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

TL;DR

This paper extends the concept of capacitary measures and their compactness properties from classical Sobolev spaces to fractional order Sobolev spaces, enabling improved optimality conditions in related minimization problems.

Contribution

It introduces the extension of capacitary measures and their compactness to fractional Sobolev spaces, broadening the scope of applications in variational problems.

Findings

Extended compactness of capacitary measures to fractional Sobolev spaces.

Applied compactness to derive finer optimality conditions.

Established foundational results for fractional capacity in variational analysis.

Abstract

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