I try to understand the last part of Interior Regularity theorem. The theorem says:
if $u\in W^{1,2}(\Omega)$ , where $\Omega$ is in $R^n$ and bounded, and $u$ is the weak solution of $\Delta u = f $ then, for every $\Omega'\subset \subset \Omega $, $u \in W^{2,2}(\Omega')$ and $\Vert u \Vert_{W^{2,2}(\Omega')} \leq C (||u||_{L^2(\Omega)} + ||f||_{L^2(\Omega)})$.
Now, I am in the last lines of proof and the question is:
Why $\int_{\Omega '}|D_i \nabla u|^2 \leq C(||u||_{L^2(\Omega)} + ||f||_{L^2(\Omega)}) $ implies that $\Vert u \Vert_{W^{2,2}(\Omega')} \leq C (||u||_{L^2(\Omega)} + ||f||_{L^2(\Omega)})$ ?
Maybe I don't understand how the Sobolev norm works? There are some properties that escapes me? I mean, I have the estimation for $||D^2 u||_{L^2(\Omega')}$ but what about $||u||_{L^2(\Omega')}$ and $||Du||_{L^2(\Omega')}$?
Thank u.
We know that $u\in W^{2,2}(\Omega')$ because $\|D^2 u\|_{L^2(\Omega')}<\infty$ and $u\in W^{1,2}(\Omega')$.
Now, take a cut-off function $\eta\in C_c^{\infty}(\Omega)$ such that $0\leq \eta \leq 1$, $\eta=1$ in $\Omega'$, $\|\nabla \eta\|_{L^{\infty}(\Omega)}\leq C_1(\Omega',\Omega)$, $\|D^2 \eta\|_{L^{\infty}(\Omega)}\leq C_2(\Omega',\Omega)$.
Then $\eta u \in W_0^{2,2}(\Omega)$. By Poincare's theorem you will have an estimate of the form $$\|\eta u\|_{W^{2,2}(\Omega)} \leq C(\Omega)\|D^2(\eta u)\|_{L^2(\Omega)}$$ On the other hand, by the properties of $\eta$, you also have $$ \|D^2(\eta u)\|_{L^2(\Omega)}\leq C(\Omega',\Omega)\|D^2 u\|_{L^2(\Omega')} $$ and $$\|u\|_{W^{2,2}(\Omega')}\leq \|\eta u\|_{W^{2,2}(\Omega)} $$ so putting together the above inequalities you obtain the desired result.