New developments on linearized problems of nonlinear elliptic equations

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04951
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Kanazawa University
• Principal Investigator: Ohtsuka Hiroshi 金沢大学, 数物科学系, 教授 (20342470)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: Gel'fand問題 / 解の爆発 / Rellichの等式 / 点渦 / 渦点 / 平衡統計力学 / 線形応答理論

Outline of Final Research Achievements

Focusing on the nonlinear partial differential equation called the Gel'fand problem, I worked on clarifying the detailed behavior of the blow-up phenomenon of the solution of nonlinear partial differential equations. In particular, I aimed to clarify the practical limits of "Rellich's equation" related to conformal field theory. Calculation results were obtained according to the research plan, but unfortunately it was not as accurate as expected. However, in order to break the situation, I conducted joint research with physicists, reconsidering equations from the viewpoint of statistical mechanics, and proceeding with consideration based on the physical theory called linear response theory, and I found a novel discrete approximation of the research object. I believe that this have opened a new way to approach the phenomenon of interest.

Free Research Field

変分問題

Academic Significance and Societal Importance of the Research Achievements

残念ながら解明を目指した事実を示すには十分な結果は得られなかったが、改めて2次元 Gel’fand問題を共形場理論や点渦系など幾何学や統計力学の観点から幅広く考察することで、豊かな構造を発見することができた。高次元Gel’fand 問題にも統計力学的な観点が存在することは知られている。今後本研究課題の対象分野を進展させる可能性がある、新たな視点を発掘できたと考えている。なお、線形応答理論に基づく数学研究は少なく、得られた観点は当初の目標以上の展開を期待できると考えている。数学者と物理学者の共同研究も必ずしも一般的ではないので、双方にとって新規性のある分野融合型の共同研究の進展も期待できる。