Development of analysis method for critical problems with logarithmic singularity

2023 Fiscal Year Final Research Report

• Project/Area Number: 19KK0349
• Research Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tohoku University
• Principal Investigator: Ioku Norisuke 東北大学, 理学研究科, 准教授 (50624607)
• Project Period (FY): 2020 – 2023
• Keywords: スケール不変性 / 対数型特異性 / 劣臨界近似 / 指数型非線形項

Outline of Final Research Achievements

As a critical problem with a logarithmic singularity, I focused on the analysis of semi-linear heat equations and semi-linear elliptic equations with exponential nonlinearity, and the analysis of critical functional inequalities. For the former research, we discovered the model case of nonlinear terms, and developed a new method to reduce the analysis of general nonlinear terms into the model case. Regarding the latter, we succeeded in converting the logarithmic singularity that appears in the critical problem into a limit problem of subcritical problems by using a power approximation called the q-logarithm function, and using this, we showed that the concentration level of the Trudinger--Moser inequality can be regarded as the limit form of the concentration level of Sobolev inequality attained by Talenti's function.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

半線形放物型・楕円型方程式は，これまでは冪乗非線形項といった理想的状況下において研究されることが多かった．冪乗非線形項はそのシンプルな見た目に反して豊富な数学的現象を提起するため，多くの関心を集めて深く理解されている．一方で，複雑なこの世界を理解するためには理想的状況の解析だけでは不十分であることも事実である．一般の指数増大度を持つ非線形項を扱うことを可能にした本研究は学術的・社会的に意義深いと考えられる．また，対数型特異性に対して体系的な研究手法はこれまでに十分に開発されてこなかった．本研究で提案した劣臨界近似法は，他の対数型特異性を伴う臨界問題にも応用可能であるため高い学術的意義を持つ．