Journal
JOURNAL OF MATHEMATICAL STUDY
Volume 53, Issue 4, Pages 402-435Publisher
GLOBAL SCIENCE PRESS
DOI: 10.4208/jms.v53n4.20.02
Keywords
conformally covariant operator; boundary operator; sigma(k)-curvature; Sobolev trace inequality; fully nonlinear PDE
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Funding
- Simons Foundation [524601]
- NSF CAREER Award [DMS-1845033]
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We classify local minimizers of integral sigma(2) + closed integral H-2 among all conformally flat metrics in the Euclidean (n+1)-ball, n= 4 or n= 5, forwhich 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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