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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions
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by Adam M. Oberman PDF
Math. Comp. 74 (2005), 1217-1230 Request permission

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
  • Email: aoberman@sfu.ca
  • 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.
  • © Copyright 2004 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 1217-1230
  • MSC (2000): Primary 35B50, 35J60, 35J70, 65N06, 65N12
  • DOI: https://doi.org/10.1090/S0025-5718-04-01688-6
  • MathSciNet review: 2137000