$\Omega$ bounded, $f\in L^2(\Omega)$ and $u\in H^{1,2} (\Omega)$ a weak solution of $$-\operatorname{div}(a(\nabla u))=f$$ with $a:\mathbb{R}^n \rightarrow \mathbb{R}^n$ lipschitz continous and there exists a $c_0$ with $$(a(p_1)-a(p_2))\cdot(p_1-p_2 \geq c_0|p_1-p_2|^2$$ for $p_1,p_2 \in \mathbb{R}^n$.
I want to show now there is
i) a $C>0$ so that, $$||\nabla u||_{L^2(\Omega)} \leq C(||u||_{L^2(\Omega)}+||f||_{L^2(\Omega)}+||a(0)||_{L^2(\Omega)})$$
ii) $u\in H^{2,2}(\Omega ')$ for all $\Omega '\subset \subset \Omega$ and there exists a $C>0$ $$||D^2 u||_{L^2(\Omega')} \leq C(||u||_{L^2(\Omega)}+||f||_{L^2(\Omega)}+||a(0)||_{L^2(\Omega)})$$
How does one see that? I even think its wrong even though it should be easy to prove. Anyone can help?