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

无界域椭圆型偏微分方程解的积分表示及其随机分析考虑的研究

基本信息

批准号：
13640153
负责人：
MIYAMOTO Ikuko
金额：
$ 2.24万
依托单位：
Chiba University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640153/
关键词：
Dirichlet Problem unbounded domain cone cylinder strip minimally thin sets rarefied sets exceptional sets

项目摘要

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.

考虑无界域上椭圆型偏微分方程边值问题的Dirichlet问题和Neumann问题的解的积分表示。关于玉米和圆筒，已经得到了结果。关于另一个无界区域带，虽然已经得到了其特解的积分表示，但其通解的具体组成和某种唯一性问题仍然没有解决。然而，调和函数作为一个解的作用却没有得到很多的研究结果。也就是说，极小薄集是Doob等人详细研究的潜在理论例外集，稀疏集是Ahlfors，Hayman等人从函数论观点研究的关于函数增加度的例外集。通常首先在光滑区域中研究函数边界附近的行为，然后在Lipschitz区域或具有更一般的复杂边界的区域中研究。在光滑区域的半空间中，对于上述两类排斥集，埃森等人研究了超调和函数在例外集外的赢家型判别准则和超调和函数的性质，并将这些结果推广到了在区域的角的无穷远点附近以及在柱面的无穷远点附近的同类结果这是一个无限大的域的尖点。此外，在硬币和圆柱的情况下，能够获得关于例外集的另一种表达式的结果。这些结果刊登在加拿大数学通报（Canadian Mathematical Bulletin）上。决定由美国《Proc.Amer.Math. Soc. and Complex Variables》杂志和捷克《Math.J.》杂志刊登。捷克斯洛伐克。