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What is the regularity of solutions of \begin{equation} \mbox{div}(A(x) \nabla u) = 0\ \mbox{in}\ B_1 \end{equation} in the weak sense (distributional sense) where the matrix $A(x)$ is Hölder continuous, this is, $A(x) \in C^{0, \gamma}(B_1)$ and $B_1$ is the unit ball in $\mathbb{R}^{n}$. \begin{equation} \lambda |\xi|^{2} \le \langle A(x)\xi , \xi \rangle \le |\xi|^{2} \quad \forall x,\xi \in \mathbb{R}^{n}. \end{equation} I know by De digiorge that if $A(x)$ is only a measurable matrix, then $u \in C^{\alpha}(B_{1/2})$ for a specific $\alpha \in (0,1)$. I want to know if the hypothesis $ A(x) \in C^{\gamma}$ implies in a regularity $u \in C^{\gamma}(B_{1/2})$. If possible I'd like to know some reference that comproves the assertion.

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    I've edited the title and body to remove `\quad`s. A simple `\` should suffice when you need a space; forcing `\quad` in there brea$k$s the typesetting for many people.2012-12-18

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It even implies $u\in C^{1,\gamma}$ (which is the natural degree of regularity, given that $A$ and $\nabla u$ participate in the equation in a similar way). This is Theorem 8.22 in Gilbarg-Trudinger, page 210. They merely sketch the proof, saying that it's similar to the Schauder approach to non-divergence equations with Hölder coefficients.