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

Institut für Mathematik, Universität Würzburg, Emil-Fischer-Straße 30, 97074 Würzburg, Germany

^* Corresponding author: anna.lentz@uni-wuerzburg.de

Received: 11 December 2024
Accepted: 8 August 2025

Abstract

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 H^s(Ω) for s ∈ (0, 1). The compactness result is applied to obtain a finer optimality condition for a class of minimization problems in H^s(Ω).