Research on potential theory for solving nonlinear problems

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03333
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Hiroshima University
• Principal Investigator: Hirata Kentaro 広島大学, 先進理工系科学研究科(理), 准教授 (30399795)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: ポテンシャル論 / 非線形楕円型方程式

Outline of Final Research Achievements

In a bounded domain with smooth or Lipschitz boundary, we established the boundary Harnack principle for positive superharmonic functions satisfying a nonlinear inequality, and applied it to obtain two-sided estimates for positive solutions of a superlinear elliptic equation with 0-Dirichlet boundary values and asymptotic estimates for positive solutions with isolated singularities at a boundary point. Furthermore, we clarified the relationship between boundary radial growth rates and the Hausdorff dimension of singular sets on the boundary for positive solutions of a superlinear elliptic equation in the unit ball, and the relationship between growth rates near interior singular sets and removability of such sets. Also, we give a necessary and sufficient condition for a sublinear elliptic equation with measure coefficients to have a positive continuous solution in a general domain.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

Bidaut-Veron氏とVivier氏は，滑らかな有界領域においてLane-Emden方程式の正値解に対する両側評価を与えたが，0-Dirichlet境界値をもつ正値解に対しては下からの評価が無意味なものであり，証明方法も積分核の具体的表示を用いた弱L1理論に基づくものであったためLipschitz領域の場合に適用することができなかった．本研究では，ポテンシャル論の結果・方法を駆使して境界Harnack原理を確立し，先行研究の不備を補完するだけでなく，新たな証明方法を構築することができた．また，解表示を有さないので，増大度と特異点集合のサイズの関係を明らかにすることも意義のあることである．