Let $\Omega$ be a domain $\Omega$ of $\mathbb{R}^d$.
Let $u \in W^{k,p}(\Omega)$ whose restriction to the boundary $\partial \Omega$ is identically zero. (To consider the "value", should we choose the superscripts $k,p$ so that $W^{k,p}$ is imbedded in the set of continuous functions by the Sobolev embedding theorem.)
Question: Is the condition $u \in W^{k,p}_0(\Omega)$ true? Namely, is there a sequnce of smooth, compact supported functions on $\Omega$ such that it converges in $W^{k,p}(\Omega)$.
My attempt:
Let $\Omega_n$ be a sequence of bounded domains such that
- $\bigcup_n \Omega_n = \Omega$,
- $\Omega_n \subset\subset \Omega_{n+1}$.
Then we consider the regularization $u_{h_n,\Omega_n}$ (see the definition blow). where $h_n$ is chosen so that $h_n < $ dist$(\Omega_n,\partial \Omega_{n+1})$ from which we infer that supp$u_{h_n,\Omega_n} \subset \Omega_{n+2}$.
Using parition of unity $\psi_n$ subordinated to the covering $\{\Omega_{n+1}-\Omega_{n-1}\}$ we can define a function $v_h:= \Sigma_n \psi_nu_{h_n,\Omega_n}$ which converges in $W^{k,p}(\Omega)$. However $v_h$ dose not have compact support. So, how to modifiy the sequece $v_h$ so that the modified one has the compact support.
Let $\phi(x) = c\exp(\frac{1}{|x|^1-1})$ for$|x| \leq 1$ and $0$ for $|x| \geq 1$ and $c$ is chosen so that $\int _{\mathbb{R} ^d}\phi(x) > dx=1$. Let $\mathbb{\Omega}$ be a dommain in $\mathbb{R}^d$. Define $u_{h,\Omega}:= \frac{1}{h^{-d}} \int_\Omega \phi(\frac{x-y}{h})u(y)dy$ for any $h < \text{dist}(x, \partial \Omega)$. Then $u_{h,\Omega}$ converges to $u$ uniformly on any domain $\Omega ' \subset \subset \Omega$. Note that $u_h \in C^\infty(\Omega ')$.