Research of the solutions of the partial differential equation of elliptic type or parabolic type in unbounded domains and its stochastic analysis consideration

本文从位势理论或概率论的角度研究了椭圆型或抛物型偏微分方程解在无界区域无穷远处的边界行为，特别是讨论了解在典型无界区域无穷远处表现不规则的例外集的定性或定量性质，例如稀疏集和a-极小薄集。关于这些例外集的定性刻画，存在“胜利者类型的判断条件”和“决定集”。虽然上一次的研究任务是将这些球推广到圆锥域，但这一次将它们推广到了圆柱域。当考虑到这些例外集的定量特征时，存在一个问题，即如何找到覆盖这些薄集的某些类型的可数无限球。将Essen关于半空间中这些薄集的结果推广到锥域上的结果，所有结果都发表在美国期刊(PROC.阿默。数学课。SoC。和复变量)，捷克期刊(捷克语。数学课。J.)和国内期刊(广岛数学)。和纯数学高级研究)，并将在国内期刊(北海道数学)上发表。关于抛物型方程的解(温度)，即热方程，虽然还没有得到与调和函数相对应的结果，但已经准备好得到它们。

In this research project the boundary behavior of the solutions of elliptic or parabolic partial differential equations at infinity of unbounded domains was investigated from the view point of potential theory or the theory of probability.In particular, qualitative or quantitative properties of the exceptional sets on which the solutions behave irregularly e.g. rarefied sets and a-minimally thin sets at infinity of typical unbounded domains were treated.About qualitative characterization of these exceptional sets, there are "the judgment condition of Winner type" and "the sets of determination" of these thin sets. Although these were made to conical domains by the last research task, these were extended to cylindrical domains this time.When quantitative characterization of these exceptional sets are considered, there is a problem to find certain kinds of countably infinite of balls with which these thin sets are covered. The result obtained about these thin sets in the half space by Essen was extended to the results in conical domains.All results were published in American journals (Proc. Amer. Math. Soc. and Complex variables), a Czech journal (Czech. Math. J.) and domestic journals (Hiroshima Math. J. and Advanced Studies in Pure Math.), and will be published in a domestic journal (Hokkaido Math. J.).With respect to solutions (temperatures) of the equations of parabolic type, i.e. heat equations, although the results corresponding to ones obtained about harmonic functions are not obtained yet, it is ready to prepare to get them.