Regularity and the Varitational Inequality

Question

Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in H_0^1(\Omega)$. So far I've shown $u\in K$ and it satisfies the variational inequality but I need to show that $u \in H^2$?

Answer 1

This result can be found in a paper by Brezis and Stampacchia, see in particular Theorem II.1. They prove under rather weak assumptions (in your case: $f \in L^2(\Omega)$), that $-\Delta u \in L^2(\Omega)$. This, in turn, gives you $u \in H^2(\Omega)$.