The Monge-Ampere system in dimension two: a further regularity improvement

TL;DR

This paper advances the regularity of convex integration solutions for the 2D Monge-Ampère system, achieving higher Hölder regularity than previous results through refined iterative methods.

Contribution

It improves the known regularity thresholds for convex integration solutions of the 2D Monge-Ampère system, extending from Hölder $rac{1}{1+4/k}$ to $rac{2^k-1}{2^{k+1}-1}$ for arbitrary codimension.

Findings

Achieves $ ext{C}^{1,1}$ regularity for $k extgreater=4$

Extends regularity results to $ ext{C}^{1,rac{2^k-1}{2^{k+1}-1}}$ for all $k$

Improves previous Hölder regularity bounds for the Monge-Ampère system

Abstract

We prove a convex integration result for the Monge-Amp\`ere system, in case of dimension $d=2$ and arbitrary codimension $k\geq 1$. Our prior result stated flexibility up to the H\"older regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$, whereas presently we achieve flexibility up to $\mathcal{C}^{1,1}$ when $k\geq 4$ and up to $\mathcal{C}^{1,\frac{2^k-1}{2^{k+1}-1}}$ for any $k$. This first result uses the approach of K\"allen, while the second result iterates on the approach of Cao-Hirsch-Inauen and agrees with it for $k=1$ at the H\"older regularity up to $\mathcal{C}^{1,1/3}$.