A deterministic-control-based approach to motion by curvature

Robert V. Kohn, Sylvia Serfaty

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)344-407
Number of pages64
JournalCommunications on Pure and Applied Mathematics
Volume59
Issue number3
DOIs
StatePublished - Mar 2006

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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