TY - JOUR
T1 - Towards a Fully Nonlinear Sharp Sobolev Trace Inequality
AU - Case , Jeffrey S.
AU - Wang , Yi
JO - Journal of Mathematical Study
VL - 4
SP - 402
EP - 435
PY - 2020
DA - 2020/12
SN - 53
DO - http://doi.org/10.4208/jms.v53n4.20.02
UR - https://global-sci.org/intro/article_detail/jms/18508.html
KW - conformally covariant operator, boundary operator, $\sigma_k$-curvature, Sobolev trace inequality, fully nonlinear PDE.
AB - <p style="text-align: justify;">We classify local minimizers of $\int\sigma_2+\oint H_2$ 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.</p>