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

本研究从位势论或概率论的观点出发，研究了无界区域无穷远处椭圆型或抛物型偏微分方程解的边界性态，特别是，定性或定量性质的例外集上的解决方案的行为不规则，例如，稀疏集和一个-讨论了典型无界域的无穷远极小薄集，关于这些例外集的定性刻画，有“赢家型判定条件”和“决定集”.虽然这些都是由上一个研究任务的锥形域，这些都是扩大到圆柱domains.When定量表征这些例外集被认为是，有一个问题，找到某些类型的可数无限的球与这些薄集覆盖。将埃森关于半空间中这类薄集的结果推广到锥域中，所有结果发表在美国期刊（Proc. Amer. Math. Soc. and Complex Variables）、捷克期刊（Czech.数学J.）和国内期刊（广岛数学杂志和纯数学高级研究），并将在国内期刊（北海道数学杂志）上发表。关于抛物型方程，即热方程的解（温度），虽然还没有得到相应于调和函数的结果，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.