Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866908747919523840 |
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| author | Pucci, Patrizia Vitillaro, Enzo |
| author_facet | Pucci, Patrizia Vitillaro, Enzo |
| contents | The paper deals with a nontrivial density result for $C^m(\overlineΩ)$ functions, with $m\in{\mathbb N}\cup\{\infty\}$, in the space $$W^{k,\ell,p}(Ω;Γ)= \left\{u\in W^{k,p}(Ω): u_{|Γ}\in W^{\ell,p}(Γ)\right\},$$ endowed with the norm of $(u,u_{|Γ})$ in $W^{k,p}(Ω)\times W^{\ell,p}(Γ)$, where $Ω$ is a bounded open subset of ${\mathbb R}^N$, $N\ge 2$, with boundary $Γ$ of class $C^m$, $k\le \ell\le m$ and $1\le p<\infty$.
Such a result is of interest when dealing with doubly elliptic problems involving two elliptic operators, one in $Ω$ and the other on $Γ$.
Moreover we shall also consider the case when a Dirichlet homogeneous boundary condition is imposed on a relatively open part of $Γ$ and, as a preliminary step, we shall prove an analogous result when either $Ω={\mathbb R}^N$ or $Ω={\mathbb R}^N_+$ and $Γ=\partial{\mathbb R}^N_+$. \keywords{Density results\and Sobolev spaces \and Smooth functions \and the Laplace--Beltrami operator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2005_10740 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems Pucci, Patrizia Vitillaro, Enzo Analysis of PDEs Functional Analysis 46E35 46.38 46M35 The paper deals with a nontrivial density result for $C^m(\overlineΩ)$ functions, with $m\in{\mathbb N}\cup\{\infty\}$, in the space $$W^{k,\ell,p}(Ω;Γ)= \left\{u\in W^{k,p}(Ω): u_{|Γ}\in W^{\ell,p}(Γ)\right\},$$ endowed with the norm of $(u,u_{|Γ})$ in $W^{k,p}(Ω)\times W^{\ell,p}(Γ)$, where $Ω$ is a bounded open subset of ${\mathbb R}^N$, $N\ge 2$, with boundary $Γ$ of class $C^m$, $k\le \ell\le m$ and $1\le p<\infty$. Such a result is of interest when dealing with doubly elliptic problems involving two elliptic operators, one in $Ω$ and the other on $Γ$. Moreover we shall also consider the case when a Dirichlet homogeneous boundary condition is imposed on a relatively open part of $Γ$ and, as a preliminary step, we shall prove an analogous result when either $Ω={\mathbb R}^N$ or $Ω={\mathbb R}^N_+$ and $Γ=\partial{\mathbb R}^N_+$. \keywords{Density results\and Sobolev spaces \and Smooth functions \and the Laplace--Beltrami operator. |
| title | Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems |
| topic | Analysis of PDEs Functional Analysis 46E35 46.38 46M35 |
| url | https://arxiv.org/abs/2005.10740 |