Nonlinear elliptic partial differential equations having variation structure

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K17623
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Keio University (2018); Kanazawa University (2016-2017)
• Principal Investigator: Ikoma Norihisa 慶應義塾大学, 理工学部(矢上), 准教授 (50728342)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 非線形楕円型方程式 / 最小化問題 / 最大化問題 / 特異摂動問題 / 幾何解析 / 臨界点理論

Outline of Final Research Achievements

In this project, the existence of solutions and their properties were studied for nonlinear elliptic partial differential equations. In particular, we treated equations which have variational structure (for instance, equations with fractional operators of elliptic operators). We proved the existence of solutions satisfying some properties and showed the existence of multiple solutions. We also studied a variant of the Trudinger-Moser inequality and found conditions when the inequality is satisfied as an equality. This inequality is related to a certain nonlinear elliptic partial differential equation.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

分数冪作用素を含む方程式に対する成果は，既存研究の結果を拡張し，その証明はこれまでの議論を整理し，様々な場合を統一的に扱えるようにするものである．また，他のテーマの成果は，更なる研究を誘発する研究成果や既存研究の枠組みに収まらないものであり，これらの成果を得るために新たな手法を開発した．このようなことから本研究成果は学術的に意義があるものである．