# American Institute of Mathematical Sciences

September  2002, 1(3): 359-378. doi: 10.3934/cpaa.2002.1.359

## Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains

 1 Departamento de Matematica, FCEyN, UBA, 1428 Buenos Aires, Argentina 2 Departamento de Matemática, F.C.E y N. UBA (1428) Buenos Aires, Argentina

Received  November 2001 Revised  April 2002 Published  June 2002

We study the asymptotic behavior for the best constant and extremals of the Sobolev trace embedding $W^{1,p} (\Omega) \rightarrow L^q (\partial \Omega)$ on expanding and contracting domains. We find that the behavior strongly depends on $p$ and $q$. For contracting domains we prove that the behavior of the best Sobolev trace constant depends on the sign of $qN-pN+p$ while for expanding domains it depends on the sign of $q-p$. We also give some results regarding the behavior of the extremals, for contracting domains we prove that they converge to a constant when rescaled in a suitable way and for expanding domains we observe when a concentration phenomena takes place.
Citation: Julián Fernández Bonder, Julio D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Communications on Pure & Applied Analysis, 2002, 1 (3) : 359-378. doi: 10.3934/cpaa.2002.1.359
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