Let $\Omega = \left( {0,1} \right) \times \left( {0,1} \right), f \in L^2 \left( \Omega \right)$ and $u$ is weak solution of boundary problem $$\left\{ \begin{gathered} - {\partial _x}\left( {\left( {2 + x} \right){u_x}} \right) - {\partial _y}\left( {\left( {2 + y} \right){u_y}} \right) = f \hfill \\ {\left. u \right|_{\partial \Omega }} = 0. \hfill \\ \end{gathered} \right.$$ Prove that $u \in {H^2}\left( \Omega \right)$.
Weak solution of that boundary problem is a function $u$ such that $$\int_\Omega {\left( {2 + x} \right){u_x}{v_x} + \left( {2 + y} \right){u_y}{v_y}} = \int_\Omega {fv} ,\forall v \in H_0^1\left( \Omega \right).$$ I dont know how to prve $u \in H^2$ i.e we must prove ${\left\| {{u_{xx}}} \right\|_{{L^2}\left( \Omega \right)}},{\left\| {{u_{yy}}} \right\|_{{L^2}\left( \Omega \right)}},{\left\| {{u_{xy}}} \right\|_{{L^2}\left( \Omega \right)}} < + \infty .$