharp trace-Sobolev inequalities on half-spaces. Equality cases are characterized for each inequality in this family by tweaking a well-known mass transportation argument and lead to a new comparison theorem for trace Sobolev inequalities. The case p=2 corresponds to a family of variational problems on conformally flat metrics which was previously settled by Carlen and Loss with their method of competing symmetries. In this case minimizers interpolate between conformally flat spherical and hyperbolic geometries, passing through the Euclidean geometry defined by the fundamental solution of the Laplacian.

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KW - Sobolev inequality

KW - Yamabe problem

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A bridge between Sobolev and Escobar inequalities and beyond

Abstract

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Keywords

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