A game approach to free boundary problems of anisotropic forced mean curvature flow equations
Takuya Sato
TL;DR
This work formulates a free boundary problem for anisotropic forced mean curvature flow as a degenerate elliptic PDE and provides a deterministic, discrete-time two-player game representation whose value functions converge to a viscosity solution of the problem. By applying a Cole–Hopf transformation, the authors translate the game dynamics into a DP framework and prove convergence of half-relaxed limits to sub- and supersolutions, enabling a robust comparison principle. The main result establishes uniqueness of solutions under time-discrete game-based analysis, and the applications connect the long-time behavior of the evolving front to the Wulff shape, yielding finite limiting behavior and geometric asymptotics. Overall, the paper integrates game-theoretic representations with viscosity-solution techniques to advance understanding of free boundary problems in anisotropic front propagation.
Abstract
We consider the free boundary problems of degenerate elliptic equations that describe the level set formulation of the interface motion evolved by anisotropic forced mean curvature flows. The type of free boundary problems in this paper was initially studied as the first-order Hamilton-Jacobi-Isaacs equations arising in pursuit-evasion differential games and applied to the models of first-order front propagation in Soravia (1994). In this paper, we consider an extension of these free boundary problems to the second-order equations and give a deterministic game representation based on a discrete approximation scheme in Kohn and Serfaty (2006). Furthermore, we prove the comparison principle for our free boundary problems by using the framework of time-discrete games.