On the sharp H\"older exponent in the De Giorgi--Nash--Moser theory

We consider solutions of uniformly elliptic equations with measurable coefficients.

We assume that the lowest eigenvalue of the coefficient matrix is at least $K^{-1}$ and the largest eigenvalue is at most $K$.

In three and higher dimensions we construct $\alpha$-Hölder continuous solutions with $\alpha = \exp(- c_n K)$.

This disproves a long-standing conjecture by showing that, except for the two-dimensional case, the Hölder exponent obtained from the Bombieri--Giusti Harnack inequality has the optimal dependence on the ellipticity constant $K$.