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

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July 2008, 10(1): 221-238. doi: 10.3934/dcdsb.2008.10.221

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

1.	Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

Received June 2007 Revised December 2007 Published April 2008

Abstract
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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: partial differential equations, convex functions, viscosity solutions, Monge-Ampère equation, convex hull., Pucci maximal equation, finite difference schemes.

Mathematics Subject Classification: 65N06, 65N12, 65M06, 65M12, 35B50, 35J60, 35R35, 35K65, 49L2.

Citation: Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221

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