Volume 53, Issue 4
Towards a Fully Nonlinear Sharp Sobolev Trace Inequality

Jeffrey S. Case & Yi Wang

J. Math. Study, 53 (2020), pp. 402-435.

Published online: 2020-12

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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, $n=4$ 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.

  • AMS Subject Headings

58J32, 53C21, 35J66, 58E11

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jscase@psu.edu (Jeffrey S. Case)

ywang@math.jhu.edu (Yi Wang)

  • BibTex
  • RIS
  • TXT
@Article{JMS-53-402, author = {Case , Jeffrey S. and Wang , Yi}, title = {Towards a Fully Nonlinear Sharp Sobolev Trace Inequality}, journal = {Journal of Mathematical Study}, year = {2020}, volume = {53}, number = {4}, pages = {402--435}, abstract = {

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.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v53n4.20.02}, url = {http://global-sci.org/intro/article_detail/jms/18508.html} }
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 -

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.

Jeffrey S. Case & Yi Wang. (2020). Towards a Fully Nonlinear Sharp Sobolev Trace Inequality. Journal of Mathematical Study. 53 (4). 402-435. doi:10.4208/jms.v53n4.20.02
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