  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.
