Suppose we have a bounded domain $\Omega \subset \mathbb{R}^N$ with sufficiently smooth boundary $\partial \Omega$. The Sobolev spaces $W^{1,p}(\mathbb{R}^N)$ and $W_0^{1,p}(\Omega)$ are defined as usual. I am wondering if there is a kind of "canonical projection" of $W^{1,p}(\mathbb{R}^N)$ onto $W_0^{1,p}(\Omega)$, which of course should be linear and continuous.
My question comes from a variational problema that I can roughly describe as follows: I have a "bunch" of differential equations, each of which is defined on a domain $\Omega_R$ with (homogeneous) Dirichlet boundary condition. These equations "tend" to a limit equation that is defined on the whole space $\mathbb{R}^N$. Since they are variational in nature, I would like to introduce a unique functional space to include all of them, and therefore my idea is to pick any element of $W^{1,p}(\mathbb{R}^N)$, project it onto $W_0^{1,p}(\Omega_R)$ (which is the natural space) and then insert this into a functional. I dont want to introduce a different functional attached to each domain $\Omega_R$, since I would lose my limiting problem.
A possible projection $u \in W_0^{1,p}(\Omega)$ of $v \in W^{1,p}(\mathbb{R}^n)$ is given by \begin{equation*} u = \operatorname{argmin}_{\tilde u \in W_0^{1,p}(\Omega)} \| \tilde u - v \|_{W^{1,p}(\Omega)}. \end{equation*} This $\operatorname{argmin}$ should exist in the reflexive case $1 < p < \infty$. Moreover, this projection is linear in case $p = 2$.
An other possibility is by using an extension operator. Let $\tau : W^{1,p}(\Omega) \to L^p(\partial\Omega)$ be the trace operator. Then, there is a bounded right-inverse $E : \tau(W^{1,p}(\Omega)) \to W^{1,p}(\Omega)$. For $v \in W^{1,p}(\mathbb{R}^n)$, you have \begin{equation*} v_{|\Omega} - E \, \tau(v_{|\Omega}) \in W_0^{1,p}(\Omega) \end{equation*} and this depends linearly on $v$. However, the extension operator is, of course, not unique and there might be no ``canonical'' one. For this second route, you also might need some conditions on $p$ for the existence of the extension operator.