On a class of linearizable Monge-Ampère equations

Daniel J. Arrigo; James M. Hill

doi:10.2991/jnmp.1998.5.2.1

paper

On a class of linearizable Monge-Ampère equations

arXiv:math/9804161 · doi:10.2991/jnmp.1998.5.2.1

Abstract

Monge-Ampère equations of the form, $u_{xx} u_{y y} - u_{x y}^{2} = F (u, u_{x}, u_{y})$ arise in many areas of fluid and solid mechanics. Here it is shown that in the special case $F = u_{y}^{4} f (u, u_{x} / u_{y})$ , where $f$ denotes an arbitrary function, the Monge-Ampère equation can be linearized by using a sequence of Ampère, point, Legendre and rotation transformations. This linearization is a generalization of three examples from finite elasticity, involving plane strain and plane stress deformations of the incompressible perfectly elastic Varga material and also relates to a previous linearization of this equation due to Khabirov [7].