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Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems

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.