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

摘要

In this paper we study homogenisation problems for Sobolev trace embedding H-1 (Omega) hooked right arrow L-q(partial derivative Omega) 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 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 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 a(epsilon)(x, del u) with non-linear Neumann boundary conditions. In particular, we can deal with the embedding W-1.P(Omega) hooked right arrow L-q(partial derivative Omega).