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A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions

Author: Adam M. Oberman
Journal: Math. Comp. 74 (2005), 1217-1230
MSC (2000): Primary 35B50, 35J60, 35J70, 65N06, 65N12
Published electronically: September 10, 2004
MathSciNet review: 2137000
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Abstract: This article considers the problem of building absolutely minimizing Lipschitz extensions to a given function. These extensions can be characterized as being the solution of a degenerate elliptic partial differential equation, the “infinity Laplacian”, for which there exist unique viscosity solutions. A convergent difference scheme for the infinity Laplacian equation is introduced, which arises by minimizing the discrete Lipschitz constant of the solution at every grid point. Existence and uniqueness of solutions to the scheme is shown directly. Solutions are also shown to satisfy a discrete comparison principle. Solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation.

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Additional Information

Adam M. Oberman
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, British Columbia Canada V5A 1S6
MR Author ID: 667376

Keywords: Finite difference, infinity Laplacian, viscosity solution
Received by editor(s): September 30, 2003
Received by editor(s) in revised form: December 29, 2003
Published electronically: September 10, 2004
Additional Notes: The author would like to thank P. E. Souganidis and A. Petrosyan for valuable discussions and L. C. Evans for his encouragement and enthusiasm.
Article copyright: © Copyright 2004 American Mathematical Society