Quantitative Global Schauder Estimates/Hölder Regularity

Question

Consider the linear second order elliptic Dirichlet problem

$$-\nabla\cdot (a\nabla u)\quad u=0 \text{ on }\partial\Omega$$

Condtion 1:$\Lambda |\xi|^2\geq\sum a_{i,j}(x) \xi_i \xi_j \geq \lambda |\xi|^2$ 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 $\|a\|_{C^\alpha}\leq S$? If so what is the best reference to see this?

Answer 1

The Schauder estimates apply for the case when you consider the equation $$a_{ij}u_{ij} + b_i u_i + c u = f$$ and in this case you need to assume that there are bounds on the Holder norm of the $a_{ij}$. Your equation is of the divergence type, and there in this case your best result is a $C^{1,\alpha}$ bound, if $a_{ij}$ and $f$ are no better than $C^\alpha$. The simplest exposition of these concepts is probably to be found in the book of Qing Han and Fanghua Lin.