The best Sobolev trace constant in a domain with oscillating boundary

In this paper we study homogenization problems for the best constant for the Sobolev trace embedding W^1,p(Ω)?L^q(∂Ω)$W^{1,p}\left(\Omega \right)?L^q\left(\partial \Omega \right)$ in a bounded smooth domain when the boundary is perturbed by adding an oscillation. We find that there exists a critical size of the amplitude of the oscillations for which the limit problem has a weight on the boundary. For sizes larger than critical the best trace constant goes to zero and for sizes smaller than critical it converges to the best constant in the domain without perturbations.