Research on the theories of viscosity solution, weak KAM, and application to the asymptotic analysis on Hamilton-Jacobi equations

2020 Fiscal Year Final Research Report

• Project/Area Number: 17KK0093
• Research Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: The University of Tokyo (2018-2020); Hiroshima University (2017)
• Principal Investigator: Mitake Hiroyoshi 東京大学, 大学院数理科学研究科, 准教授 (90631979)
• Project Period (FY): 2018 – 2020
• Keywords: ハミルトン・ヤコビ方程式 / 均質化問題 / 長時間挙動 / 粘性解理論 / Aubry-Mather理論

Outline of Final Research Achievements

The main subject of my research is nonlinear Partial Differential Equations, and in particular I have made some special effort to the study on the Hamilton-Jacobi (HJ) equation. This equation is an important fundamental equation for various branches of science like classical mechanics, geometric optics, calculus of variations, optimal control and differential games. During the project period, I focused on problems related to various properties of viscosity solutions of Hamilton-Jacobi equations appearing in the context of classical mechanics and crystal growth. In particular, I have worked on the following topics: (a) Asymptotic analysis on Hamilton-Jacobi-Bellman equations (the large time behavior, homogenization), (b) Analysis on the birth-and-spread model equation appearing in the crystal growth, (c) Selection problems for the mean field game. I got several new and important results and published 9 (peer-reviewed) papers.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

補助事業期間中に，最適確率制御問題に現れる退化粘性ハミルトン・ヤコビ方程式と呼ばれるクラスの方程式に適用できるよう，弱 KAM 理論の一般化に取り組んだ．従来の弱 KAM 理論は決定論的な力学系しか扱えないため，新しい道具立てを必要とした．この点を偏微分方程式論から見直すことで決定論及び確率論を 統一する一つの新しい枠組みを幾つか作ることに成功してきた．応用として，漸近解析(長時間挙動，均質化問題)ついて解決した．これらの成果は，偏微分方程式論における粘性解理論，弱KAM理論において，特に重要な学術的意義を持つと期待できる．