Viscosity solutions on metric spaces

2015 Fiscal Year Final Research Report

• Project/Area Number: 25610025
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: The University of Tokyo
• Principal Investigator: GIGA Yoshikazu 東京大学, 大学院数理科学研究科, 教授 (70144110)
• Research Collaborator: ASAI Tomoro ; OHTSUKA Takeshi ; GIGA Mi-Ho ; KURODA Hirotoshi ; NAKAYASU Atsushi ; HAMAMUKI Nao
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 粘性解 / 距離空間 / ハミルトン・ヤコビ方程式 / アイコナール方程式 / クリスタライン曲率

Outline of Final Research Achievements

We consider the Eikonal equation in a space such as network or fractal, where the gradient of function is not well-defined in canonical way. We establish the theory of viscosity solutions in a general metric space. We also establish the theory of viscosity solutions for a curvature flow equation describing motion of a surface of a crystal or a grain boundary, especially a crystalline curvature flow, which has a strong anisotropy, when the surface is regarded as a curve. A curvature flow with strong anisotropy is regarded at least formally as a gradient flow of area measured by non-Euclidean metric in a suitable metric space. However, a general theory is not yet established so we study the problem individually.

Free Research Field

非線形解析