Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations
Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations
复制标题
程函方程、积分曲率流和非线性抛物型积分微分方程的重复博弈
DOI：
10.3934/dcds.2011.29.1517
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
S. Serfaty
中科院分区：
文献类型：
--
作者：
C. Imbert;S. Serfaty
本文的主要目的是在Kohn和第二作者（2006和2009）以前的工作的精神下，用零和重复博弈来近似几个非局部发展方程：一方面是一般的完全非线性抛物型积分微分方程，另一方面是界面的积分曲率流（英伯特，2008）。为了做到这一点，我们开始构建这样一个游戏的程函方程的速度有一个非常数的符号。这提供了一个（离散）确定性控制解释这些发展方程。在我们所有的博弈中，两个参与人都是依次选择位置的，他们的最终收益取决于他们的位置和其他选择参数。由于近似问题的非局部性，与局部问题相比，它们的选择必须“收集”远离当前位置的信息。对于积分曲率流，玩家选择整个空间中的超曲面以及这些超曲面上的位置。对于抛物型积分微分方程，玩家选择整个空间上的光滑函数。
The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an interface (Imbert, 2008) on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations. In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces. For parabolic integro-differential equations, players choose smooth functions on the whole space.