i want to prove that the followinf function is bijection: $$A: H_{0}^{1}(\Omega)\cap H^{2}(\Omega) \to L^{2}(\Omega)$$
$\Omega$ is $C^{1}$ bounded domain of $R^{n}$ and the hint was to use lax milgram theorem
so i consider \begin{array}{ll} -\Delta u=f, & \hbox{$\ \in \Omega$,} \\ u=0\in \partial \Omega. & \hbox{} \end{array} and $f \in L^{2}(\Omega)$
first i used this theorem with V$=( H_{0}^{1}(\Omega),\lvert. \rvert_{H_{0}^{1}})$ since with elliptic regularity i will have that u $\in H^{2}(\Omega)$ but im not sure that is the best answer and im thinking to use the lax milgram with W$=(H_{0}^{1}(\Omega)\cap H^{2}(\Omega),\lvert \Delta .\rvert_{2})$ because W is banach space (as you can see Sobolov Space $W^{2,2}\cap W^{1,2}_0$ norm equivalence) so what space i should consider V or W thanks
