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

Adam M. Oberman

doi:10.3934/dcdsb.2008.10.221

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2008, Volume 10, Issue 1: 221-238. Doi: 10.3934/dcdsb.2008.10.221

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

Adam M. Oberman^1,

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

Received: May 31, 2007

Revised: November 30, 2007

Published: December 2007

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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.

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Mathematics Subject Classification: 65N06, 65N12, 65M06, 65M12, 35B50, 35J60, 35R35, 35K65, 49L25.

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

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