Comportamiento de la mejor constante de trazas de Sobolev y extremales en dominios angostos

FERNÁNDEZ BONDER, JULIÁN; MARTÍNEZ, SANDRA RITA; ROSSI, JULIO D

Universidad Catolica de hile, Chile

Workshop; First Workshop on Elliptic and Parabolic Partial Differential Equations; 2003

In this paper we study the asymptotic behavior of the best Sobolev trace constant andextremals for the immersion $W^{1,p}(Omega)hookrightarrow L^q(partialOmega)$in a bounded smooth domain when it is contracted in one direction. We find thatthe limit problem, when rescaled in a suitable way, is a Sobolev type immersion inweighted spaces over a projection of $Omega$,$W^{1,p} (P(Omega), alpha) hookrightarrow L^q (P(Omega), eta)$. For the special case $p=q$, this problem leads to an eigenvalueproblem with a nonlinear boundary condition. We also study theconvergence of the eigenvalues and eigenvectors in this case.