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Towards a Fully Nonlinear Sharp Sobolev Trace Inequality

Towards a Fully Nonlinear Sharp Sobolev Trace Inequality
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Year:    2020

Author:    Jeffrey S. Case, Yi Wang

Journal of Mathematical Study, Vol. 53 (2020), Iss. 4 : pp. 402–435

Abstract

We classify local minimizers of $\int\sigma_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $n=4$ or $n=5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank-Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v53n4.20.02

Journal of Mathematical Study, Vol. 53 (2020), Iss. 4 : pp. 402–435

Published online:    2020-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    34

Keywords:    conformally covariant operator boundary operator $\sigma_k$-curvature Sobolev trace inequality fully nonlinear PDE.

Author Details

Jeffrey S. Case

Yi Wang

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