De Giorgi-Nash-Moser theorem
= De Giorgi-Nash-Moser theorem
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If $u\in W^{1,2}(B_1)$ weakly solves $D_i(a^{ij}D_ju)=0$ with bounded measurable uniformly elliptic coefficients, then $u$ is locally Hölder continuous. For $0<\theta<1$, some $\alpha\in(0,1)$ and $C$ depending only on the dimension, ellipticity bounds, and $\theta$ satisfy
$$
\|u\|_{C^{0,\alpha}(B_\theta)}\leq C\|u\|_{L^2(B_1)}.
$$