Fractional Sobolev space with zero boundary values

Question

For $0<s<1$ and $1<p<\infty$, and a given bounded open set $\Omega$ of $\mathbb{R}^n$, we define $$ W_0^{s,p}(\Omega)=\{u\in W^{s,p}(\mathbb{R}^n):u=0\text{ in }\mathbb{R}^n\setminus\Omega\}, $$ where $$ W^{s,p}(\mathbb{R}^n)=\{u:\mathbb{R}^n\to\mathbb{R}\text{ measurable }:[u]<\infty\}, $$ where $$ [u]=\Big(\iint_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx dy\Big)^\frac{1}{p}. $$ Also, I saw at few places the space $W_0^{s,p}(\Omega)$ is defined as the closure of $C_c^{\infty}(\Omega)$ in $(W^{s,p}(\Omega),\|\|)$ under the norm $$ \|u\|=\|u\|_{L^p(\Omega)}+\Big(\iint_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx dy\Big)^\frac{1}{p}. $$

My question is are the two definitions of $W_0^{s,p}(\Omega)$ above equivalent or is there some inclusion?

Any help would be highly appreciated. Thanks in advance.

Answer 1

Define $$ W^{s,p}_0(\Omega)= \text{ closure of }C_c^\infty(\Omega) \text{ wrt } \|\cdot\|, $$ $$ \tilde W^{s,p}(\Omega) = \{u \in W^{s,p}(\mathbb R^d): \ u|_{\mathbb R^d\setminus \Omega}=0\}. $$ Since $C_c^\infty(\Omega) \subset \tilde W^{s,p}(\Omega)$ it follows $W^{s,p}_0(\Omega)\subset \tilde W^{s,p}(\Omega)$.

If $\Omega$ is a bounded Lipschitz domain and $s-\frac1p$ is not integer then both spaces coincide. This is Corollary 1.4.4.5 in the book ``Elliptic problems in nonsmooth domains'' by Grisvard. More such basic results can be found in Chapter 1 of this book.