Min–max formulas for nonlocal elliptic operators
Min–max formulas for nonlocal elliptic operators
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DOI：
10.1007/s00526-019-1631-z
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发表时间：
2016-06
影响因子：
2.1
通讯作者：
Nestor Guillen;Russell W. Schwab
中科院分区：
文献类型：
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作者：
Nestor Guillen;Russell W. Schwab
在这项工作中，我们给出了Lipschitz算子在函数空间（也是，）上的一个特征，这些函数服从全局比较性质，即那些在其图形可能接触的任何点上保持输入函数的全局顺序的函数，通常被称为“椭圆”算子。这里M是一个完备的黎曼流形。特别是，我们表明，所有这样的运营商可以写为一个最小最大的线性算子，是漂移扩散和积分微分部分的组合。在线性（和非局部）的情况下，这些算子在20世纪60年代就已经被刻画出来了，而在局部但非线性的情况下--例如局部Hamilton-Jacobi-Bellman算子--这种刻画也是从大约20世纪60年代或70年代开始就已经被知道和使用了。我们的主要定理包含这两个结果作为特殊情况。证明了任何非线性标量椭圆型方程都可以表示为适当的微分对策的Isaacs方程。我们的方法是“项目”的运营商的一个作用于大型有限图形的函数，近似的流形，使用非光滑分析，推导出一个最小最大公式在这个有限维水平，然后通过限制，以解除公式原来的运营商。
In this work, we give a characterization of Lipschitz operators on spaces offunctions (also,,,) that obey the global comparison property—i.e. those that preserve the global ordering of input functions at any points where their graphs may touch, often called “elliptic” operators. HereMis a complete Riemannian manifold. In particular, we show that all such operators can be written as a min–max over linear operators that are a combination of drift–diffusion and integro-differential parts. In thelinear(and nonlocal) case, these operators had been characterized in the 1960s, and in thelocal, but nonlinearcase—e.g. local Hamilton–Jacobi–Bellman operators—this characterization has also been known and used since approximately since 1960s or 1970s. Our main theorem contains both of these results as special cases. It also shows any nonlinear scalar elliptic equation can be represented as an Isaacs equation for an appropriate differential game. Our approach is to “project” the operator to one acting on functions on large finite graphs that approximate the manifold, use non-smooth analysis to derive a min–max formula on this finite dimensional level, and then pass to the limit in order to lift the formula to the original operator.