## A. Figalli - Y. Jhaveri - C. Mooney

# Nonlinear bounds in Hölder spaces for the Monge-Ampère equation

created by figalli on 12 Nov 2015

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BibTeX]

*Accepted Paper*

**Inserted:** 12 nov 2015

**Last Updated:** 12 nov 2015

**Journal:** J. Funct. Anal.

**Year:** 2015

**Abstract:**

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.

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