The best Sobolev trace constant in a domain with oscillating boundary

Abstract

Abstract. In this paper we study the best constant in the Sobolev trace embedding H 1 (Ω) → L q (∂Ω) in a bounded smooth domain for critical or subcritical q, that is 1 < q ≤ 2 * = 2(N − 1)/(N − 2). First, we consider a domain with periodically distributed holes inside where 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 other critical exists and its extra term appears on the boundary.