Study for Hamilton-Jacobi equations and logarithmic Sobolev inequality

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540165
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic 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・総合科学学術院, 教授 (70102887) ; ISHII Katsuyuki 神戸大学, 海事科学部, 教授 (40232227)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: Hamilton-Jacobi方程式 / 対数型Sobolevの不等式 / Lipschitz正則効果 / 超縮小性

Outline of Final Research Achievements

Throughout this study, I wrote three papers. First, I gave a new and complete proof of Lp logarithmic Sobolev inequality by using the hypercontractivity of the solution of some Hamilton-Jacobi equation. Next, I clarified many properties of the inequality which is obtained by letting p→∞ in this Lp logarithmic Sobolev inequality. This was done by using Laplace transforms and the theory of regular variations. Finally, I investigated the set which determines the structure of solutions of Hamilton-Jacobi equations. This set plays the same role as the uniqueness set for boundary value problems.

Free Research Field

函数方程式論