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Studies on approximation, regularity and singularity for mean curvature flow

Research Project

Project/Area Number 20540117
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKobe University

Principal Investigator

ISHII Katsuyuki  Kobe University, 海事科学研究科, 准教授 (40232227)

Co-Investigator(Kenkyū-buntansha) MARUO Kenji  神戸大学, 大学院・海事科学研究科, 教授 (90028225)
NAITO Yuki  愛媛大学, 大学院・理工学研究科, 教授 (10231458)
KUWAMURA Masataka  神戸大学, 大学院・人間環境発達学研究科, 准教授 (30270333)
Co-Investigator(Renkei-kenkyūsha) TAKAHASHI Futoshi  大阪市立大学, 大学院・理学研究科, 教授 (10374901)
ISHIWATA Tetsuya  芝浦工業大学, システム理工学部, 准教授 (50334917)
OHMUNA Masaki  徳島大学, ソシオ・アーツ・サイエンス研究部, 准教授 (90304500)
Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2008: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords非線形偏微分方程式 / 平均曲率流 / 粘性解 / 自由境界 / 数理ファイナンス / 非線形楕円型偏微分方程式 / 非線形放物型偏微分方程式 / パターン形成
Research Abstract

We consider an approximation scheme for anisotropic curvature flow, arising from the image processing., and prove the convergence of this scheme. We also study the parabolic free boundary problems arising from mathematical finance. We introduce a time-discrete scheme for a parabolic variational inequality and obtain the convergences of approximate solutions and approximate free boundaries. We also prove the existence and uniqueness of solutions and of free boundary for an investment problem.

Report

(4 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (66 results)

All 2011 2010 2009 2008 Other

All Journal Article (28 results) (of which Peer Reviewed: 25 results) Presentation (36 results) Remarks (2 results)

  • [Journal Article] An eigenvalue problem related to the critical obolev exponent : variable coefficient case2010

    • Author(s)
      F.Takahashi
    • Journal Title

      Differential Integral Equations 23

      Pages: 827-860

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Spectral estimates of least energy solutions to the Brezis-Nirenberg problem with a variable coefficient2010

    • Author(s)
      F.Takahashi
    • Journal Title

      J.Differential Equations 248

      Pages: 2497-2527

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Non-degeneracy of least-energy solutions for an elliptic problem with nearly critical nonlinearity2010

    • Author(s)
      F.Takahashi
    • Journal Title

      Proc.Roy.Soc.Edinburgh Sect.A 140

      Pages: 203-222

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Mathematical modelling and experiments for the proliferation and differentiation of Drosophila intestinal stem cells I2010

    • Author(s)
      M.Kuwamura
    • Journal Title

      J.Bio.Dyms.

      Volume: 4 Pages: 248-257

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence and non-existence of sign-changing solutions for a class of two-point boundary value problems involving one-dimensional p-Laplacian2010

    • Author(s)
      Y.Naito
    • Journal Title

      Mathematica Bohemia (印刷中)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A nondegeneracy result for least energy solutions to a biharmonic problem with nearly critical exponent, Mathematical analysis on the self-organization and self-similarity2009

    • Author(s)
      T.Sato, F.Takahashi
    • Journal Title

      RIMS Kokyuroku Bessatsu, B 15

      Pages: 73-85

    • NAID

      110007480950

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Global existence of solutions for a nonlinearly perturbed Keller-Segel system in ${\bf R}^22009

    • Author(s)
      M.Kurokiba, F.Takahashi, T.Ogawa
    • Journal Title

      Z.Angew.Math.Phys. 60

      Pages: 840-866

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic uniqueness for a biharmonic equation with nearly critical growth on symmetric convex domains2009

    • Author(s)
      T.Sato, F.Takahashi
    • Journal Title

      Funkcial.Ekvac. 52

      Pages: 213-232

    • NAID

      130000140520

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in $ {\bf R}^2 $ Electron.2009

    • Author(s)
      M.Ishiwata, F.Takahashi, T.Ogawa
    • Journal Title

      J.Differential Equations 32

      Pages: 10-10

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators2009

    • Author(s)
      M.Kuwamura, H.Chiba
    • Journal Title

      Chaos 19

      Pages: 043121-10

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment2009

    • Author(s)
      M.Kuwamura, T.Nakazawa, T.Ogawa
    • Journal Title

      J.Math.Biol. 58, no.3

      Pages: 459-479

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment2009

    • Author(s)
      M.Kuwamura
    • Journal Title

      Journal of Mathematical Biology 58

      Pages: 459-479

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators2009

    • Author(s)
      M.Kuwamura
    • Journal Title

      Chaos 19

      Pages: 1-10

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Self-similar blow-up for a chemotaxis system in higher dimensional domains2009

    • Author(s)
      Y.Naito, T.Senba
    • Journal Title

      RIMS Kokyuroku Bessatsu B15

      Pages: 87-99

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2009

    • Author(s)
      Y.Naito, T.Sato
    • Journal Title

      RIMS Kokyuroku 1671

      Pages: 91-95

    • Related Report
      2009 Annual Research Report
  • [Journal Article] Deviation from the predicted wavenumber in a mode selection problem for the Turing patterns, Japan2008

    • Author(s)
      M.Kuwamura
    • Journal Title

      J.Indust.Appl.Math. 25

      Pages: 281-303

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Motion of essential admissible polygons by crystalline curvatureNonlinear phenomena with energy dissipation, GAKUTO Internat.2008

    • Author(s)
      T.Ishiwata
    • Journal Title

      Ser.Math.Sci.Appl.(Gakkotosho) 29

      Pages: 201-210

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Motion of non-convex polygons by crystalline curvature and almost convexity phenomena,Japan2008

    • Author(s)
      T.Ishiwata
    • Journal Title

      J.Indust.Appl.Math. 25

      Pages: 233-253

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Numerical study and examples on singularities of solutions to anisotropic crystalline curvature flows of nonconvex polygonal curves, Asymptotic analysis and singularities-elliptic and parabolic PDEs and related problems, Adv. Stud. Pure Math.2008

    • Author(s)
      C.Hirota, T.Ishiwata, S.Yazaki
    • Journal Title

      Math.Soc.Japan, Tokyo 47-2

      Pages: 233-253

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Sharp conditions for the existence of sign-changing solutions to equations involving the one-dimensional p-Laplacian2008

    • Author(s)
      Y. Naito, s. Tanaka
    • Journal Title

      Nonlinear Analysis 69

      Pages: 3070-3083

    • NAID

      120001384683

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Self-similar solutions for a semilinear heat equation with criticalexponent2008

    • Author(s)
      Y. Naito
    • Journal Title

      Indiana Univ. Math. J. 57

      Pages: 1283-1315

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Self-similarity in chomt taxis system2008

    • Author(s)
      Y. Nailo, T. Suzuki
    • Journal Title

      Colloq. Math 111

      Pages: 11-34

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence of non-radially syminetric viscosoity solutions to semilinear degenerate elliptic equations with radially symmetric coefficients in theplane, Part I2008

    • Author(s)
      K. Maruo, N. Yamadn
    • Journal Title

      J. Math. Anal. Appl. 345

      Pages: 743-753

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Deviation from the predicted wavenumber in a mode selection problem for the Turing patterns2008

    • Author(s)
      M. Kuwamura
    • Journal Title

      J. J. Indust. Appl. Math. 25

      Pages: 281-303

    • NAID

      10023986000

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent

    • Author(s)
      Y.Naito, T.Sato
    • Journal Title

      Ann.Mat.Pura Appl.(to appear)

    • Related Report
      2010 Final Research Report
  • [Journal Article] Existence and non-existence of sign-changing solutions for a class of two-point boundary value problems involving one-dimensional p-Laplacian

    • Author(s)
      Y.Naito, S.Tanaka
    • Journal Title

      Mathematica Bohemia (to appear)

    • Related Report
      2010 Final Research Report
  • [Journal Article] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent

    • Author(s)
      Y.Naito
    • Journal Title

      Ann.Mat.Pura Appl.

      Volume: (To appear)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Existence and non-existence of sign-changing solutions for a class of two-point boundary value problems involving one-dimensional p-Laplacian

    • Author(s)
      Y.Naito
    • Journal Title

      Mathematica Bohemia

      Volume: (To appear)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] 臨界Sobolev指数をもつ非斉次半線形楕円型偏微分方程式2011

    • Author(s)
      内籐雄基
    • Organizer
      首都大・変分問題セミナー
    • Place of Presentation
      首都大学東京(東京都)(招待講演)
    • Year and Date
      2011-01-14
    • Related Report
      2010 Annual Research Report
  • [Presentation] 臨界Sobolev指数をもつ非斉次半線形楕円型偏微分方程式2010

    • Author(s)
      内籐雄基
    • Organizer
      日本数学会2010年度秋期総合分科会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-24
    • Related Report
      2010 Final Research Report
  • [Presentation] 臨界Sobolev指数をもつ非斉次半線形楕円型偏微分方程式2010

    • Author(s)
      内籐雄基
    • Organizer
      日本数学会2010年度秋期総合分科会
    • Place of Presentation
      名古屋大学(愛知県)
    • Year and Date
      2010-09-24
    • Related Report
      2010 Annual Research Report
  • [Presentation] Dormancy of predators dependent on the rate of variation in the prey density2010

    • Author(s)
      桑村雅隆、仲澤剛史
    • Organizer
      第20回日本数理生物学会大会
    • Place of Presentation
      北海道大学
    • Year and Date
      2010-09-15
    • Related Report
      2010 Final Research Report
  • [Presentation] Dormancy of predators dependent on the rate of variation in the prey density2010

    • Author(s)
      桑村雅隆、仲澤剛史
    • Organizer
      第20回日本数理生物学会大会
    • Place of Presentation
      北海道大学(北海道)
    • Year and Date
      2010-09-15
    • Related Report
      2010 Annual Research Report
  • [Presentation] Asymptotic behavior of solutions to the Cauchy problem for a semilinear heat equation2010

    • Author(s)
      内籐雄基
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Puan National University, Korea.
    • Year and Date
      2010-08-30
    • Related Report
      2010 Final Research Report
  • [Presentation] Asymptotic behavior of solutions to the Cauchy problem for a semilinear heat equation2010

    • Author(s)
      Yuki Naito
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Puan National University, Korea(招待講演)
    • Year and Date
      2010-08-30
    • Related Report
      2010 Annual Research Report
  • [Presentation] A time-discretization to the American option pricing2010

    • Author(s)
      石井克幸
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Universita di Napoli, Italy
    • Year and Date
      2010-04-16
    • Related Report
      2010 Final Research Report
  • [Presentation] A time-discretization to the American option pricing2010

    • Author(s)
      Katsuyuki Ishii
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Universita di Napoli, Italy(招待講演)
    • Year and Date
      2010-04-16
    • Related Report
      2010 Annual Research Report
  • [Presentation] A time-discretization to the American option pricing2010

    • Author(s)
      石井克幸
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Sapienza Universit|'a di Roma, Italy
    • Year and Date
      2010-04-14
    • Related Report
      2010 Final Research Report
  • [Presentation] A time-discretization to the American option pricing2010

    • Author(s)
      Katsuyuki Ishii
    • Organizer
      Nonlinear PDE seminar
    • Place of Presentation
      Sapienza Universit'a di Roma, Italy(招待講演)
    • Year and Date
      2010-04-14
    • Related Report
      2010 Annual Research Report
  • [Presentation] A time-discretization to the American option pricing2010

    • Author(s)
      石井克幸
    • Organizer
      広島応用解析セミナー
    • Place of Presentation
      広島大学(広島県)
    • Year and Date
      2010-01-13
    • Related Report
      2009 Annual Research Report
  • [Presentation] The role of self-similarity in the Cauchy problems for semilinear heat equations2010

    • Author(s)
      内籐雄基
    • Organizer
      The 4th Euro-Japanese Workshop on Blow-up
    • Place of Presentation
      Lotentz Center, Leiden.
    • Related Report
      2010 Final Research Report
  • [Presentation] The role of self-similarity in the Cauchy problems for semilinear heat equations2010

    • Author(s)
      Yuki Naito
    • Organizer
      The 4th Euro-Japanese Workshop on Blow-up
    • Place of Presentation
      Lotentz Center, Leiden(招待講演)
    • Related Report
      2010 Annual Research Report
  • [Presentation] A time-discretization to the American option pricing2010

    • Author(s)
      石井克幸
    • Organizer
      松山解析セミナー
    • Place of Presentation
      愛媛大学(愛媛県)
    • Related Report
      2009 Annual Research Report
  • [Presentation] A time-discretization to the American option pricing2010

    • Author(s)
      石井克幸
    • Organizer
      非線形偏微分方程式ワークショップ
    • Place of Presentation
      石田環境改善センター(長崎県)
    • Related Report
      2009 Annual Research Report
  • [Presentation] Critical phenomena in linear elliptic problem with Hardy potential2009

    • Author(s)
      内藤雄基
    • Organizer
      京都大学数理解析研究所研究集会「現象解析と関数方程式の新展望」
    • Place of Presentation
      京都大学数理解析研究所(京都府)
    • Year and Date
      2009-11-18
    • Related Report
      2009 Annual Research Report
  • [Presentation] A variational approach for self-similar solutions to nonlinear heat equations2009

    • Author(s)
      内藤雄基
    • Organizer
      POSTEC Analysis seminar
    • Place of Presentation
      POSTEC, Korea
    • Year and Date
      2009-10-24
    • Related Report
      2009 Annual Research Report
  • [Presentation] アメリカンオプションに対する近似問題について2009

    • Author(s)
      石井克幸
    • Organizer
      神戸大学解析セミナー
    • Place of Presentation
      神戸大学
    • Year and Date
      2009-10-13
    • Related Report
      2010 Final Research Report
  • [Presentation] アメリカンオプションに対する近似問題について2009

    • Author(s)
      石井克幸
    • Organizer
      神戸大学解析セミナー
    • Place of Presentation
      神戸大学(兵庫県)
    • Year and Date
      2009-10-13
    • Related Report
      2009 Annual Research Report
  • [Presentation] 高次元領域における走化性方程式系の解の自己相似爆発2009

    • Author(s)
      仙葉隆、内藤雄基
    • Organizer
      日本数学会函数方程式分科会
    • Place of Presentation
      大阪大学(大阪府)
    • Year and Date
      2009-09-26
    • Related Report
      2009 Annual Research Report
  • [Presentation] 捕食者の休眠を伴うprey-predetor 系について2009

    • Author(s)
      桑村雅隆
    • Organizer
      日本数学会2009年度秋期総合分科会応用数学分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-25
    • Related Report
      2010 Final Research Report
  • [Presentation] 捕食者の休眠を伴うprey-predetor系について2009

    • Author(s)
      桑村雅隆
    • Organizer
      日本数学会2009年度秋期総合分科会応用数学分科会
    • Place of Presentation
      大阪大学(大阪府)
    • Year and Date
      2009-09-25
    • Related Report
      2009 Annual Research Report
  • [Presentation] Blowup rate for a parabolic-elliptic system in higher dimensional domain2009

    • Author(s)
      内籐雄基
    • Organizer
      Equadiff 12 (2009)
    • Place of Presentation
      Brno, Czech Republic
    • Year and Date
      2009-07-21
    • Related Report
      2010 Final Research Report
  • [Presentation] Blowup rate for a parabolic-elliptic system in higher dimensional domains2009

    • Author(s)
      内藤雄基
    • Organizer
      Equadiff 12(2009)
    • Place of Presentation
      Brno, Czech Republic
    • Year and Date
      2009-07-21
    • Related Report
      2009 Annual Research Report
  • [Presentation] Self-similar blowup for a chemotaxis system2009

    • Author(s)
      内籐雄基
    • Organizer
      09International Workshop on Nonlinear PDE and Applications
    • Place of Presentation
      Puan National Univ. Korea
    • Year and Date
      2009-06-29
    • Related Report
      2010 Final Research Report
  • [Presentation] Self-similar blowup for a chemotaxis system2009

    • Author(s)
      内藤雄基
    • Organizer
      09 International Workshop on Nonlinear PDE and Applications
    • Place of Presentation
      Puan National Univ.Korea
    • Year and Date
      2009-06-29
    • Related Report
      2009 Annual Research Report
  • [Presentation] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2009

    • Author(s)
      内藤雄基
    • Organizer
      京都大学数理解析研究所研究集会「変分問題とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2009-06-09
    • Related Report
      2010 Final Research Report
  • [Presentation] Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent2009

    • Author(s)
      内藤雄基
    • Organizer
      京都大学数理解析研究所研究集会「変分問題とその周辺」
    • Place of Presentation
      京都大学数理解析研究所(京都府)
    • Year and Date
      2009-06-09
    • Related Report
      2009 Annual Research Report
  • [Presentation] A time-discretization to the American option pricing2009

    • Author(s)
      石井克幸
    • Organizer
      Second International Conference on Reaction-Diffusion systems and Viscosity solutions
    • Place of Presentation
      Taichung, Taiwan
    • Related Report
      2010 Final Research Report
  • [Presentation] A time-discretization to the American option pricing2009

    • Author(s)
      石井克幸
    • Organizer
      Second International Conference on Reaction-Diffusion systems and Viscosity solutions
    • Place of Presentation
      Taichung(台湾)
    • Related Report
      2009 Annual Research Report
  • [Presentation] Optimal rate of convergence of Allen - Calm equation to motion by mean curvature2009

    • Author(s)
      K. Ishii
    • Organizer
      Workshop on viscosity solutions and related Topics
    • Place of Presentation
      埼玉大学東京ステーションカレッジ
    • Related Report
      2008 Annual Research Report
  • [Presentation] R^2での球対称な係数を持つ非線形退化楕円型偏微分方程式の小さな非球対称解の存在について2009

    • Author(s)
      丸尼健二
    • Organizer
      振動理論ワークショップ
    • Place of Presentation
      愛媛大学理学部
    • Related Report
      2008 Annual Research Report
  • [Presentation] R^2での球対称な係数を持つ非線形退化楕円型偏微分方程式の小さな非球対称解の存在について2008

    • Author(s)
      丸尾健二
    • Organizer
      北九州地区における偏微分方程式研究集会
    • Place of Presentation
      九州工業大学戸畑キャンパス
    • Year and Date
      2008-11-30
    • Related Report
      2008 Annual Research Report
  • [Presentation] Optimal rate of convergence of Allen - Cahn equation to motion by mean curvature2008

    • Author(s)
      K. Ishii
    • Organizer
      Viscosity, metric and control theoretic methods in nonlinear PDEs
    • Place of Presentation
      Departimento do Matematica, Sapienza Universitadi Roma (イタリア)
    • Related Report
      2008 Annual Research Report
  • [Presentation] Proliferation and differentiation of Drosophia intestinal sem cells2008

    • Author(s)
      M. Kuwamura
    • Organizer
      第2回日中数理生物学コロキューム
    • Place of Presentation
      岡山大学
    • Related Report
      2008 Annual Research Report
  • [Remarks] ホームページ等

    • Related Report
      2010 Final Research Report
  • [Remarks]

    • URL

      http://wwwmain.h.kobe-u.ac.jp/~kuwamura/kuwaj.html

    • Related Report
      2010 Annual Research Report

URL: 

Published: 2008-04-01   Modified: 2016-04-21  

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