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2019 Fiscal Year Research-status Report

楕円型過剰決定問題の発展方程式的解析

Research Project

Project/Area Number 16K17628
Research InstitutionTokyo Institute of Technology

Principal Investigator

小野寺 有紹  東京工業大学, 理学院, 准教授 (70614999)

Project Period (FY) 2016-04-01 – 2021-03-31
Keywords自由境界問題 / 過剰決定問題 / 発展方程式
Outline of Annual Research Achievements

昨年度に引き続き,Bernoulli の自由境界問題の発展方程式による解析を行った.
函数解析的な統一的視点によって過剰決定問題に対する発展方程式的解析手法を定式化し,それを本問題に適用できることを線型化作用素のスペクトル解析によって証明した.さらに,その大域的解挙動について球面調和関数を用いた保存量に着目して調べた.

また,Serrin の対称性の結果が著名な Saint-Venent 問題の剛性の安定性について Alexandra Gilsbach 氏と共同研究を開始し,その線形化作用素のスペクトル解析を行った.
平行移動不変性から現れる線形化作用素の退化性を扱うため,リャプノフ-シュミット法によって有限次元の解析へと帰着した.

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

Saint-Venent 問題が当初計画していたよりも数学的に面白い構造があることが判明し,本年2月に来日した外国人学振特別研究員の Alexandra Gilsbach 氏とともに現在解析を行っている.

Strategy for Future Research Activity

Bernoulliの自由境界問題については,Elie Cartan 研究所の Antoine Henrot 教授と共同研究を進め,双曲型解の大域挙動を調べる予定である.
また,Saint-Venen 問題の安定性については Alexandra Gilsbach 氏と線型化による解析を押し進める予定である.

Causes of Carryover

感染症拡大のため,当初予定した研究打ち合わせのための出張を行うことを見合わせた.
したがって,研究計画遂行を一年延期し,本年度後半に出張等にて使用する予定である.

  • Research Products

    (10 results)

All 2020 2019 Other

All Int'l Joint Research (1 results) Presentation (9 results) (of which Int'l Joint Research: 8 results,  Invited: 9 results)

  • [Int'l Joint Research] University of Florence/University of Naples Federico II(イタリア)

    • Country Name
      ITALY
    • Counterpart Institution
      University of Florence/University of Naples Federico II
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2020

    • Author(s)
      Michiaki Onodera
    • Organizer
      Seminar on Calculus of Variations & Partial Differential Equations
    • Int'l Joint Research / Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2020

    • Author(s)
      Michiaki Onodera
    • Organizer
      Analysis Seminar
    • Int'l Joint Research / Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2019

    • Author(s)
      Michiaki Onodera
    • Organizer
      6th Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDE's
    • Int'l Joint Research / Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2019

    • Author(s)
      Michiaki Onodera
    • Organizer
      RIMS Conference on "Viscosity solution approach to asymptotic problems in front propagation, dynamical system and related topics"
    • Int'l Joint Research / Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2019

    • Author(s)
      Michiaki Onodera
    • Organizer
      12th ISAAC Congress, Session 10: Geometric & Regularity Properties of Solutions to Elliptic and Parabolic PDEs
    • Int'l Joint Research / Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2019

    • Author(s)
      Michiaki Onodera
    • Organizer
      日本数学会秋季総合分科会函数方程式分科会 (特別講演)
    • Invited
  • [Presentation] Foliated hyperbolic solutions to\\Bernoulli's free boundary problem2019

    • Author(s)
      Michiaki Onodera
    • Organizer
      The 2nd Taiwan-Japan Joint Conference on Differential Geometry
    • Int'l Joint Research / Invited
  • [Presentation] Foliated solutions to Bernoulli's free boundary problem2019

    • Author(s)
      Michiaki Onodera
    • Organizer
      Geometric Analysis and General Relativity
    • Int'l Joint Research / Invited
  • [Presentation] Dynamical approach to an overdetermined problem2019

    • Author(s)
      Michiaki Onodera
    • Organizer
      PDE & Analysis Seminar
    • Int'l Joint Research / Invited

URL: 

Published: 2021-01-27  

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