The behavior of the best Sobolev trace constant and extremals in thin domains. (Q1428437)

The goal of the authors is to study the dependence of the best constant \(S_{p,q}(\Omega)\) and extremals on the domain, where \[ S_{p,q}(\Omega)= \inf_{u\in W^{1,p}(\Omega)\setminus\{0\}} {\int_\Omega(|\nabla u|^p+|u|^p)\,dx\over (\int_{\partial\Omega}|u|^q d\sigma)^{p,q}}. \] The authors focus their attention on this domain. They show that, when the domain \(\Omega\) (\(\Omega\) is a smooth bounded domain in \(\mathbb R^N\)) is very narrow, the problem of looking at the trace of a function is ``equivalent'' to the problem of the immersion of the function in the projection of the domain over the transversal variables.