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

本课题从势论或概率论的角度研究了无界域上椭圆型或抛物型偏微分方程无穷远处解的边界行为。特别地，讨论了典型无界域上解具有不规则行为的例外集的定性或定量性质，例如：稀有集和a-最小薄集。关于这些例外集的定性表征，有“赢家型判断条件”和“确定集”。虽然在上一个研究任务中，这些区域被做成了锥形区域，但这次这些区域被扩展成了圆柱形区域。当考虑这些例外集的定量表征时，有一个问题是找到覆盖这些薄集的若干种可数无限球。埃森在半空间中关于这些薄集的结果推广到圆锥域上的结果。所有研究结果均发表在美国期刊上（Proc. Amer）。数学。Soc。和复杂变量)，捷克期刊(捷克。数学。J.)和国内期刊(广岛数学。J.和《纯数学高等研究》)，并将在国内期刊（北海道数学）上发表。j .)。对于抛物型方程即热方程的解（温度），虽然目前还没有得到调和函数的解所对应的结果，但已经准备好了。

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.