The Best Constant and Extremals of the Sobolev Embeddings in Domains with Holes: the L∞ Case

Let Ω ⊂ R be a bounded, convex 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 = inf ‖u‖W1,∞(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 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 and the best holes are of the form A∗ = Ω \B(x0, r∗). 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(x0, r∗).