Convex integration for the Monge–Ampère equation in two dimensions

Abstract

This paper concerns the questions of flexibility and rigidity of solutions to the Monge–Ampère equation, which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on degenerate, i.e., “flexible”, weak solutions that can be constructed through methods of convex integration à la Nash and Kuiper and establish the related $h$-principle for the Monge–Ampère equation in two dimensions.