### 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 language | English (US) |
---|---|

Pages (from-to) | 344-407 |

Number of pages | 64 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 59 |

Issue number | 3 |

State | Published - Mar 2006 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*59*(3), 344-407.

**A deterministic-control-based approach to motion by curvature.** / Kohn, Robert; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 59, no. 3, pp. 344-407.

}

TY - JOUR

T1 - A deterministic-control-based approach to motion by curvature

AU - Kohn, Robert

AU - Serfaty, Sylvia

PY - 2006/3

Y1 - 2006/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33644591594&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33644591594&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33644591594

VL - 59

SP - 344

EP - 407

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 3

ER -