Necessity of Ladyzhenskaya \& Ural'tseva's sufficient conditions in nonuniform ellipticity

Abstract

The sufficient conditions of Ladyzhenskaya \& Ural'tseva \cite{lu70} for interior gradient estimates in nonuniformly elliptic problems are found to be also necessary. Specifically, nonparametric minimal surfaces exhibit the maximal nonuniformity rate compatible with the local Lipschitz regularity of solutions, and the classical theory of Finn \cite{fin54} and Bombieri \& De Giorgi \& Miranda \cite{bdm69} doesn't extend beyond Bernstein genre $2$.

24 pages, comments are welcome