Fattening and Comparison Principle for Level-set Equation of Mean Curvature Type

In this paper, we give several applications of the discrete game approach to partial differential equations. We first present a rigorous game-theoretic proof of fattening phenomenon for motion by curvature with figure-eight shaped initial curves without using parabolic PDE theory. The proof is based on a comparison between the game value and its inverse one. Accompanied with the example of figure eight, our second result shows, for the stationary equation of mean curvature type in an arbitrary region Ω, that fattening of positive curvature flow with initial surface ∂Ω causes loss of the weak comparison principle, which partially answers an open question posed by R. V. Kohn and S. Serfaty in 2006. In addition, we prove the existence of solutions of the stationary problem and its game approximation in the absence of comparison principles but under regularity conditions of the flow. The main difference between our games and those in other papers is that we take the domain perturbation into consideration.