Research of an integration representation of the solutions of the partial differential equation of elliptic type in unbounded domains and its stochastic analysis consideration

目的是考虑Dirichlet问题和Neumann问题的解在无界域上对椭圆型偏微分方程边值问题的积分表示。关于玉米和圆筒，已经得到了结果。关于另一个无界域带，虽然已经得到了一个特解的积分表示，但通解的具体组成和一类唯一性问题仍未得到解决。然而，许多关于谐波函数作用的研究结果是一种解。即，最小薄集是潜在的理论例外集，Doob等人对此进行了详细的研究。稀薄集是Ahlfors、Hayman等人从函数理论的角度研究函数递增度的例外集。通常首先在光滑区域研究函数边界附近的行为，然后在Lipschitz区域或具有更一般复杂边界的区域进行研究。在光滑半空间上，Essen等研究了上述两类不相容集的Winner型判据和超调和函数在不相容集外的行为。这些结果被推广到类似的结果，在玉米的无穷点附近，即定义域的角度，以及在圆柱体的无穷点附近，即定义域的尖端。此外，在硬币和圆柱的情况下，可以得到例外集的另一种表达式。这些结果发表在加拿大的《加拿大数学公报》上。这篇文章决定由一家美国杂志（procer . amer . math . soc）发表。和复变量)和捷克斯洛伐克期刊（Czecho.Math.J.）。

Consideration of an integration representation of the solution of the Dirichlet problem and the Neumann problem with respect to the boundary value problem of an elliptic partial differential equation on unbounded domains was the purpose. About the corn and the cylinder, the result has already been obtained. About another unbounded domain strip, although the integration representation of a special solution has already been obtained also the concrete composition of a general solution and the problem of uniqueness of a certain kind are still unsolved. However, many re-search results of the action of the harmonic function which is a solution were able to be obtained. Namely, minimally thin sets are the potential theoretical exceptional sets which were studied in detail by Doob etc. Rarefied sets are the exceptional sets studied by Ahlfors, Hayman, etc. from the function theoretical viewpoint about the degree of increase of a function. It is usual that the behavior near the boundary of the function is first studied in a smooth domain and next studied in a Lipschitz domain or a doma with a still more general complicated boundary. In the half-space which is a smooth domain, with respect to two kinds of exclusion sets above-mentioned, the Winner type criteria and the behavior of superharmonic functions in the outside of the exceptional sets were researched by Essen etc. These results were extended to the results of the same kind near the infinite point of a corn which is the angle of the domain and also near the infinite point of the cylinder prolonged infinitely which is the cusp of the domain. Moreover, the results in the case of a coin and a cylinder were able to be obtained about another expression of the exceptional sets. These results were carried by the journal (Canadian Mathematical Bulletin) of Canada. It is decided to be carried by an American journal (Proc.Amer.Math.Soc. and Complex variables) and the journal (Czecho.Math.J.) of Czechoslovakia.