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.