Nonlinear Bounds in H\"older Spaces for the Monge-Amp\`ere Equation

TL;DR

This paper investigates the nonlinear dependence of $C^{2,eta}$ estimates for solutions to the Monge-Ampère equation on the regularity parameter and the right-hand side, establishing polynomial bounds and sharpness through explicit examples.

Contribution

It provides new quantitative bounds on the $C^{2,eta}$ regularity of solutions, revealing nonlinear dependence on the data and demonstrating sharpness with constructed solutions.

Findings

Polynomial dependence of $C^{2,eta}$ norm on the right-hand side's $C^{eta}$ norm.

Exponential-type bounds as $eta o 0$, showing sharpness.

Constructed solutions illustrating the bounds are optimal.

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.