The Monge-Ampere system: convex integration with improved regularity in dimension two and arbitrary codimension

TL;DR

This paper advances convex integration techniques for the Monge-Ampere system in two dimensions, achieving improved regularity results up to Hölder exponent 1/(1+4/k), surpassing previous bounds and aligning with known results for specific cases.

Contribution

It provides a new convex integration result for the Monge-Ampere system in 2D with arbitrary codimension, improving the Hölder regularity of solutions compared to prior work.

Findings

Achieves Hölder regularity up to 1/(1+4/k) in 2D for the Monge-Ampere system.

Improves previous regularity bounds from 1/(1+6/k) to 1/(1+4/k).

Results match known regularity for specific cases like k=1 and as k approaches infinity.

Abstract

We prove a convex integration result for the Monge-Ampere system in dimension $d=2$ and arbitrary codimension $k\geq 1$. We achieve flexibility up to the Holder regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$, improving hence the previous $\mathcal{C}^{1,\frac{1}{1+ 6/k}}$ regularity that followed from flexibility up to $\mathcal{C}^{1,\frac{1}{1+d(d+1)/k}}$ in our previous work, valid for any $d,k\geq 1$. The present result agrees with flexibility up to $\mathcal{C}^{1,\frac{1}{5}}$ for $d=2, k=1$ obtained by Conti, Delellis, Szekelyhidi, as well as with the $\mathcal{C}^{1,\alpha}$ result where $\alpha\to 1$ as $k\to\infty$, due to Kallen.