# Very weak solutions to the two dimensional Monge-Amp\'ere equation

## TL;DR

This paper explores a convex integration method to construct very weak solutions to the 2D Monge-Ampère equation, extending previous techniques with a new diagonalization approach that bypasses conformal coordinates.

## Contribution

It introduces a novel diagonalization procedure combined with existing methods to construct very weak solutions with Hölder-continuous derivatives for the 2D Monge-Ampère equation.

## Key findings

- Successfully constructs very weak solutions with Hölder continuity of derivatives.
- Avoids conformal coordinates using a new diagonalization technique.
- Extends convex integration methods to the Monge-Ampère context.

## Abstract

In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Amp\'ere equation with H\"older-continuous first derivatives of exponent $\beta<1/5$. Our approach is based on combining the approach of Lewicka-Pakzad \cite{Lewicka:2015gz} with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by the second author with De Lellis and Inauen in \cite{DeLellis:2015wm} for the isometric immersion problem.