From Pierre Grisvard "Elliptic Problems in Nonsmooth Domains":

Theorem[Theorem 1.4.2.1] Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a continuous boundary, then $C_c^\infty(\overline{\Omega})$ is dense in $W^{s,p}(\Omega)$ for all $s>0$.

But this Theorem has not been proved therein. I am barely looking for suitable proof.

Moreover, $C_c^\infty(\mathbb{R}^d)$ is dense in $W^{s,p}(\mathbb{R}^d)$ for all $s$ and consequently $C_c^\infty(\overline{\Omega})$ is dense in $W^{s,p}(\overline{\Omega})$ without any assumption on $\Omega$.

Theorem[Theorem 1.4.2.4] Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a Lipschitz boundary, then $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$ for $0\leq s\leq 1/p$

As proof Theorem 1.4.2.4 here i quote the author:

"In view of Theorem 1.4.2.1 we only have to approximate functions in $C^\infty(\overline{\Omega})$ by function in $C_c^\infty(\Omega)$. This is easily achieved by means of a sequence of cutt-off functions."

Question For which type of cut-off functions $(q_j)_j$ supported in $\Omega$ do we have

$$|u-q_ju|_{W^{s,p}(\Omega)}\xrightarrow[]{j\to \infty}0?$$

Recall that,

$$|u|^p_{W^{s,p}(\Omega)}= \iint\limits_{\Omega\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dx dy$$ and

$$W^{s,p}(\Omega)= \{u\in L^p(\Omega): |u|^p_{W^{s,p}(\Omega)}<\infty\}$$

equipped with the Banach norm

$$\|u\|^p_{W^{s,p}(\Omega)}= |u|^p_{L^{p}(\Omega)}+|u|^p_{W^{s,p}(\Omega)}$$

From Pierre Grisvard "Elliptic Problems in Nonsmooth Domains":

Theorem[Theorem 1.4.2.1] Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a continuous boundary, then $C_c^\infty(\overline{\Omega})$ is dense in $W^{s,p}(\Omega)$ for all $s>0$.

But this Theorem has not been proved therein. I am barely looking for suitable proof.

Moreover, $C_c^\infty(\mathbb{R}^d)$ is dense in $W^{s,p}(\mathbb{R}^d)$ for all $s$ and consequently $C_c^\infty(\overline{\Omega})$ is dense in $W^{s,p}(\overline{\Omega})$ without any assumption on $\Omega$.

Theorem[Theorem 1.4.2.4] Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a Lipschitz boundary, then $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$ for $0\leq s<1/p$

As proof Theorem 1.4.2.4 here i quote the author:

"In view of Theorem 1.4.2.1 we only have to approximate functions in $C^\infty(\overline{\Omega})$ by function in $C_c^\infty(\Omega)$. This is easily achieved by means of a sequence of cutt-off functions."

Question For which type of cut-off functions $(q_j)_j$ supported in $\Omega$ do we have

$$|u-q_ju|_{W^{s,p}(\Omega)}\xrightarrow[]{j\to \infty}0?$$

Recall that,

$$|u|^p_{W^{s,p}(\Omega)}= \iint\limits_{\Omega\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dx dy$$ and

$$W^{s,p}(\Omega)= \{u\in L^p(\Omega): |u|^p_{W^{s,p}(\Omega)}<\infty\}$$

equipped with the Banach norm

$$\|u\|^p_{W^{s,p}(\Omega)}= |u|^p_{L^{p}(\Omega)}+|u|^p_{W^{s,p}(\Omega)}$$

From Pierre Grisvard "Elliptic Problems in Nonsmooth Domains":

Theorem[Theorem 1.4.2.1] Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a continuous boundary, then $C_c^\infty(\overline{\Omega})$ is dense in $W^{s,p}(\Omega)$ for all $s>0$.

Moreover, $C_c^\infty(\mathbb{R}^d)$ is dense in $W^{s,p}(\mathbb{R}^d)$ for all $s$ and consequently $C_c^\infty(\overline{\Omega})$ is dense in $W^{s,p}(\overline{\Omega})$ without any assumption on $\Omega$.

Theorem[Theorem 1.4.2.4] Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a Lipschitz boundary, then $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$ for $0\leq s\leq 1/p$

As proof Theorem 1.4.2.4 here i quote the author:

"In view of Theorem 1.4.2.1 we only have to approximate functions in $C^\infty(\overline{\Omega})$ by function in $C_c^\infty(\Omega)$. This is easily achieved by means of a sequence of cutt-off functions."

Question For which type of cut-off functions $(q_j)_j$ supported in $\Omega$ do we have

$$|u-q_ju|_{W^{s,p}(\Omega)}\xrightarrow[]{j\to \infty}0?$$

Recall that,

$$|u|^p_{W^{s,p}(\Omega)}= \iint\limits_{\Omega\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dx dy$$ and

$$W^{s,p}(\Omega)= \{u\in L^p(\Omega): |u|^p_{W^{s,p}(\Omega)}<\infty\}$$

equipped with the Banach norm

$$\|u\|^p_{W^{s,p}(\Omega)}= |u|^p_{L^{p}(\Omega)}+|u|^p_{W^{s,p}(\Omega)}$$

From Pierre Grisvard "Elliptic Problems in Nonsmooth Domains":

Theorem[Theorem 1.4.2.1] Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a continuous boundary, then $C_c^\infty(\overline{\Omega})$ is dense in $W^{s,p}(\Omega)$ for all $s>0$.

But this Theorem has not been proved therein. I am barely looking for suitable proof.

Moreover, $C_c^\infty(\mathbb{R}^d)$ is dense in $W^{s,p}(\mathbb{R}^d)$ for all $s$ and consequently $C_c^\infty(\overline{\Omega})$ is dense in $W^{s,p}(\overline{\Omega})$ without any assumption on $\Omega$.

Theorem[Theorem 1.4.2.4] Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a Lipschitz boundary, then $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$ for $0\leq s\leq 1/p$

As proof Theorem 1.4.2.4 here i quote the author:

"In view of Theorem 1.4.2.1 we only have to approximate functions in $C^\infty(\overline{\Omega})$ by function in $C_c^\infty(\Omega)$. This is easily achieved by means of a sequence of cutt-off functions."

Question For which type of cut-off functions $(q_j)_j$ supported in $\Omega$ do we have

$$|u-q_ju|_{W^{s,p}(\Omega)}\xrightarrow[]{j\to \infty}0?$$

Recall that,

$$|u|^p_{W^{s,p}(\Omega)}= \iint\limits_{\Omega\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dx dy$$ and

$$W^{s,p}(\Omega)= \{u\in L^p(\Omega): |u|^p_{W^{s,p}(\Omega)}<\infty\}$$

equipped with the Banach norm

$$\|u\|^p_{W^{s,p}(\Omega)}= |u|^p_{L^{p}(\Omega)}+|u|^p_{W^{s,p}(\Omega)}$$