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

• Published in: The ANZIAM journal Vol. 49; no. 2; pp. 213 - 230
• Main Authors: Fernandezbonder, J., Orive, R., Rossi, J. D.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01-10-2007
• Subjects: Domains; homogenization; nonlinear boundary conditions; Sobolev trace embedding; homogenization; nonlinear boundary conditions; Sobolev trace embedding
• ISSN: 1446-1811; 1446-8735
• DOI: 10.1017/S1446181100012797

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.