| Sumario: | LetΩ⊂ R N be a bounded domain. We study the best constant of the Sobolev trace embedding W 1,∞ (Ω) {right arrow, hooked} L ∞ (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole. That is we deal with the minimization problem S T A = inf ||u|| W 1,∞ (Ω)/||u||L ∞ (∂Ω) for functions that verify u|A = 0. We find that there exists an optimal hole that minimizes the best constant S T A among subsets of Ω of prescribed volume and we give a geometrical characterization of this optimal hole. In fact, minimizers associated to these holes are cones centered at some points x* 0 on ∂Ω with respect to the arc-length metric in Ω and the best holes are of the form A* =Ω\\B d (x* 0 , r*) where the ball is taken again with respect of the arc-length metric. A similar analysis can be performed for the best constant of the embedding W 1,∞ (Ω) {right arrow, hooked} L ∞ (Ω) with holes. In this case, we also find that minimizers associated to optimal holes are cones centered at some points x* 0 on ∂Ω and the best holes are of the form A*=Ω\\ B d (x* 0 ,r*). © 2009 University of Illinois.
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