Schauder estimate

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The Schauder estimates refer to an interior regularity result assuming that the coefficients of a uniformly elliptic equation are Holder continuous. More precisely, for the equation \(a_{ij}(x) u_{ij} = f \qquad \text{in a domain } D \) with $a_{ij}$ in $C^\alpha$ and $f \in C^\alpha$, then $u$ is in $C^{2,\alpha}$ in the interior of $D$.

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