On the sharp Hölder exponent in the De Giorgi--Nash--Moser theory

Abstract

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 $K$. Using the same construction as a starting point, we disprove a conjecture of De Giorgi about continuity of solutions to non-uniformly elliptic equations.

16 pages. v2: added a new section on an AI-assisted counter-example to a conjecture of De Giorgi for non-uniformly elliptic equations