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

The paper deals with a nontrivial density result for $C^m(\overline{\Omega})$ functions, with $m\in{\mathbb N}\cup\{\infty\}$, in the space $$W^{k,\ell,p}(\Omega;\Gamma)= \left\{u\in W^{k,p}(\Omega): u_{|\Gamma}\in W^{\ell,p}(\Gamma)\right\},$$ endowed with the norm of $(u,u_{|\Gamma})$ in $W^{k,p}(\Omega)\times W^{\ell,p}(\Gamma)$, where $\Omega$ is a bounded open subset of ${\mathbb R}^N$, $N\ge 2$, with boundary $\Gamma$ 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 $\Omega$ and the other on $\Gamma$. Moreover we shall also consider the case when a Dirichlet homogeneous boundary condition is imposed on a relatively open part of $\Gamma$ and, as a preliminary step, we shall prove an analogous result when either $\Omega={\mathbb R}^N$ or $\Omega={\mathbb R}^N_+$ and $\Gamma=\partial{\mathbb R}^N_+$. \keywords{Density results\and Sobolev spaces \and Smooth functions \and the Laplace--Beltrami operator.