Towards a Fully Nonlinear Sharp Sobolev Trace Inequality (Journal of Mathematical Study, 2020)
We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $n=4$ or $n=5$, 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.
Keywords: conformally covariant operator; boundary operator; $σ_k$-curvature; Sobolev trace inequality; fully nonlinear PDE.