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A deterministic‐control‐based approach motion by curvature
Author(s) -
Kohn Robert,
Serfaty Sylvia
Publication year - 2006
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20101
Subject(s) - mathematics , curvature , degenerate energy levels , bellman equation , mathematical analysis , domain (mathematical analysis) , motion (physics) , function (biology) , regular polygon , mean curvature , initial value problem , discrete time and continuous time , boundary (topology) , mathematical optimization , geometry , classical mechanics , statistics , physics , quantum mechanics , evolutionary biology , biology
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.