Poincaré inequation [closed]

Suppose $\Omega \subset \mathbb{R}^n$ is bounded Lipschitz domain, $C>0$ prove that $\forall u\in H^1_0(\Omega)\cap H^2(\Omega)$: $||u||_{H^2(\Omega)}\leq C||D^2u||_{L^2(\Omega)}$.

I shall use two Poincaré inequations. Ansatz: Known $||u||_{L^2}\leq C ||\nabla u||_{L^2}$ and $||u||^2_{H^1}\leq C (||u||^2_{L^2}+(⨍u)^2)$.

Definition: $||u||^2_{H^2}=||u||^2_{L^2}+||Du||^2_{L^2}+||D^2u||^2_{L^2}$. I literally have no idea how to use known Poincaré inequations... Thanks for any help!!!

The condition $u\in H^1_0\cap H^2$ tells you that your first Poincaré inequality (the one without the average) is the one you want to use. This tells you that $\| u\|_{H^2}^2\lesssim \| \nabla u\|_{L^2}^2 + \| D^2 u\|_{L^2}^2$.

Throughout I'll assume that $u\in H_0^1\cap C^\infty(\overline{\Omega})$ (the general case is an approximation argument). $$ \int_\Omega |\nabla u|^2\, dx = -\int_\Omega u \Delta u \, dx, $$ since the boundary terms vanish. On the other hand, by Cauchy-Schwarz, $$ \left| \int_\Omega u\Delta u\, dx \right| \leq \| u\|_{L^2}\| D^2 u\|_{L^2} \leq \dfrac{\epsilon}{2}\| u\|_{L^2}^2 + \dfrac{1}{2\epsilon}\| D^2 u\|_{L^2}^2, $$ for any $\epsilon>0$ (this follows from $ab\leq a^2/2+b^2/2$).

Now if $C_1$ is the constant in the Poincaré inequality we get $$ \|\nabla u\|_{L^2}^2 \leq \dfrac{\epsilon}{2}\| u\|_{L^2}^2 + \dfrac{1}{2\epsilon}\| D^2 u\|_{L^2}^2 \leq \dfrac{C_1^2\epsilon}{2}\| \nabla u\|_{L^2}^2 + \dfrac{1}{2\epsilon}\| D^2 u\|_{L^2}^2, $$ and so, if $\epsilon<1/C_1^2$, we get $$ \| \nabla u\|_{L^2}^2 \leq \dfrac{1}{\epsilon}\| D^2 u\|_{L^2}^2. $$