Capacitary measures in fractional order Sobolev spaces: Compactness and applications to minimization problems
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917870172110848 |
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| author | Lentz, Anna |
| author_facet | Lentz, Anna |
| contents | 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 $H^s(Ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11876 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Capacitary measures in fractional order Sobolev spaces: Compactness and applications to minimization problems Lentz, Anna Analysis of PDEs Optimization and Control 49K30, 28A33, 31A15 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 $H^s(Ω)$. |
| title | Capacitary measures in fractional order Sobolev spaces: Compactness and applications to minimization problems |
| topic | Analysis of PDEs Optimization and Control 49K30, 28A33, 31A15 |
| url | https://arxiv.org/abs/2412.11876 |