On a fully nonlinear sharp Sobolev trace inequality
Abstract
We classify local minimizers of $\int\sigma_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$ball, $4\leq n\leq 5$, 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 FrankLieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obatatype arguments.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.14232
 Bibcode:
 2019arXiv191014232C
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 58J32;
 53C21;
 35J66;
 58E11
 EPrint:
 25 pages