正则树上的拟凸函数
Quasiconvex functions on regular trees
AI总结：
本文在无限有向正则树上基于以根为中点的区间定义拟凸函数，证明给定连续边界数据存在唯一拟凸包络，并刻画其满足的含中位数的均值方程，同时建立其与障碍问题的联系。
摘要：
我们引入了一个定义在无限有向正则树上的拟凸函数概念，该概念依赖于我们对树上区间的理解。我们的定义基于将区间视为以根为区间中点的子树。树中的凸集则是指这样的子集：它包含所有以该集合内节点为端点的区间的每一个中点。于是，树上的拟凸函数是一个实值映射，使得每一个水平集都是凸集。针对这一树上拟凸函数概念，我们证明：给定一个连续边界数据，树上存在唯一的拟凸包络，并且我们刻画了该包络所满足的方程。结果表明，该方程是一个均值性质，它涉及给定顶点的后继节点上函数值的中位数。我们还将定义在树内部的函数的拟凸包络与该特征方程的障碍问题的解联系起来。
英文摘要：
We introduce a definition of a quasiconvex function on an infinite directed regular tree that depends on what we understood by a segment on the tree. Our definition is based on thinking on segments as sub-trees with the root as the midpoint of the segment. A convex set in the tree is then a subset such that it contains every midpoint of every segment with terminal nodes in the set. Then a quasiconvex function is a real map on the tree such that every level set is a convex set. For this concept of quasiconvex functions on a tree, we show that given a continuous boundary datum there exists a unique quasiconvex envelope on the tree and we characterize the equation that this envelope satisfies. It turns out that this equation is a mean value property that involves a median among values of the function on successors of a given vertex. We also relate the quasiconvex envelope of a function defined inside the tree with the solution of an obstacle problem for this characteristic 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。
