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, $4\le n\le 5$, 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.

25 pages

Cited by in corpus (1)

• The Frank-Lieb approach to sharp Sobolev inequalities