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arXiv · 2412.11876

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

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

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Anna Lentz. 2024-12-16. Capacitary measures in fractional order Sobolev spaces: Compactness and applications to minimization problems. https://arxiv.org/abs/2412.11876

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