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Qualitative study of nonlinear elliptic equations based on refinement of classical inequalities

Research Project

Project/Area Number 16K05189
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionIbaraki University

Principal Investigator

Horiuchi Toshio  茨城大学, 理工学研究科(理学野), 教授 (80157057)

Co-Investigator(Kenkyū-buntansha) 下村 勝孝  茨城大学, 理工学研究科(理学野), 教授 (00201559)
中井 英一  茨城大学, 理工学研究科(理学野), 教授 (60259900)
安藤 広  茨城大学, 理工学研究科(理学野), 准教授 (60292471)
Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords加藤の不等式 / CKN型不等式 / p-ラプラシアン / 重み付きハーディ不等式 / 重み付きソボレフ不等式 / 強最大値原理 / 逆最大値原理 / 非線型変分問題 / CKN 型不等式 / 重み付きソボレフの不等式 / 非線形楕円型変分問題 / 退化楕円型偏微分方程式
Outline of Final Research Achievements

Basic research on the refinement of the weighted classical inequalities centered on Hardy inequalities, Sobolev inequalities, Kato's inequalities and CKN-type inequalities for the generalized p-Laplace operator A was energetically carried out. Specifically, the following researches were conducted: (1) Establishment of Kato's inequality of measure values for the operator A, Strong Maximum and Inverse Maximum Principles for quasi-linear elliptic operators with A as the principal part, (2) Kato's inequalities including the boundary of the domain for p-Laplacian, (3) Establishment of CKN-type inequalities for p = 1 and critical case, (4) Establishment of one-dimensional Hardy inequalities under one-sided boundary conditions. We introduced weak Hardy type inequalities whose weight are the power of the distance from the boundary and applied them to study variational problem.

Academic Significance and Societal Importance of the Research Achievements

学術的意義を具体的に述べる:(1)適切に許容空間を設定することにより準線型楕円型作用素に対する強最大値原理と逆最大値原理の証明、境界値問題の解の一意存在性、解の特異性が調べられた。また、許容空間解とRenormalized解との関係が明らかにされた。(2) 準線形の場合に境界まで込めた加藤の不等式が初めて議論された。 (3)p=1の場合のCKN型不等式が等周不等式であることに着目し、その証明と対称性の崩れの研究が行われた。(4)すべてのベキ型の重みに対してHardy不等式が1次元の片側境界条件の下で確立された。応用として、境界からの距離のべきを重みとする弱Hardy型の不等式が導入された。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (23 results)

All 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (4 results) Journal Article (10 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 10 results,  Open Access: 8 results,  Acknowledgement Compliant: 3 results) Presentation (9 results) (of which Int'l Joint Research: 3 results,  Invited: 2 results)

  • [Int'l Joint Research] Chalmers University(スウェーデン)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Chalmers University(スウェーデン)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Chalmers University(Sweden)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Goteborg University(Sweden)

    • Related Report
      2016 Research-status Report
  • [Journal Article] One dimensional weighted Hardy’s inequalities and application2020

    • Author(s)
      Hiroshi Ando, Toshio Horiuchi, Xiaojing Liu
    • Journal Title

      Journal of Mathematical Inequalities

      Volume: 1-16

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] On the Caffarelli-Kohn-Niremberg type inequalities and related topics [translation]2019

    • Author(s)
      Toshio Horiuchi
    • Journal Title

      Sugaku Expositions (Amer. Math. Soc.)

      Volume: 32-2 Pages: 233-259

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Kato's inequalities up to the boundary for a quasilinear operator2019

    • Author(s)
      Toshio Horiuchi, Peter Kumlin
    • Journal Title

      Scientiae Mathematicae Japonicae in Editione Electronica

      Volume: 32 Pages: 1-14

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Kato&apos;s inequalities for admissible functions to quasilinear elliptic operators <i>A </i>2019

    • Author(s)
      Toshio Horiuchi, Xiaojing Liu
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 51 Issue: 0 Pages: 49-64

    • DOI

      10.5036/mjiu.51.49

    • NAID

      130007687047

    • ISSN
      1343-3636, 1883-4353
    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Kato’s inequalities for admissible functions to quasilinear elliptic operators A2019

    • Author(s)
      Xiaojing Liu, Toshio Horiuchi
    • Journal Title

      Mathematical journal of ibaraki University

      Volume: 51

    • NAID

      130007687047

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The equivalences among <i>p</i>-capacity, <i>p</i>-Laplace-capacities and Hausdorff measure2018

    • Author(s)
      Liu Xiaojing、Horiuchi Toshio
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 50 Issue: 0 Pages: 5-13

    • DOI

      10.5036/mjiu.50.5

    • NAID

      130007465434

    • ISSN
      1343-3636, 1883-4353
    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The equivalences among p-capacity, $p$-Laplace-capacities and Hausdorff measure.2018

    • Author(s)
      Xiaojing Liu, Toshio Horiuchi
    • Journal Title

      Mathematical Journal of Ibaraki University Volume

      Volume: 50

    • NAID

      130007465434

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Radial symmetry and its breaking in the Caffarelli-Kohn- Nirenberg type inequalities2016

    • Author(s)
      Naoki. Chiba, Toshio. Horiuchi
    • Journal Title

      Proceedings of the Japan academy

      Volume: 92 (4) Issue: 4 Pages: 51-55

    • DOI

      10.3792/pjaa.92.51

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Remarks on Kato's inequality when $\Delta_p u$ is a measure2016

    • Author(s)
      Xiaojing Liu, Toshio Horiuchi
    • Journal Title

      Mathematical Journal of Ibaraki University

      Volume: 48 Pages: 45-61

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Remarks on the strong maximum principle involvingp-Laplacian2016

    • Author(s)
      Xiaojing Liu, Toshio Horiuchi
    • Journal Title

      Hiroshima Mathematical Journal

      Volume: 46 Pages: 311-331

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] One dimensional weighted Hardy's inequalities and application2019

    • Author(s)
      堀内利郎
    • Organizer
      日本数学会、函数方程式分科会
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] One dimensional Weighted Hardy’s Inequalities and application2019

    • Author(s)
      Xiaojing Liu
    • Organizer
      調和解析セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Weighted Hardy's inequalities with compact perturbations2019

    • Author(s)
      Hiroshi Ando
    • Organizer
      調和解析セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Weighted Hardy's inequalities with compact perturbations2019

    • Author(s)
      Hiroshi Ando
    • Organizer
      日本数学会 函数方程式論分科会
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Improved Kato's inequalities for a quasilinear elliptic operator and relating topics2018

    • Author(s)
      Toshio Horiuchi
    • Organizer
      ICM 2018 (Rio De Janeiro, Brasil)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] The equivalences among p- capacity, p-Laplace-capacities and Hausdorff measure.2018

    • Author(s)
      Xiaojing Liu, Toshio Horiuchi
    • Organizer
      日本数学会、函数方程式分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] p-ラプラシアンを含む精密化された加藤の不等式とその応用2017

    • Author(s)
      劉暁静、堀内利郎
    • Organizer
      日本数学会、函数方程式分科会
    • Place of Presentation
      首都大学東京
    • Year and Date
      2017-03-24
    • Related Report
      2016 Research-status Report
  • [Presentation] Remarks on Kato's inequality when $\Delta_pu$ is a measure2016

    • Author(s)
      劉暁静、堀内利郎
    • Organizer
      日本数学会、函数方程式分科会
    • Place of Presentation
      関西大学
    • Year and Date
      2016-09-15
    • Related Report
      2016 Research-status Report
  • [Presentation] 加藤の不等式とその周辺2016

    • Author(s)
      堀内利郎
    • Organizer
      「偏微分方程式および関連する諸問題」研究集会
    • Place of Presentation
      茨城大学
    • Year and Date
      2016-09-10
    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2021-02-19  

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