Let $\Omega$ be a bounded domain of $\mathbb{R}^n$, $\partial \Omega$ is of class $C^{2}$ . Assume that $a_{ij}(x)\in C^{0,1} (\bar{\Omega}), \Lambda|\xi|^2> a_{i j}(x) \xi_{i} \xi_{j} >\lambda|\xi|^{2} \quad$ for any $x \in \Omega$ and any $\xi \in \mathbb{R}^{n}$ for some positive constant $\Lambda,\lambda$. $f\in C^1(\Omega \times \mathbb{R} \times \mathbb{R}^n)$, and
$$|f(x,u,Du)| \leq M(1+|DU|), for \ some\ M>0$$ If $u \in W^{1,2}(\Omega), u-\phi \in W_0^{1,2}(\Omega)$, and in $\Omega$ we have
$$(a_{i j} (x)D_{j} u)_{i}=f(x, u, D U)$$
in the sense of distribution. Prove that $u\in W^{2,2}(\Omega)$, and
$$||u||_{W^{2,2}\ (\Omega)} \leq c(||u||_{L^2(\Omega)} +||\phi||_{W^{2,2}\ (\Omega)})$$
for some $c=c(n,\lambda, \partial \Omega,\Lambda,||a_{ij}||_{C^{0,1}\ (\bar{\Omega})})$
I want to use two theorems in G-T to solve this problem
Theorem 8.8. Let $u \in W^{1,2}(\Omega)$ be a weak solution of the equation $Lu =f$ in $\Omega$ where $L$ is strictly elliptic in $\Omega$ , the coefficients $a^{i j}, b^{i}, i, j=1, \ldots, n$ are uniformly Lipschitz continuous in $\Omega$ , the coefficients $c^{i}, d, i=1, \ldots , n$ are essentially bounded in $\Omega$ and the function $f$ is in $L^{2}(\Omega)$ . Then for any subdomain $\Omega^{\prime} \subset \subset \Omega$ , we have $u \in W^{2,2}\left(\Omega^{\prime}\right)$ and
for $C=C\left(n, \lambda, K, d^{\prime}\right)$ , where $\lambda$ is given by (8.5) ,
$K=\max \left\{\left\|a^{i j}, b^{i}\right\|_{C^{0,1}\ (\bar{\Omega})},\left\|c^{i}, d\right\|_{\left.L^{\infty}(\Omega)\right\}}\right\} \quad \text { and } \quad d^{\prime}=\operatorname{dist}\left(\Omega^{\prime}, \partial \Omega\right) $.
Theorem 8.12. Let us assume, in addition to the hypotheses of Theorem 8.8, that $\partial \Omega$ is of class $C^{2}$ and that there exists a function $\varphi \in W^{2,2}(\Omega)$ for which $u-\varphi \in W_{0}^{1,2}(\Omega)$ . Then we have also $u \in W^{2,2}(\Omega)$ and
$\|u\|_{W^{2,2}\ (\Omega)} \leqslant C\left(\|u\|_{L^{2}(\Omega)}+\|f\|_{L^{2}(\Omega)}+\|\varphi\|_{W^{2,2}\ (\Omega)}\right)$
where $C=C(n, \lambda, K, \partial \Omega)$
But if I directly apply these two theorems, I cannot control the $\|f\|_{L^{2}(\Omega)}$, if I want to get the required inequality, I need to control the norm of $1, |\bigtriangledown (u)|,|\bigtriangledown (u)|^2$ by $||u||_{L^2(\Omega)} +||\phi||_{W^{2,2}\ (\Omega)}$, I don't know how to achieve this. I tried to get stronger results then Theorem 8.8 in G-T and then get stronger results then Theorem 8.12 but failed. In particular, I have no ideas about how to control the norm of $1$ by $||u||_{L^2(\Omega)} +||\phi||_{W^{2,2}\ (\Omega)}$, could you give me some help, thank you very much!!! Any help would be appreciated.
The estimate is $$\|u\|_{W^{2,2}} \le c(\|u\|_{L^2}+ \|\phi\|_{W^{2,2}})$$ is not true, the correct estimate is $$\|u\|_{W^{2,2}} \le c(1+\|u\|_{L^2}+ \|\phi\|_{W^{2,2}}).$$ To see this, consider the following simple problem: $-u''=f$ on $(0,1)$ with $\phi=0$ and $f(x,u,Du) = \sin(n\pi x)$. So the assumption is satisfied with $M=1$. Then $u = \frac1{n^2\pi^2} \sin(n\pi x)$, hence $u\to0$ in $L^2$ while $\|u''\|_{L^2} \not\to0$ for $n\to\infty$.