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THE BEST SOBOLEV TRACE CONSTANT IN PERIODIC MEDIA FOR CRITICAL AND SUBCRITICAL EXPONENTS

Published online by Cambridge University Press:  01 September 2009

JULIÁN FERNÁNDEZ BONDER ,
RAFAEL ORIVE  and
JULIO D. ROSSI
JULIÁN FERNÁNDEZ BONDER
Affiliation:
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (1428), Buenos Aires, Argentina e-mail: jfbonder@dm.uba.ar, web page: http://mate.dm.uba.ar/~jfbonder
RAFAEL ORIVE
Affiliation:
Departamento de Matemáticas, Universidad Autonoma de Madrid, Crta. Colmenar Viejo km. 15, 28049 Madrid, Spain e-mail: rafael.orive@uam.es, web page: http://www.uam.es/rafael.orive
JULIO D. ROSSI
Affiliation:
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (1428), Buenos Aires, Argentina e-mail: jrossi@dm.uba.ar, web page: http://mate.dm.uba.ar/~jrossi
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Abstract

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In this paper we study homogenisation problems for Sobolev trace embedding H1(Ω) ↪ Lq(∂Ω) in a bounded smooth domain. When q = 2 this leads to a Steklov-like eigenvalue problem. We deal with the best constant of the Sobolev trace embedding in rapidly oscillating periodic media, and we consider H1 and Lq spaces with weights that are periodic in space. We find that extremals for these embeddings converge to a solution of a homogenised limit problem, and the best trace constant converges to a homogenised best trace constant. Our results are in fact more general; we can also consider general operators of the form aɛ(x, ∇u) with non-linear Neumann boundary conditions. In particular, we can deal with the embedding W1,p(Ω) ↪ Lq(∂Ω).

Keywords

35B2735J6546E35
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 3 , September 2009 , pp. 619 - 630
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

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