In this manuscript we study the following optimization problem: given a bounded and regular domain Ω ⊂ R N we look for an optimal shape for the "W−vanishing window" on the boundary with prescribed measure over all admissible profiles in the framework of the Orlicz-Sobolev spaces associated to constant for the "Sobolev trace embedding". In this direction, we establish existence of minimizer profiles and optimal sets, as well as we obtain further properties for such extremals. Finally, we also place special emphasis on analyzing the corresponding optimization problem involving an "A−vanishing hole" (inside the domain) with volume constraint.

Abstract. In this paper we give sufficient conditions on the approximating domains in order to obtain the continuity of solutions for the fractional p−laplacian. These conditions are given in terms of the fractional capacity of the approximating domains.

Abstract. In this paper we study two optimal design problems associated to fractional Sobolev spaces W s,p (Ω). Then we find a relationship between these two problems and finally we investigate the convergence when s ↑ 1.