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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2409.09880 |
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| _version_ | 1866912127504089088 |
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| author | Del Nin, Giacomo Wu, Bian |
| author_facet | Del Nin, Giacomo Wu, Bian |
| contents | Given any divergence-free vector field of Sobolev class $W^{m,p}_0(Ω)$ in a bounded open subset $Ω\subset \mathbb{R}^2$, we are interested in approximating it in the $W^{m,p}$ norm with divergence-free smooth vector fields compactly supported in $Ω$. We show that this approximation property holds in the following cases: For $p>2$, this holds given that $\partial Ω$ has zero Lebesgue measure (a weaker but more technical condition is sufficient); For $p \leq 2$, this holds if $Ω^c$ can be decomposed into finitely many disjoint closed sets, each of which is connected or $d$-Ahlfors regular for some $d\in[0,2)$. This has links to the uniqueness of weak solutions to the Stokes equation in $Ω$. For Hölder spaces, we prove this approximation property in general bounded domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09880 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation of divergence-free vector fields vanishing on rough planar sets Del Nin, Giacomo Wu, Bian Analysis of PDEs 46E35, 41A30, 31A99, 76D07 Given any divergence-free vector field of Sobolev class $W^{m,p}_0(Ω)$ in a bounded open subset $Ω\subset \mathbb{R}^2$, we are interested in approximating it in the $W^{m,p}$ norm with divergence-free smooth vector fields compactly supported in $Ω$. We show that this approximation property holds in the following cases: For $p>2$, this holds given that $\partial Ω$ has zero Lebesgue measure (a weaker but more technical condition is sufficient); For $p \leq 2$, this holds if $Ω^c$ can be decomposed into finitely many disjoint closed sets, each of which is connected or $d$-Ahlfors regular for some $d\in[0,2)$. This has links to the uniqueness of weak solutions to the Stokes equation in $Ω$. For Hölder spaces, we prove this approximation property in general bounded domains. |
| title | Approximation of divergence-free vector fields vanishing on rough planar sets |
| topic | Analysis of PDEs 46E35, 41A30, 31A99, 76D07 |
| url | https://arxiv.org/abs/2409.09880 |