# Fractional Sobolev space with zero boundary values

For $$0 and $$1, 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.

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.