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The best Sobolev trace constant in domains with holes for critical or subcritical exponents

Published online by Cambridge University Press:  17 February 2009

J. Fernandezbonder ,
R. Orive  and
J. D. Rossi
J. Fernandezbonder
Affiliation:
Departamento de Matematica FCEYN Universidad de Buenos Aires Pabellon I Ciudad Universitaria (1428), Buenos Aires Argentina; email: jfbonder@dm.uba.ar.
R. Orive
Affiliation:
Departamento de Matematicas Universidad Autonoma de Madrid Crta Colmenar Viejo km 15 28049Madrid Spain; email: rafael.orive@uam.es.
J. D. Rossi
Affiliation:
Instituto de Matematicas y Fi′sica Fundamental Consejo Superior de Investigaciones Cientfficas Serrano 123 Madrid Spain on leave from Departamento de Matematica FCEyN UBA (1428), Buenos Aires Argentina email: jrossi@dm.uba.ar.
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Abstract

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In this paper we study the best constant in the Sobolev trace embedding H1 (Ω) →Lq(∂Ω) in a bounded smooth domain for 1 < q < 2+ = 2(N - 1)/(N - 2), that is, critical or subcritical q. First, we consider a domain with periodically distributed holes inside which we impose that the involved functions vanish. There exists a critical size of the holes for which the limit problem has an extra term. For sizes larger than critical the best trace constant diverges to infinity and for sizes smaller than critical it converges to the best constant in the domain without holes. Also, we study the problem with the holes located on the boundary of the domain. In this case another critical exists and its extra term appears on the boundary.

Keywords

homogenizationnonlinear boundary conditionsSobolev trace embedding.
Type
Articles
Information
The ANZIAM Journal , Volume 49 , Issue 2 , October 2007 , pp. 213 - 230
Copyright
Copyright © Australian Mathematical Society 2007

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