Geometry of solutions of partial differential equations and the inverse problems accompanied by it

2017 Fiscal Year Final Research Report

• Project/Area Number: 26287020
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Tohoku University
• Principal Investigator: SAKAGUCHI SHIGERU 東北大学, 情報科学研究科, 教授 (50215620)
• Co-Investigator(Kenkyū-buntansha): 倉田 和浩 首都大学東京, 理工学研究科, 教授 (10186489) ; 川上 竜樹 龍谷大学, 理工学部, 准教授 (20546147) ; 宮本 安人 東京大学, 大学院数理科学研究科, 准教授 (90374743) ; 池畠 優 広島大学, 工学(系)研究科(研究院), 教授 (90202910)
• Co-Investigator(Renkei-kenkyūsha): ISHIGE Kazuhiro 東京大学, 大学院数理科学研究科, 教授 (90272020) ; MIKAMI Toshio 津田塾大学, 学芸学部, 教授 (70229657) ; MISAWA Masashi 熊本大学, 大学院自然科学研究科, 教授 (40242672)
• Research Collaborator: CAVALLINA Lorenzo 東北大学, 大学院情報科学研究科, D3
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 熱拡散方程式 / 不変等温面 / 複合媒質 / 不変等熱流面 / 反応拡散方程式 / 半線形楕円型方程式 / 囲い込み法 / 逆問題

Outline of Final Research Achievements

We studied mainly the relationship between the behavior of solutions and the geometry of domains in the problems widely described by partial differential equations from the point of view of inverse problems. First, the topology of unbounded stationary isothermic surfaces in Euclidean 3-space was completely determined, and the almost complete characterization of the hyperplanes and the circular cylinders involving a stationary isothermic surface was obtained. Secondly, in the conductivity equation over composite media in Euclidean 3-space, we succeeded in characterizing the concentric balls by means of the neutral conductors without any influence on outside uniform electric fields. Thirdly, in the heat equation over composite media we obtained the characterization of the concentric balls involving either a stationary isothermic surface or a surface with the constant heat flow property among two-phase heat conductors.

Free Research Field

偏微分方程式論