Consider the linear second order elliptic Dirichlet problem
$$-\nabla\cdot (a\nabla u)\quad u=0 \text{ on }\partial\Omega$$
Condtion 1:$\Lambda |\xi|^2\geq\sum a_{i,j}(x) \xi_i \xi_j \geq \lambda |\xi|^2</math> for all <math>x\in \Omega, \xi \in \mathbb{R}^n$$\Lambda |\xi|^2\geq\sum a_{i,j}(x) \xi_i \xi_j \geq \lambda |\xi|^2$ for all $x\in \Omega$, $\xi \in \mathbb{R}^n$
The Schauder Estimates Boundary estimates yield
$|u|_{2,\alpha;\Omega} \leq C(n,\alpha,\lambda,\Lambda,\Omega) (|u|_{0,\Omega} + |f|_{0,\alpha;\Omega} + |\phi|_{2,\alpha;\partial\Omega}).$
Does this estimate hold uniformly for all $a$ that satisfy Condition 1 for a fixed $\lambda$ and $\Lambda$ and $||a||_{C^\alpha}\leq S$$\|a\|_{C^\alpha}\leq S$? If so what is the best reference to see this?
