 Solution

The De Giorgi-Nash-Moser theorem gives numbers $\alpha =\alpha (n,\lambda ,\Lambda )\in (0,1)$ and $C=C(n,\lambda ,\Lambda ,\theta )$ such that every weak solution satisfiesfor each fixed $0<\theta <1$. In particular, every such weak solution has a locally Hölder-continuous representative despite the coefficients being only bounded and measurable.