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arXiv 0811.0208math.APmath.OCmath.PR
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有偏拔河，有偏无穷拉普拉斯算子，以及与指数锥的比较

Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

Yuval Peres, Gábor Pete, Stephanie Somersille

备注：
27 pages, 1 figure. Some typos corrected
发表信息：
Calc. Var. and PDE 38 (2010), 541-564
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摘要：

证明了如果U是开域，其闭包\subset在路径度量上是紧的，F是\partial{U}上的Lipschitz函数，则对每个β-有偏无穷Laplacian方程β存在唯一粘性解|\nabla使用|+ Δ_ \infty u=0在U上延伸F，其中Δ_ \infty u=| \nabla使用|^{-2} \sum_{i,j} u_{x_i} u_{x_ix_j} u_{x_j}。在证明中，我们扩展了Peres，Schramm，Sheffield和Wilson的拔河思想，并定义了β偏\eps -博弈如下。起始位置为x_0 \in U。在第k^ \text{th}步，两个参与者掷出一个适当偏向的硬币（在我们的关键例子中，参与者I以\exp（β \eps）为1的赔率获胜），获胜者选择x_k，d（x_k，x_{k-1}）< \eps。当x_k \in \partial{U}，参与者II支付给参与者I金额F（x_k）时，游戏结束。我们证明了这个对策的值u^{ \eps }（x_0）存在，且|u^ \eps - u | \infty \to 0作为\eps \to 0，其中u是F到\overline{U}的唯一扩张，满足与β-指数锥的比较.与指数锥的比较是我们在这里引入的一个概念，推广了Crandall，Evans和Garibaldi关于与线性锥比较的定理，我们证明了一个连续函数满足与β-指数锥的比较当且仅当它是β-有偏无穷拉普拉斯方程的粘性解。

英文摘要：

We prove that if U\subset\R^n is an open domain whose closure \overline{U} is compact in the path metric, and F is a Lipschitz function on \partial{U}, then for each β\in\R there exists a unique viscosity solution to the β-biased infinity Laplacian equation β|\nabla u| + Δ_\infty u=0 on U that extends F, where Δ_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased \eps-game as follows. The starting position is x_0 \in U. At the k^\text{th} step the two players toss a suitably biased coin (in our key example, player I wins with odds of \exp(β\eps) to 1), and the winner chooses x_k with d(x_k,x_{k-1}) < \eps. The game ends when x_k \in \partial{U}, and player II pays the amount F(x_k) to player I. We prove that the value u^{\eps}(x_0) of this game exists, and that \|u^\eps - u\|_\infty \to 0 as \eps \to 0, where u is the unique extension of F to \overline{U} that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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