Nonlinear partial differential equations on sub-Riemannian manifolds based on viscosity solution theory

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03574
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Okinawa Institute of Science and Technology Graduate University (2022-2023); Fukuoka University (2019-2021)
• Principal Investigator: LIU Qing 沖縄科学技術大学院大学, 幾何学的偏微分方程式ユニット, 准教授 (70753771)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 非線形偏微分方程式 / サブリーマン多様体 / 粘性解 / 凸関数 / 凸集合

Outline of Final Research Achievements

In recent years, the importance of mathematical analysis has been recognized not only in Euclidean spaces but also on sub-Riemannian manifolds with more complex geometric structures. In this research project, we studied partial differential equations in sub-Riemannian manifolds arising from various fields including biology, optimal control theory and image processing, focusing on the　analysis of their geometric properties. We extended the well-established viscosity solution theory for fully nonlinear equations in Euclidean spaces to more general geometric settings. Additionally, we investigated new notions of convex sets, convex functions and convex envelopes that align with the space geometric properties, leading to a deeper understanding from the perspective of partial differential equations.

Free Research Field

数学解析

Academic Significance and Societal Importance of the Research Achievements

サブリーマン多様体における偏微分方程式の数学解析は，数学のみならず，生物学や工学などの問題にも様々な応用がある重要なテーマです．我々の研究は、複雑な幾何学的構造を持つ空間を理解するための枠組みを提供し，そのような空間における偏微分方程式及びその幾何学的性質の研究において基本的な数学的ツールを確立しました．現実世界の応用に現れる様々な数学モデルを研究するための数学的基盤を構築しました．