Vol. 10, No. 3, 2017

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Convex integration for the Monge–Ampère equation in two dimensions

Vol. 10 (2017), No. 3, 695–727
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.

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Monge–Ampére equation, convex integration, $h$-principle, rigidity and flexibility, developable surfaces