A deterministic-control-based approach motion by curvature

✍ Scribed by Robert Kohn; Sylvia Serfaty

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 454 KB
• Volume: 59
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.20101

✦ Synopsis

Abstract

The level‐set formulation of motion by mean curvature is a degenerate parabolic equation. We show that its solution can be interpreted as the value function of a deterministic two‐person game. More precisely, we give a family of discrete‐time, two‐person games whose value functions converge in the continuous‐time limit to the solution of the motion‐by‐curvature PDE. For a convex domain, the boundary's “first arrival time” solves a degenerate elliptic equation; this corresponds, in our game‐theoretic setting, to a minimum‐exit‐time problem. For a nonconvex domain the two‐person game still makes sense; we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the “positive part of the curvature.” These results are unexpected, because the value function of a deterministic control problem is normally the solution of a first‐order Hamilton‐Jacobi equation. Our situation is different because the usual first‐order calculation is singular. © 2005 Wiley Periodicals, Inc.