2019 Fiscal Year Final Research Report

Geometric analysis on Kuramochi boundaries

Research Project

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Project/Area Number	16K05124
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Research Field	Geometry
Research Institution	Kanazawa University
Principal Investigator	Kasue Atsushi 金沢大学, 数物科学系, 教授 (40152657)
Co-Investigator(Kenkyū-buntansha)	服部多恵石川工業高等専門学校, 一般教育科, 准教授 (40569365)
Project Period (FY)	2016-04-01 – 2020-03-31
Keywords	非再帰的ネットワーク / 倉持境界 / ディリクレ・ノイマン写像 / モスコ型収束 / ディリクレ境界値問題 / ペロン法 / リウヴィユ性 / 大森-ヤウ型の弱最大値原理
Outline of Final Research Achievements	We study the Kuramochi boundary of a connected nonparabolic network in connection with the Hilbert space consisting of functions with finite Dirichlet sum. It is proved that the Dirichlet form induced on the boundary of a connected finite subset of the network Mosco-converges to that on the Kuramochi boundary as the connected finite subsets increases to exhaust the network. When we consider a nonparabolic weighted Riemannian manifold, we find the similar result hold. We deal also with a nonlinear resistive network in the framework of modular sequence spaces, and study the Kuramochi type boundary. We show that Perron method is applicable to solve Dirichlet boundary value problems, and the regularity of the boundary is investigated. Moreover the equivalence of the Liouville property, the Khas'minskii condition and the weak Omori-Yau maximum principle for operators of Laplacian with potential term is proved. A number of criteria for these properties are given.
Free Research Field	数物科学
Academic Significance and Societal Importance of the Research Achievements	ネットワークの収束に関する過去に類似の結果がないオリジナルな発見は、非再帰的ネットワークのエネルギー有限な調和関数全体のなす空間を表現する倉持コンパクト化と倉持境界に関して新しい見識を提供する。非再帰的重み付きリーマン多様体にも適用できる内容で、倉持境界に関する課題解決の重要なステップとなる。また、モジュラー列空間の枠組みにおける非線形抵抗ネットワークの非線形ポテンシャル論の研究成果は、先駆け的内容で、今後の非線形ネットワークの幾何解析の基礎となる。