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 $\gamma$-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 $H^1(\mathrm{\Omega })$-capacity. This paper extends it to the fractional capacity defined for fractional order Sobolev spaces $H^s(\mathrm{\Omega })$ for $s\in (0,1)$. The compactness result is applied to obtain a finer optimality condition for a class of minimization problems in $H^s(\mathrm{\Omega })$.