Aronsson方程的最大解和最小解：$L^{\infty}$变分问题与对策论
Maximal and minimal solutions of an Aronsson equation: $L^{\infty}$ variational problems versus the game theory
摘要：
狄利克雷问题$$ \begin{cases} Δ_{\infty}u-|Du|^2=0 \quad \text{on $Ω\subset \Rset ^n$} u|_{\partial Ω}=g \end{cases} $$可能有许多解，其中$Δ_{\infty}u=\sum_{1\leq i,j\leq n}u_{x_i}u_{x_j}u_{x_ix_j}$。本文利用$L^{\infty}$的变分法证明了最大解是$H(p,z)={1\over 2}|p|^2-z$的唯一绝对极小元，最小解是拔河游戏的连续值函数.我们还将刻画既不是绝对极小值也不是值函数的解的图。剩下的一个有趣的问题是如何解释这些中间解决方案。我们的大多数方法都是基于Barles-Busca [BB]的思想。
英文摘要：
The Dirichlet problem $$ \begin{cases} Δ_{\infty}u-|Du|^2=0 \quad \text{on $Ω\subset \Rset ^n$} u|_{\partial Ω}=g \end{cases} $$ might have many solutions, where $Δ_{\infty}u=\sum_{1\leq i,j\leq n}u_{x_i}u_{x_j}u_{x_ix_j}$. In this paper, we prove that the maximal solution is the unique absolute minimizer for $H(p,z)={1\over 2}|p|^2-z$ from calculus of variations in $L^{\infty}$ and the minimal solution is the continuum value function from the "tug-of-war" game. We will also characterize graphes of solutions which are neither an absolute minimizer nor a value function. A remaining interesting question is how to interpret those intermediate solutions. Most of our approaches are based on an idea of Barles-Busca [BB].
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。
