open access

https://doi.org/10.14943/105482

https://hdl.handle.net/2115/87822

A game-theoretic approach to the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation

英語

Deterministic game

2023-02-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

1149

1-40

We consider the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation by using a game-theoretic approximation, which we extend from that of Kohn and Serfaty [13]. The paper [13] gives a deterministic twoperson zero-sum game whose value functions approximate the solution to the level set mean curvature flow equation without obstacle functions. We prove that moving curves governed by the mean curvature flow converge in time to the boundary of the convex hull of obstacles under some assumptions on the initial curves and obstacles. Convexity of the initial set, as well as smoothness of the initial curves and obstacles, are not needed. In these proofs, we utilize the properties of the game trajectories given by very elementary game strategies and consider reachability of each player. Also, when the equation has driving force, we present several examples of computation of the asymptotic behavior, including a problem dealt in [8].

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