$C^{1,\frac{1}{3}-}$ very weak solutions to the two dimensional Monge-Amp\'ere equation

TL;DR

This paper demonstrates that very weak solutions with $C^{1,\theta}$ regularity for the 2D Monge-Ampère equation are dense among continuous functions, using a convex integration approach with a refined defect decomposition.

Contribution

It introduces a novel convex integration scheme that simplifies the perturbation process by a refined defect decomposition, advancing the understanding of weak solutions to the Monge-Ampère equation.

Findings

Very weak $C^{1,\theta}$ solutions are dense in continuous functions for $\theta<\frac{1}{3}$.

A new convex integration method reduces the number of perturbations needed.

The defect decomposition diagonalizes the defect and incorporates leading-order error terms.

Abstract

For any $\theta<\frac{1}{3}$, we show that very weak solutions to the two-dimensional Monge-Amp\`ere equation with regularity $C^{1,\theta}$ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one.