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

Julián Fernández Bonder; Julio D. Rossi

doi:10.3934/cpaa.2002.1.359

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2002, Volume 1, Issue 3: 359-378. Doi: 10.3934/cpaa.2002.1.359

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

Julián Fernández Bonder^1, and
Julio D. Rossi^2,

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

Received: November 2001

Revised: April 2002

Published: September 2002

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Abstract

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.

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Mathematics Subject Classification: 35J65, 35J20, 35P30, 35P15.

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