The Best Sobolev Trace Constant as Limit of the Usual Sobolev Constant for Small Strips near the Boundary

In this paper we prove that the best constant in the Sobolev trace embedding H(Ω) ↪→ L(∂Ω) in a bounded smooth domain can be obtained as the limit as ε → 0 of the best constant of the usual Sobolev embedding H(Ω) ↪→ L(ωε, dx/ε) where ωε = {x ∈ Ω : dist(x, ∂Ω) < ε} is a small neighborhood of the boundary. We also analyze symmetry properties of extremals of this last embedding when Ω is a ball.