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The local Sobolev space,defined as $W^{k,p}_{loc}(\Omega)$, is the space such that for any $u \in W^{k,p}_{loc}(\Omega)$ and any compact $V\subset \Omega$, $u \in W^{k,p}(V)$. I am just wondering if for any $u\in W^{k,p}_{loc}(\Omega)$, there exists ${u_n}\in C^\infty(\Omega)$, such that $u_n \rightarrow u$ in $W^{k,p}(V)$ for any compact $V\subset \Omega$. ($C^\infty(\Omega)$ is dense in $W^{k,p}(\Omega)$, I am just wondering if this is also true for $W^{k,p}_{loc}(\Omega)$)

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2 Answers 2

The standard mollifier construction should still hold in this space. Let $\phi_n$ be standard mollifiers, $\operatorname{supp}{\phi_n} \subset B_{\frac{1}{n}}(0), f \in W^{k, p}_{loc}(\Omega)$. Then $f * \phi_n$ converges to $f$ in any $W^{k, p}_{loc}(K)$ (by the standard argument on $K \backslash B_{\frac{1}{n}}(\partial K)$ and everywhere since this strip around the boundary shrinks to measure 0 and $f * \phi_n$ is bounded independently of n).

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The answer is yes for $V\subset\Omega$, for which $\bar V\subset\Omega$.

Let $U_1\subset \bar{U}_1 \subset U_2 \subset \bar U_2\subset \cdots\subset U_n\subset\bar U_n\subset \cdots\subset \Omega$, where $\bar U_n$ compact, for all $n$, and $\bigcup U_n=\Omega$. Clearly, every such $V$ there exists an $n$, such that $\bar V\subset U_n$. Such sequence of domains can be readily obtained, i.e., $U_n=B(0,n)\cap \{x\in\Omega: \mathrm{dist}(x,\partial\Omega)<1/n\}$.

Now fixing a $u\in W_{\mathrm{loc}}^{k,p}(\Omega)$ and a $n_0\in \mathbb N$, we can find a sequence $\{\varphi_{n_0,n}\}_{n\in\mathbb N}\subset C^\infty(\Omega)$, using for example mollifiers, converging to $u|_{U_{n_0}}$ in the norm of $W^{k,p}(U_{n_0})$. The desired sequence is then obtained using a standard diagonal argument.

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