论文标题
Quasiconvex在常规树上起作用
Quasiconvex functions on regular trees
论文作者
论文摘要
我们在无限的定期树上介绍了Quasiconvex函数的定义，该函数取决于我们在树上的细分市场所理解的内容。我们的定义是基于将段作为词根作为段中点的子树的思考。然后，树中的凸集是一个子集，因此它包含每个段的每个中点，并包含集合中的终端节点。然后，Quasiconvex函数是树上的真实地图，因此每个级别集都是凸集。对于Quasiconvex在树上起作用的概念，我们显示给定一个连续的边界基准，存在着树上唯一的Quasiconvex信封，我们表征了该信封所满足的方程式。事实证明，该方程是一个平均值属性，它涉及给定顶点后继器功能值之间的中位数。我们还将树内定义的函数的Quasiconvex信封与该特征方程的障碍物问题的解决方案联系起来。
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.
