Asymptotic Behavior of the Best Sobolev Trace Constant in Expanding and Contracting Domains

We study the asymptotic behavior for the best constant and extremals of the Sobolev trace embedding W 1,p(Ω) ↪→ Lq(∂Ω) on expanding and contracting domains. We find that the behavior strongly depends on p and q. For contracting domains we prove that the behavior of the best Sobolev trace constant depends on the sign of qN − pN + p while for expanding domains it depends on the sign of q − p. We also give some results regarding the behavior of the extremals, for contracting domains we prove that they converge to a constant when rescaled in a suitable way and for expanding domains we observe when a concentration phenomena takes place.