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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

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

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.
Keywords: Sobolev trace constants, p-Laplacian, eigenvalue problems., nonlinear boundary conditions.
Mathematics Subject Classification: 35J65, 35J20, 35P30, 35P1.
Citation: Julián Fernández Bonder, Julio D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Communications on Pure and Applied Analysis, 2002, 1 (3) : 359-378. doi: 10.3934/cpaa.2002.1.359
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