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

Adam M. Oberman
2004 Mathematics of Computation  
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
more » ... 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.
doi:10.1090/s0025-5718-04-01688-6 fatcat:3vywq377wvh3zfg22jqeaqqgda