On Sobolev spaces of bounded subanalytic manifolds
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2021
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917644011044864 |
|---|---|
| author | Valette, Guillaume |
| author_facet | Valette, Guillaume |
| contents | We focus on the Sobolev spaces of bounded subanalytic submanifolds of $\mathbb{R}^n$.
We prove that if $M$ is such a manifold then the space $\mathscr{C}_0^\infty(M)$ is dense in $W^{1,p}(M,\partial M)$ (the kernel of the trace operator) for all $p\le p_M$, where $p_M$ is the codimension in $M$ of the singular locus of $\overline{M}\setminus M$. In the case where $M$ is normal, i.e. when $B(x_0,\varepsilon)\cap M$ is connected for every $x_0\in\overline{M}$ and $\varepsilon>0$ small, we show that $\mathscr{C}^\infty(\overline{M})$ is dense in $W^{1,p}(M)$ for all such $p$. This yields some duality results between $W^{1,p}(Ω,\partial Ω)$ and $W^{-1,p'}(Ω)$ in the case where $1< p\le p_Ω$ and $Ω$ is a bounded subanalytic open subset of $\mathbb{R}^n$, and consequently that $W^{1,p}(Ω,\partial Ω)$ is reflexive for such $p$. As a byproduct, we deduce uniqueness of the (weak) solution of the Dirichlet problem associated with the Laplace equation. We then prove a version of Sobolev's Embedding Theorem for subanalytic bounded manifolds, show Gagliardo-Nirenberg's inequality (for all $p\in [1,\infty)$), and derive some versions of Poincaré-Friedrichs' inequality. We finish with a generalization of Morrey's Embedding Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_12338 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On Sobolev spaces of bounded subanalytic manifolds Valette, Guillaume Analysis of PDEs Algebraic Geometry 46E35, 32B20, 14P10 We focus on the Sobolev spaces of bounded subanalytic submanifolds of $\mathbb{R}^n$. We prove that if $M$ is such a manifold then the space $\mathscr{C}_0^\infty(M)$ is dense in $W^{1,p}(M,\partial M)$ (the kernel of the trace operator) for all $p\le p_M$, where $p_M$ is the codimension in $M$ of the singular locus of $\overline{M}\setminus M$. In the case where $M$ is normal, i.e. when $B(x_0,\varepsilon)\cap M$ is connected for every $x_0\in\overline{M}$ and $\varepsilon>0$ small, we show that $\mathscr{C}^\infty(\overline{M})$ is dense in $W^{1,p}(M)$ for all such $p$. This yields some duality results between $W^{1,p}(Ω,\partial Ω)$ and $W^{-1,p'}(Ω)$ in the case where $1< p\le p_Ω$ and $Ω$ is a bounded subanalytic open subset of $\mathbb{R}^n$, and consequently that $W^{1,p}(Ω,\partial Ω)$ is reflexive for such $p$. As a byproduct, we deduce uniqueness of the (weak) solution of the Dirichlet problem associated with the Laplace equation. We then prove a version of Sobolev's Embedding Theorem for subanalytic bounded manifolds, show Gagliardo-Nirenberg's inequality (for all $p\in [1,\infty)$), and derive some versions of Poincaré-Friedrichs' inequality. We finish with a generalization of Morrey's Embedding Theorem. |
| title | On Sobolev spaces of bounded subanalytic manifolds |
| topic | Analysis of PDEs Algebraic Geometry 46E35, 32B20, 14P10 |
| url | https://arxiv.org/abs/2111.12338 |