标题： 图上的离散无穷拉普拉斯方程和拔河游戏 显示英文标题

标题： Discrete infinity Laplace equations on graphs and tug-of-war games

摘要： 我们研究在具有有限宽度的子图$$\Delta_{\infty} u(x) = \inf_{y \sim x}u(y)+\sup_{y \sim x}u(y)-2u(x) = f(x).$$上以下离散无穷拉普拉斯方程的狄利克雷问题。我们称一个子图具有有限宽度，如果所有顶点到边界距离是统一有界的。 通过佩龙方法，我们证明了有界解的存在性。 我们还通过建立一个比较结果证明了当$f$非负时的唯一性，从而得到 Peres, Schramm, Sheffield 和 Wilson (2009) 引入的相应拔河游戏的游戏值的存在性。 作为应用，我们展示了无穷调和函数的一个强刘维尔性质。 通过 Arzela-Ascoli 的论证，我们证明了解在$\varepsilon$拔河游戏中当$\varepsilon$趋于 0 时的收敛性。 相应地，我们得到了在具有有限宽度的欧几里得域上归一化无穷拉普拉斯方程的有界解的存在性。 显示英文摘要

摘要： We study the Dirichlet problem of the following discrete infinity Laplace equation on a subgraph with finite width $$\Delta_{\infty} u(x) = \inf_{y \sim x}u(y)+\sup_{y \sim x}u(y)-2u(x) = f(x).$$ We say that a subgraph has finite width if the distances from all vertices to the boundary are uniformly bounded. By Perron's method, we show the existence of bounded solutions. We also prove the uniqueness if $f$ is nonnegative by establishing a comparison result, and hence obtain the existence of game values for corresponding tug-of-war games introduced by Peres, Schramm, Sheffield, and Wilson (2009). As an application we show a strong Liouville property for infinity harmonic functions. By an argument of Arzela-Ascoli, we prove the convergence of solutions of $\varepsilon$-tug-of-war games as $\varepsilon$ tends to 0. Correspondingly, we obtain the existence of bounded solutions to normalized infinity Laplace equations on Euclidean domains with finite width.