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 Frank--Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2019
- DOI:
- 10.48550/arXiv.1910.14232
- arXiv:
- arXiv:1910.14232
- Bibcode:
- 2019arXiv191014232C
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Differential Geometry;
- 58J32;
- 53C21;
- 35J66;
- 58E11
- E-Print:
- 25 pages