What is $W_{loc}^{2,2}(\Omega )$?

Question

I know that $$W^{2,2}(\Omega )=\left\{u\in L^2(\Omega )\mid \exists v_i\in L^2(\Omega ):\forall \varphi\in \mathcal C_0^\infty (\Omega ), \int_\Omega fD^\alpha \varphi=-\int_\Omega v_i\varphi, |\alpha |\leq 2 \right\},$$ but what is $W_{loc}^{2,2}(\Omega )$ ? is it $$\left\{u\in L^2_{loc}(\Omega )\mid \exists v_i\in L^2(\Omega ):\forall \varphi\in \mathcal C_0^\infty (\Omega ), \int_\Omega fD^\alpha \varphi=-\int_\Omega v_i\varphi, |\alpha |\leq 2 \right\}$$ or $$\left\{u\in L^2_{loc}(\Omega )\mid \exists v_i\in L^2_{loc}(\Omega ):\forall \varphi\in \mathcal C_0^\infty (\Omega ), \int_\Omega fD^\alpha \varphi=-\int_\Omega v_i\varphi, |\alpha |\leq 2 \right\}$$

Answer 1

One way to define this is

$$W_{{\rm loc}}^{2,2}(\Omega)=\left\{u\in L^2_{{\rm loc}}(\Omega)\Big{|}\,\forall\phi\in C_0^{\infty}(\Omega),\,u\phi\in W^{2,2}(\Omega)\right\}.$$