Advances in Differential Equations

A billiard-based game interpretation of the Neumann problem for the curve shortening equation

Yoshikazu Giga and Qing Liu

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Abstract

This paper constructs a family of discrete games, whose value functions converge to the unique viscosity solution of the Neumann boundary problem of the curve shortening flow equation. To derive the boundary condition, a billiard semiflow is introduced and its basic properties near the boundary are studied for convex and more general domains. It turns out that Neumann boundary problems of mean curvature flow equations can be intimately connected with purely deterministic game theory.

Article information

Source
Adv. Differential Equations Volume 14, Number 3/4 (2009), 201-240.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867265

Mathematical Reviews number (MathSciNet)
MR2493561

Zentralblatt MATH identifier
1170.35437

Subjects
Primary: 35K20: Initial-boundary value problems for second-order parabolic equations 49L25: Viscosity solutions 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 91A05: 2-person games

Citation

Giga, Yoshikazu; Liu, Qing. A billiard-based game interpretation of the Neumann problem for the curve shortening equation. Adv. Differential Equations 14 (2009), no. 3/4, 201--240. https://projecteuclid.org/euclid.ade/1355867265.


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