#### Vol. 10, No. 3, 2017

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Convex integration for the Monge–Ampère equation in two dimensions

### Marta Lewicka and Mohammad Reza Pakzad

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.

##### Keywords
Monge–Ampére equation, convex integration, $h$-principle, rigidity and flexibility, developable surfaces
##### Mathematical Subject Classification 2010
Primary: 35M10, 76B03, 76F02
##### Milestones
Received: 1 September 2016
Revised: 30 December 2016
Accepted: 13 February 2017
Published: 17 April 2017
##### Authors
 Marta Lewicka Department of Mathematics University of Pittsburgh 139 University Place Pittsburgh, PA 15260 United States Mohammad Reza Pakzad Department of Mathematics University of Pittsburgh 139 University Place Pittsburgh, PA 15260 United States