Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains

Abstract

Abstract. We study the asymptotic behavior for the best constant and extremals of the Sobolev trace embedding W 1,p (Ω) → L q (∂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.

Asymptotic Behavior of the Steklov Eigenvalues For the p−Laplace Operator

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“…functions where the infimum is attained. These extremals are strictly positive in Ω (see [14]) and smooth up to the boundary (see [6]). When one normalize the extremals with…”

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Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

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