Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian

Pages: 221 - 238, Volume 10, Issue 1, July 2008

doi:10.3934/dcdsb.2008.10.221       Abstract        Full Text (2040.6K)       Related Articles

Adam M. Oberman - Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada (email)

Abstract: Certain fully nonlinear elliptic Partial Differential Equations can be written as functions of the eigenvalues of the Hessian. These include: the Monge-Ampère equation, Pucci’s Maximal and Minimal equations, and the equation for the convex envelope. In this article we build convergent monotone finite difference schemes for the aforementioned equations. Numerical results are presented.

Keywords:  finite difference schemes, partial differential equations, viscosity solutions, Monge-Ampère equation, Pucci maximal equation, convex functions, convex hull.
Mathematics Subject Classification:  65N06, 65N12, 65M06, 65M12, 35B50, 35J60, 35R35, 35K65, 49L25.

Received: June 2007;      Revised: December 2007;      Published: April 2008.