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arXiv · 1910.14232

On a fully nonlinear sharp Sobolev trace inequality

Jeffrey S. Case·Yi Wang
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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.

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Jeffrey S. Case, Yi Wang. 2019-10-31. On a fully nonlinear sharp Sobolev trace inequality. https://arxiv.org/abs/1910.14232

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