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Deepening and application of a theory for the logarithmic Sobolev inequality

Research Project

Project/Area Number 15K04949
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionUniversity of Toyama

Principal Investigator

Fujita Yasuhiro  富山大学, 大学院理工学研究部(理学), 教授 (10209067)

Co-Investigator(Renkei-kenkyūsha) Ishii Hitoshi  早稲田大学, 教育・総合学術院 (70102887)
Ishii Katsuyuki  神戸大学, 海事科学部, 教授 (40232227)
Project Period (FY) 2015-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords対数型 Sobolev の不等式 / Hamilton-Jacobi方程式 / 下からの評価 / gradient / Cauchy 問題 / 完全な証明 / 減衰率の最適性 / Lipschitz 定数 / Borell-Brascamp-Lieb / GNS の不等式 / 精密化 / 対数型のソボレフの不等式 / リプシッツ定数 / 放物型方程式 / ハミルトン-ヤコビ方程式
Outline of Final Research Achievements

I was able to achieve an important aim that I planned in this study at first. It is to provide a lower estimate of the sup-norm of the gradient of a function by using the logarithmic Sobolev inequality with index p which is equal to infinity. This estimate is applied to show the optimality of the decay rate of the sup-norm of the gradients to solutions of several Cauchy problems. This result has been published in an appropriate mathematical journal. On the other hand, through workshops, I let many researchers know widely about my complete proof of the logarithmic Sobolev inequality with index p which is greater than 1. The paper of this proof was published in the beginning of this study. In these senses, the result of this study was able to be accomplished in a satisfactory form.

Report

(4 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (19 results)

All 2018 2017 2016 2015 Other

All Int'l Joint Research (3 results) Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results,  Open Access: 1 results,  Acknowledgement Compliant: 1 results) Presentation (10 results) (of which Int'l Joint Research: 4 results,  Invited: 7 results) Remarks (3 results)

  • [Int'l Joint Research] リヨン大学(フランス)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] リヨン大(フランス)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] リヨン第1大学(フランス)

    • Related Report
      2015 Research-status Report
  • [Journal Article] A lower bound of L∞-norm of gradients for Cauchy problems2018

    • Author(s)
      Yasuhiro Fujita
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 458 Issue: 2 Pages: 910-924

    • DOI

      10.1016/j.jmaa.2017.08.045

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the sets of maximum points for generalized Takagi functions2017

    • Author(s)
      Yasuhiro Fujita and Yusuke Saito
    • Journal Title

      Toyama Mathematical Journal

      Volume: 39 Pages: 87-94

    • NAID

      120006457876

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A supplementary proof of Lp--logarithmic Sobolev inequality2015

    • Author(s)
      Yasuhiro Fujita
    • Journal Title

      Annales de la Faculte des Sciences de Toulouse

      Volume: 24 Issue: 1 Pages: 119-132

    • DOI

      10.5802/afst.1444

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] 病的函数を初期値とする Hamilton-Jacobi flow について2018

    • Author(s)
      藤田安啓
    • Organizer
      日本数学会2018年度年会 函数方程式分科会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Hamilton-Jacobi 方程式と高木函数の間の対応構造について2017

    • Author(s)
      藤田 安啓
    • Organizer
      日本数学会2017年度年会
    • Place of Presentation
      首都大学東京
    • Year and Date
      2017-03-25
    • Related Report
      2016 Research-status Report
  • [Presentation] 1.病的函数を初期値とする Hamilton-Jacobi flow の幾何学的性質2017

    • Author(s)
      藤田安啓
    • Organizer
      九州関数方程式セミナー
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] 2.On a geometrical property of Hamilton-Jacobi ow starting from some pathological function2017

    • Author(s)
      Yasuhiro Fujita
    • Organizer
      Nonlinear PDE for Future Applications  - Optimal Control and PDE -,
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On Hamilton-Jacobi equation with initial data of the Takagi function2016

    • Author(s)
      藤田 安啓
    • Organizer
      北大 PDE セミナー
    • Place of Presentation
      北海道大学
    • Year and Date
      2016-10-28
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 一様放物型方程式の解の gradient の下からの評価について,2016

    • Author(s)
      藤田 安啓
    • Organizer
      日本数学会2016年度秋季総合分科会, 函数方程式分科会
    • Place of Presentation
      関西大学
    • Year and Date
      2016-09-16
    • Related Report
      2016 Research-status Report
  • [Presentation] 一般化された高木函数が最大値を取る集合について,2016

    • Author(s)
      藤田 安啓
    • Organizer
      第45回 金沢解析セミナー
    • Place of Presentation
      金沢大学
    • Year and Date
      2016-09-02
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] On a Gagliardo-Nirenberg type inequality for log-concave functions2015

    • Author(s)
      Yasuhiro Fujita
    • Organizer
      RIMS研究集会「偏微分方程式の漸近問題と粘性解」
    • Place of Presentation
      京都大学
    • Year and Date
      2015-12-02
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] pplication of a logarithmic Sobolev inequality to gradient bounds of solutions of parabolic equations2015

    • Author(s)
      Yasuhiro Fujita
    • Organizer
      研究集会「確率論と幾何学」
    • Place of Presentation
      東京工業大学
    • Year and Date
      2015-11-09
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On a Sobolev-type inequality for log-concave functions via a Hamilton-Jacobi equation2015

    • Author(s)
      Yasuhiro Fujita
    • Organizer
      早稲田大学における講演会(講演者 3 名)
    • Place of Presentation
      早稲田大学教育・総合科学学術院
    • Year and Date
      2015-08-03
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] http://www.sci.u-toyama.ac.jp/~yfujita/index.html

    • Related Report
      2017 Annual Research Report
  • [Remarks] Ivan Gentil - Institut Camille Jordan

    • Related Report
      2016 Research-status Report
  • [Remarks] 藤田研究室のホームページ

    • URL

      http://www.sci.u-toyama.ac.jp/~yfujita/index.html

    • Related Report
      2015 Research-status Report

URL: 

Published: 2015-04-16   Modified: 2019-03-29  

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