Convex integration for the Monge-Amp\`ere equation in two dimensions

TL;DR

This paper explores the flexibility of weak solutions to the Monge-Ampère equation in two dimensions using convex integration, demonstrating an h-principle that highlights the equation's potential for highly irregular solutions.

Contribution

It establishes the h-principle for the Monge-Ampère equation in two dimensions, showing the existence of flexible weak solutions via convex integration methods.

Findings

Existence of highly irregular weak solutions to the Monge-Ampère equation.

Application of convex integration techniques to a geometric PDE.

Demonstration of the h-principle indicating flexibility in solutions.

Abstract

This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Amp\`ere equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration \`a la Nash & Kuiper and establish the related h-principle for the Monge-Amp\`ere equation in two dimensions