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

Abstract

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.

This is a pre-print of an article published in Boll. Unione Mat Ital. (2020). The final authenticated version is available online at: https://doi.org/10.1007/s40574-020-00225-w

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