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 | References | Similar Articles | Additional Information

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

Email:
aoberman@sfu.ca

DOI:
https://doi.org/10.1090/S0025-5718-04-01688-6

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