Calculus of Variations and Geometric Measure Theory

A. Figalli - Y. Jhaveri - C. Mooney

Nonlinear bounds in Hölder spaces for the Monge-Ampère equation

created by figalli on 12 Nov 2015


Accepted Paper

Inserted: 12 nov 2015
Last Updated: 12 nov 2015

Journal: J. Funct. Anal.
Year: 2015


We demonstrate that $C^{2,\alpha}$ estimates for the Monge-Amp\`{e}re equation depend in a highly nonlinear way both on the $C^{\alpha}$ norm of the right-hand side and $1/\alpha$. First, we show that if a solution is strictly convex, then the $C^{2,\alpha}$ norm of the solution depends polynomially on the $C^{\alpha}$ norm of the right-hand side. Second, we show that the $C^{2,\alpha}$ norm of the solution is controlled by $\exp((C/\alpha)\log(1/\alpha))$ as $\alpha \to 0$. Finally, we construct a family of solutions in two dimensions to show the sharpness of our results.