An Interior A Priori Estimate for Solutions to Monge-Amp\`ere Equations with Right-Hand Side Close to One

TL;DR

This paper derives an interior a priori estimate for solutions to Monge-Ampère equations with right-hand sides near one, providing explicit control on the second derivatives' modulus of continuity and the shape of shrinking sections.

Contribution

It introduces a new exponential bound on the second derivatives' modulus of continuity for Monge-Ampère solutions with near-constant right-hand side, extending previous regularity results.

Findings

Established an exponential upper bound for the second derivatives' modulus of continuity.

Provided explicit control on the shape of shrinking sections.

Revealed the nature of exponential dependence on a Dini-like semi-norm.

Abstract

We consider Monge-Amp\'ere equations with the right hand side function close to a constant and from a function class that is larger than any H\"older class and smaller than the Dini-continuous class. We establish an upper bound for the modulus of continuity of the solution's second derivatives. This bound depends exponentially on a quantity similar to but larger than the Dini semi-norm. We establish explicit control on the shape of the sequence of shrinking sections, hence revealing the nature of such exponential dependence.