Capacitary measures in fractional order Sobolev spaces: Compactness and applications to minimization problems

Capacitary measures form a class of measures that vanish on sets of capacity zero. These measures are compact with respect to so-called γ-convergence, which relates a sequence of measures to the sequence of solutions of relaxed Dirichlet problems. This compactness result is already known for the classical H1(Ω)-capacity. This paper extends it to the fractional capacity defined for fractional order Sobolev spaces Hs(Ω) for s ∈ (0, 1). The compactness result is applied to obtain a finer optimality condition for a class of minimization problems in Hs(Ω).