• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Studies on nonlinear partial differential equations by potential analysis

Research Project

Project/Area Number 19740062
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Basic analysis
Research InstitutionAkita University

Principal Investigator

HIRATA Kentaro  Akita University, 教育文化学部, 講師 (30399795)

Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥3,860,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywords実解析 / 関数方程式 / 優調和関数 / 境界挙動 / 半線形楕円型方程式 / Helmholtz方程式 / 調和測度 / John領域 / Green関数 / p-Laplace方程式 / 一様領域 / 準一様領域 / Martin核 / 境界増大度 / 非線形楕円型方程式
Research Abstract

In a bounded smooth domain, we established a suitable boundary growth estimate for positive superharmonic functions satisfying a nonlinear inequality. Moreover, we proved the existence of positive solutions of semilinear elliptic equations that are comparable to the Poisson kernel. Also, we gave a sufficient condition for Dirichlet boundary data to guarantee the existence of positive solutions of singular semilinear elliptic equations. In regard to potential theory, we studied a doubling property of harmonic measure, an estimate for the product of the Green function and the Martin kernel, and the boundary behavior of Martin kernels.

Report

(4 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (39 results)

All 2010 2009 2008 2007

All Journal Article (20 results) (of which Peer Reviewed: 11 results) Presentation (19 results)

  • [Journal Article] Boundary behavior of solutions of the Helmholtz equation2009

    • Author(s)
      K. Hirata
    • Journal Title

      Canadian Mathematical Bulletin Vol.52

      Pages: 555-563

    • NAID

      120006459514

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Boundary behavior of superharmonic functions satisfying nonlinear inequalities in a planar smooth domain2009

    • Author(s)
      K. Hirata
    • Journal Title

      Journal of the Australian Mathematical Society Vol.87

      Pages: 253-261

    • Related Report
      2009 Final Research Report
  • [Journal Article] Limits at infinity of superharmonic functions and solutions of semilinear elliptic equations of Matukuma type2009

    • Author(s)
      K. Hirata
    • Journal Title

      Potential Analysis Vol.30

      Pages: 165-177

    • Related Report
      2009 Final Research Report
  • [Journal Article] Limits at infinity of superharmonic functions and solutions of semilinear elliptic equations of Matukuma type2009

    • Author(s)
      Kentaro Hirata
    • Journal Title

      Potential Analysis 30

      Pages: 165-177

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Boundary behavior of superharmonic functions satisfying nonlinear inequalities in a planar smooth domain2009

    • Author(s)
      Kentaro Hirata
    • Journal Title

      Journal of the Australian Mathematical Society 87

      Pages: 253-261

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Boundary behavior of solutions of the Helmholtz equation2009

    • Author(s)
      Kentaro Hirata
    • Journal Title

      Canadian Mathematical Bulletin 52

      Pages: 555-563

    • NAID

      120006459514

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global estimates for non- symmetric Green type functions with applications to the p-Laplace equation2008

    • Author(s)
      K. Hirata
    • Journal Title

      Potential Analysis Vol.29

      Pages: 221-239

    • Related Report
      2009 Final Research Report
  • [Journal Article] Doubling conditions for harmonic measure in John domains2008

    • Author(s)
      H. Aikawa, K. Hirata
    • Journal Title

      Annales de l'Institut Fourier Vol.58

      Pages: 429-445

    • NAID

      120006459500

    • Related Report
      2009 Final Research Report
  • [Journal Article] On the existence of positive solutions of singular nonlinear elliptic equations with Dirichlet boundary conditions2008

    • Author(s)
      K. Hirata
    • Journal Title

      Journal of Mathematical Analysis and Applications Vol.338

      Pages: 885-891

    • Related Report
      2009 Final Research Report
  • [Journal Article] The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations2008

    • Author(s)
      K. Hirata
    • Journal Title

      Mathematische Annalen Vol.340

      Pages: 625-645

    • NAID

      120006459518

    • Related Report
      2009 Final Research Report
  • [Journal Article] Doubling conditions for harmonic measure in John domains2008

    • Author(s)
      Hiroaki Aikawa Kentaro Hirata
    • Journal Title

      Annales de l'institut Fourier(Grenoble) 58

      Pages: 429-445

    • NAID

      120006459500

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global estimates for non-symmetric Green type functions with applications to the p-Laplace equation2008

    • Author(s)
      Kentaro Hirata
    • Journal Title

      Potential Analysis 29

      Pages: 221-239

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the existence of positive solutions of singular nonlinear elliptic equations with Dirichlet boundary conditions2008

    • Author(s)
      Kentaro, Hirata
    • Journal Title

      Journal of Mathematical Analysis and Applications 338

      Pages: 885-891

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations2008

    • Author(s)
      Kentaro, Hirata
    • Journal Title

      Mathematische Annalen 340

      Pages: 625-645

    • NAID

      120006459518

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Estimates for the products of the Green function and the Martin kernel2007

    • Author(s)
      K. Hirata
    • Journal Title

      Nagoya Mathematical Journal Vol.188

      Pages: 40561-40561

    • Related Report
      2009 Final Research Report
  • [Journal Article] Martin boundary points of cones generated by spherical John regions2007

    • Author(s)
      K. Hirata
    • Journal Title

      Annales Academiae Scientiarium Fennicae. Mathematica Vol.32

      Pages: 289-300

    • NAID

      120006459435

    • Related Report
      2009 Final Research Report
  • [Journal Article] Boundary behavior of quotients of Martin kernels2007

    • Author(s)
      K. Hirata
    • Journal Title

      Proceedings of the Edinburgh Mathematical Society Vol.50

      Pages: 377-388

    • NAID

      120006459433

    • Related Report
      2009 Final Research Report
  • [Journal Article] Boundary behaviour of quotients of Martin kernels2007

    • Author(s)
      Kentaro, Hirata
    • Journal Title

      Proceedings of the Edinburgh Mathematical Society 50

      Pages: 377-388

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Martin boundary points of cones generated by spherical John regions2007

    • Author(s)
      Kentaro, Hirata
    • Journal Title

      Annales Academiae Scientiarium Fennicae, Mathematica 32

      Pages: 289-300

    • NAID

      120006459435

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Estimates for the products of the Green function and the Martin kernel2007

    • Author(s)
      Kentaro, Hirata
    • Journal Title

      Nagoya Mathematical Journal 188

      Pages: 1-18

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] Boundary behavior of superharmonic functions satisfying nonlinear inequalities2010

    • Author(s)
      K. Hirata
    • Organizer
      The 11th Northeastern Symposium on Mathematical Analysis
    • Place of Presentation
      北海道大学
    • Year and Date
      2010-02-23
    • Related Report
      2009 Annual Research Report 2009 Final Research Report
  • [Presentation] 非線形不等式を満たす優調和関数の境界挙動2010

    • Author(s)
      平田賢太郎
    • Organizer
      2009年度日本数学会東北支部会
    • Place of Presentation
      秋田大学
    • Year and Date
      2010-02-20
    • Related Report
      2009 Final Research Report
  • [Presentation] 非線形不等式を満たす優調和関数の境界挙動2010

    • Author(s)
      平田賢太郎
    • Organizer
      日本数学会東北支部会
    • Place of Presentation
      秋田大学
    • Year and Date
      2010-02-20
    • Related Report
      2009 Annual Research Report
  • [Presentation] 非線形不等式を満たす優調和関数の境界挙動2009

    • Author(s)
      平田賢太郎
    • Organizer
      日本数学会2009年度秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-26
    • Related Report
      2009 Final Research Report
  • [Presentation] 非線形不等式を満たす優調和関数の境界挙動2009

    • Author(s)
      平田賢太郎
    • Organizer
      日本数学会秋季総合分科会(函数論分科会)
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-26
    • Related Report
      2009 Annual Research Report
  • [Presentation] Boundary behavior of superharmonic functions satisfying nonlinear inequalities2009

    • Author(s)
      K. Hirata
    • Organizer
      The XXIst Rolf Nevanlinna Colloquium
    • Place of Presentation
      京都大学百周年時計台記念館
    • Year and Date
      2009-09-10
    • Related Report
      2009 Final Research Report
  • [Presentation] Boundary behavior of superharmonic functions satisfying nonlinear inequalities2009

    • Author(s)
      平田賢太郎
    • Organizer
      XXIst Rolf Nevanlinna Colloquium
    • Place of Presentation
      京都大学百周年時計台記念館
    • Year and Date
      2009-09-10
    • Related Report
      2009 Annual Research Report
  • [Presentation] Upper and lower estimates for parabolic Green functions in Lipschitz domains2009

    • Author(s)
      K. Hirata
    • Organizer
      International Workshop on Potential Theory 2009
    • Place of Presentation
      北海道大学
    • Year and Date
      2009-09-01
    • Related Report
      2009 Annual Research Report 2009 Final Research Report
  • [Presentation] Limits at infinity of superharmonic functions and solutions of semilinear elliptic equations of Matukuma typ2009

    • Author(s)
      平田賢太郎
    • Organizer
      RIMS研究集会「ポテンシャル論とその関連分野」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2009-02-17
    • Related Report
      2009 Final Research Report
  • [Presentation] Limits at infinity of superharmonic functions and solutions of semilinear elliptic equations of Matukuma type2009

    • Author(s)
      平田賢太郎
    • Organizer
      RIMS研究集会「ポテンシャル論とその関連分野」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2009-02-17
    • Related Report
      2008 Annual Research Report
  • [Presentation] 非線形不等式を満たす優調和関数に対する調和増大評価, 逆平均値の不等式とHarnack原理2008

    • Author(s)
      平田賢太郎
    • Organizer
      2008年度ポテンシャル論研究集会
    • Place of Presentation
      秋田カレッジプラザ
    • Year and Date
      2008-11-01
    • Related Report
      2009 Final Research Report
  • [Presentation] 非線形不等式を満たす優調和関数に対する調和増大評価,逆平均値の不等式とHarnack原理2008

    • Author(s)
      平田賢太郎
    • Organizer
      2008年度ポテンシャル論研究集会
    • Place of Presentation
      秋田カレッジプラザ
    • Year and Date
      2008-11-01
    • Related Report
      2008 Annual Research Report
  • [Presentation] 非線形不等式を満たす優調和関数の境界挙動に関する注意2008

    • Author(s)
      平田賢太郎
    • Organizer
      日本数学会2008年度春季年会
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-23
    • Related Report
      2009 Final Research Report
  • [Presentation] 非線形不等式を満たす優調和関数の境界挙動に関する注意2008

    • Author(s)
      平田賢太郎
    • Organizer
      日本数学会2008年度春季年会(関数論分科会)
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-23
    • Related Report
      2007 Annual Research Report
  • [Presentation] Boundary behavior of superharmonic functions satisfying nonlinear inequalities in the unit disk2008

    • Author(s)
      K. Hirata
    • Organizer
      The 11th Conference on Real and Complex Analysis
    • Place of Presentation
      広島大学
    • Year and Date
      2008-02-18
    • Related Report
      2009 Final Research Report
  • [Presentation] Boundary behavior of superharmonic functions satisfying nonlinear inequalities in the unit disk2008

    • Author(s)
      平田賢太郎
    • Organizer
      The Eleventh Conference on Real and Complex Analysis
    • Place of Presentation
      広島大学
    • Year and Date
      2008-02-18
    • Related Report
      2007 Annual Research Report
  • [Presentation] 平面領域において非線形不等式を満たす優調和関数の非接境界極限2007

    • Author(s)
      平田賢太郎
    • Organizer
      2007年度ポテンシャル論研究集会
    • Place of Presentation
      広島大学
    • Year and Date
      2007-11-02
    • Related Report
      2009 Final Research Report 2007 Annual Research Report
  • [Presentation] Dirichlet境界条件を満たす非線形楕円型方程式の正値解の存在2007

    • Author(s)
      平田賢太郎
    • Organizer
      日本数学会2007年度秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Related Report
      2009 Final Research Report
  • [Presentation] Dirichlet境界条件を満たす非線形楕円型方程式の正値解の存在2007

    • Author(s)
      平田賢太郎
    • Organizer
      日本数学会2007年度秋季年会(函数論分科会)
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Related Report
      2007 Annual Research Report

URL: 

Published: 2007-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi