ANZIAM J. 49 (2007), no. 2, pp. 213–230.
The best Sobolev trace constant in domains with holes for critical or subcritical exponents
J. Fernández Bonder J. D. Rossi†
Departamento de Matemática
FCEyN, Universidad de Buenos Aires
Pabellon I
Ciudad Universitaria (1428)
Buenos Aires
Argentina
jfbonder@dm.uba.ar		 Instituto de Matemáticas y Física Fundamental
Consejo Superior de Investigaciones Científicas
Serrano 123
Madrid
Spain

on leave from

Departamento de Matemática
FCEyN UBA (1428)
Buenos Aires
Argentina
jrossi@dm.uba.ar
R. Orive
Departamento de Matemáticas
Universidad Autonoma de Madrid Crta.
Colmenar Viejo km. 15
28049 Madrid
Spain
rafael.orive@uam.es
Received 8 November, 2006

Abstract

In this paper we study the best constant in the Sobolev trace embedding H^1(\Omega ) \hookrightarrow L^q (\partial \Omega ) in a bounded smooth domain for 1 < q\le 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.

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2000 Mathematics Subject Classification: primary 35B27, 35J65; secondary 46E35
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2376???
†indicates author for correspondence

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