ASYMPTOTIC BEHAVIOR OF THE BEST SOBOLEV TRACE CONSTANT IN EXPANDING AND CONTRACTING DOMAINS

We study the asymptotic behavior for the best constant and ex- tremals of the Sobolev trace embedding W 1,p () ,! L q (@) 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 be- havior 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.

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