The Best Sobolev Trace Constant in Periodic Media for Critical and Subcritical Exponents

Abstract

Abstract. In this paper we study homogenization problems for the Sobolev trace embedding H 1 (Ω) → L q (∂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, we consider H 1 and L q spaces with weights that are periodic in space. We find that extremals for these embeddings converge to a solution of an homogenized limit problem and the best trace constant converges to a homogenized best trace constant. Our results are in fact more general, we can also consider general operators of the form a ε (x, ∇u) with nonlinear Neumann boundary conditions. In particular, we can deal with the embedding W 1,p (Ω) → L q (∂Ω).