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