Deepening and application of a theory for the logarithmic Sobolev inequality

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K04949
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: University of Toyama
• Principal Investigator: Fujita Yasuhiro 富山大学, 大学院理工学研究部(理学), 教授 (10209067)
• Co-Investigator(Renkei-kenkyūsha): Ishii Hitoshi 早稲田大学, 教育・総合学術院 (70102887) ; Ishii Katsuyuki 神戸大学, 海事科学部, 教授 (40232227)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 対数型 Sobolev の不等式 / Hamilton-Jacobi方程式 / 下からの評価 / gradient / Cauchy 問題 / 完全な証明

Outline of Final Research Achievements

I was able to achieve an important aim that I planned in this study at first. It is to provide a lower estimate of the sup-norm of the gradient of a function by using the logarithmic Sobolev inequality with index p which is equal to infinity. This estimate is applied to show the optimality of the decay rate of the sup-norm of the gradients to solutions of several Cauchy problems. This result has been published in an appropriate mathematical journal. On the other hand, through workshops, I let many researchers know widely about my complete proof of the logarithmic Sobolev inequality with index p which is greater than 1. The paper of this proof was published in the beginning of this study. In these senses, the result of this study was able to be accomplished in a satisfactory form.

Free Research Field

数学基礎解析