The behavior of the best Sobolev trace constant and extremals in thin domains

Abstract

In this paper, we study the asymptotic behavior of the best Sobolev trace constant and extremals for the immersion W 1;p ðOÞ+L q ð@OÞ in a bounded smooth domain when it is contracted in one direction. We find that the limit problem, when rescaled in a suitable way, is a Sobolev-type immersion in weighted spaces over a projection of O; W 1;p ðPðOÞ; aÞ+ L q ðPðOÞ; bÞ:For the special case p ¼ q; this problem leads to an eigenvalue problem with a nonlinear boundary condition. We also study the convergence of the eigenvalues and eigenvectors in this case. r