Nonlinear Bounds in Hölder Spaces for the Monge-Ampère Equation

Abstract

We demonstrate that $C^{2,\alpha }$ estimates for the Monge-Ampère 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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