The best constant and extremals of the Sobolev embeddings in domains with holes: The L∞ case

Abstract

Let Ω⊂ℝ^N be a bounded domain. We study the best constant of the Sobolev trace embedding W^1,∞(Ω)↪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_A^T=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_A^T 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 x₀^* on ∂Ω with respect to the arc-length metric in Ω and the best holes are of the form A^*=Ω∖B_d(x₀^*, 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)↪L^∞(Ω) with holes. In this case, we also find that minimizers associated to optimal holes are cones centered at some points x₀^* on ∂Ω and the best holes are of the form A^*=Ω∖B_d(x₀^*, r^*).