Inequality on elliptic pde

Question

Let $\Omega=\overline{N(0,1)}$.

Consider linear elliptic pde $au_{xx}+2bu_{xy}+cu_{yy}+r=0$ on $\Omega$, where $a,b,c,r\in C^\infty(\Omega)$ and $ac-b^2\equiv 1$, $\frac{1}{100}<a,b,c<100$.

For all $p<\infty$, is there $C\in\mathbb{R}$ such that for all $u\in C^2(\Omega)$ that is a solution to the pde above and $u|_{\partial\Omega}=0$ satisfies

$$\|u_{xx}\|_p\le C(\|r\|_\infty+\|u_x\|_\infty+\|u_y\|_\infty)$$

Answer 1

Hint: The matrix $A := \begin{pmatrix}a & b \\ b & c\end{pmatrix}$ is symmetric and (uniformly) positive definite (why?). Now use elliptic regularity theory.