Monotonicity estimates and local regularity and singularity for the p-harmonic flows

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K04962
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Kumamoto University
• Principal Investigator: Masashi Misawa 熊本大学, 大学院先端科学研究部(理), 教授 (40242672)
• Co-Investigator(Renkei-kenkyūsha): YAMAURA Yoshihiko 日本大学, 文理学部, 教授 (90255597)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 偏微分方程式 / 退化特異放物型方程式系 / 正則性特異性 / 調和写像 / 調和写像熱流 / p調和写像 / p調和写像熱流

Outline of Final Research Achievements

The p-harmonic flow is the heat flow for the p-harmonic maps,the critical
points for p-energy for maps beteween two smooth compact Riemannian manifolds, and given by the degenerate and singular parabolic system of 2nd ordered partial differential equations of so-called p-Laplcian type. We study the regularity and singularity of solutions of the p-harmonic flows. Our main result is the local regularity theorem, which holds true uniformly for the p-energy bounded, regular solutions of the p-harmonic flows. Based on this local regularity theorem, we prove that the weak limits of p-energy bounded, regular p-harmonic flows are partial regular weak solutions of the p-harmonic flows, with the almost optimal size estimation by Hausdorff measure, of their singular sets, the exceptional sets of regularity.

Free Research Field

偏微分方程式