Research activity information

• Quasilinear Parabolic Equations Associated with Semilinear Parabolic Equations

  Katsuyuki Ishii, Michel Pierre, Takashi Suzuki

  We formulate a quasilinear parabolic equation describing the behavior of the global-in-time solution to a semilinear parabolic equation. We study this equation in accordance with the blow-up and quenching patterns of the solution to the original semilinear parabolic equation. This quasilinear equation is new in the theory of partial differential equations and presents several difficulties for mathematical analysis. Two approaches are examined: functional analysis and a viscosity solution.

  MDPI AG, Feb. 2023, Mathematics, 11(3) (3), 758 - 758

  Scientific journal


• Convergence of a threshold-type algorithm for curvature-dependent motions of hypersurfaces

  Katsuyuki Ishii

  Dec. 2020, Advanced Studies in Pure Mathematics, 85, 181 - 191, English

  [Refereed][Invited]

  International conference proceedings


• Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces

  Katsuyuki Ishii, Takahiro Izumi

  American Institute of Mathematical Sciences, Mar. 2018, Discrete and Continuous Dynamical Systems- Series A, 38(3) (3), 1103 - 1125, English

  [Refereed]

  Scientific journal


• Remarks on viscosity solutions for mean curvature flow with obstacles

  K. Ishii, H. Kamata, S. Koike

  Springer New York LLC, 2017, Springer Proceedings in Mathematics and Statistics, 215, 83 - 103, English

  [Refereed][Invited]

  International conference proceedings


• Convergence of a threshold-type algorithm using the signed distance function

  Katsuyuki Ishii, Masato Kimura

  2016, INTERFACES AND FREE BOUNDARIES, 18(4) (4), 479 - 522, English

  [Refereed][Invited]

  Scientific journal


• An approximation scheme for the anisotropic and nonlocal mean curvature flow

  Katsuyuki Ishii

  Apr. 2014, NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 21(2) (2), 219 - 252, English

  [Refereed]

  Scientific journal


• Quasilinear Parabolic Equations Associated with Semilinear Parabolic Equations

  Katsuyuki Ishii, Michel Pierre, Takashi Suzuki

  Jan. 2013, Mathematics, 11, English

  [Refereed]

  Scientific journal


• An area-minimizing scheme for anisotropic mean-curvature flow.

  ISHII KATSUYUKI, ETO TOKUHIRO, GIGA YOSHIKAZU

  Jul. 2012, Advances in Differential Equations, 17(11-12) (11-12), 1031 - 1084, English

  [Refereed]

  Scientific journal


• An area minimizing scheme for anisotropic mean curvature flow

  Tokuhiro Eto, Yoshikazu Giga, Katsuyuki Ishii

  Jan. 2012, PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES, 88(1) (1), 7 - 10, English

  [Refereed]

  Scientific journal


• Convergence of the Approximation Scheme to American Option Pricing via the Discrete Morse Semiflow

  Katsuyuki Ishii, Seiro Omata

  Dec. 2011, APPLIED MATHEMATICS AND OPTIMIZATION, 64(3) (3), 363 - 415, English

  [Refereed]

  Scientific journal


• Mathematical analysis to an approximation scheme for mean curvature flow

  ISHII KATSUYUKI

  This paper presents some mathematical analysis on the Bence - Merriman - Osher algorithm for mean curvature flow, which was proposed in 1992.

  GAKKO TOSHO, Oct. 2011, International Syposium on Computational Science 2011, GAKUTO International Series, Mathematical Sciences and Applications, 34, 67 - 85, English

  [Refereed][Invited]

  International conference proceedings


• Optimal rate of convergence to the motion by mean curvature with a driving force

  Katsuyuki Ishii

  Aug. 2007, Advances in Differential Equations, Vol. 12 No. 5 481 - 514, English

  [Refereed]

  Scientific journal


• Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature

  Y Goto, K Ishii, T Ogawa

  Jun. 2005, COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 4(2) (2), 311 - 339, English

  [Refereed]

  Scientific journal


• Optimal rate of convergence of the Bence-Merriman-Osher algorithm for motion by mean curvature

  K Ishii

  2005, SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 37(3) (3), 841 - 866, English

  Scientific journal

  Link to Kobe Univ. Repository (Kernel)


• Nonlinear second order elliptic PDEs with subdifferential

  ISHII KATSUYUKI, YAMADA NAOKI

  Gakkōtosho, Jun. 2002, Advances in Mathematical Sciences and Applications, 12(1) (1), 477 - 497, English

  [Refereed]

  Scientific journal


• An approximation scheme for motion by mean curvature with right-angle boundary condition

  Hitoshi Ishii, Katsuyuki Ishii

  Society for Industrial and Applied Mathematics Publications, 2001, SIAM Journal on Mathematical Analysis, 33(2) (2), 369 - 389, English

  [Refereed]

  Scientific journal


• Unbounded viscosity solutions of nonlinear second order PDE's

  ISHII KATSUYUKI, TOMITA YOSHIHITO

  Gakkōtosho, Dec. 2000, Advances in Mathematical Sciences and Applications, 10(2) (2), 689 - 710, English

  [Refereed]

  Scientific journal


• Regularity and convergence of crystalline motion

  Katsuyuki Ishii, Halil Mete Soner

  Society for Industrial and Applied Mathematics Publications, 1998, SIAM Journal on Mathematical Analysis, 30(1) (1), 19 - 37, English

  [Refereed]

  Scientific journal


• Regularity and convergence of crystalline motion in the plane

  ISHII KATSUYUKI

  Aarhus Univ., Denmark, 1997, Proceedings of the Korea-Japan Partial Differential Equations Conference, 7 - 14, English

  [Refereed][Invited]

  International conference proceedings


• On unbounded viscosity solutions of nonlinear second order partial differential equations

  ISHII KATSUYUKI, TOMITA YOSHIHITO

  Gakkōtosho, 1996, GAKUTO International Series. Mathematical Sciences and Applications., 201 - 214, English

  [Refereed]

  Scientific journal


• Viscosity solutions of nonlinear second-order elliptic PDEs associated with impulse control problems. II

  ISHII KATSUYUKI

  Division of Functional Equations, Mathematical Society of Japan, Aug. 1995, Funkcialaj Ekvacioj, 38(2) (2), 297 - 328, English

  [Refereed]

  Scientific journal


• Some results on the viscosity solutions of fully nonlinear equations containing non-local terms

  K Ishii, N Yamada

  1995, WORLD CONGRESS OF NONLINEAR ANALYSIS '92, VOLS 1-4, 2605 - 2616, English

  [Refereed]

  International conference proceedings


• Viscosity solutions of nonlinear elliptic PDEs with nonlocal terms

  ISHII KATSUYUKI

  Gakkōtosho, Dec. 1994, Advances in Mathematical Sciences and Applications, 4(2) (2), 345 - 356, English

  [Refereed]

  Scientific journal


• Viscosity solutions of nonlinear second order elliptic PDEs involving nonlocal operators

  ISHII KATSUYUKI, YAMADA NAOKI

  Department of Mathematics, Osaka University, Aug. 1993, Osaka Journal of Mathematics, 30(3) (3), 439 - 455, English

  [Refereed]

  Scientific journal


• Viscosity solutions of nonlinear second order elliptic PDEs associated with impulse control problems

  ISHII KATSUYUKI

  Division of Functional Equations, Mathematical Society of Japan, Apr. 1993, Funkcialaj Ekvacioj, 36(1) (1), 123 - 141, English

  [Refereed]

  Scientific journal


• On the rate of convergence of solutions for the singular perturbations of gradient obstacle problems

  ISHII KATSUYUKI, YAMADA NAOKI

  Division of Functional Equations, Mathematical Society of Japan, Dec. 1990, Funkcialaj Ekvacioj, 33(3) (3), 551 - 562, English

  [Refereed]

  Scientific journal

• Rate of convergence of an algorithm for curvature-dependent motions of hypersurfaces (Reconsideration of the method of estimates on partial differential equations from a point of view of the theory on abstract evolution equations)

  Ishii Katsuyuki

  Kyoto University, Feb. 2016, RIMS Kokyuroku, 1984, 152 - 163, English


• On the convergence of an area minimizing scheme for the anisotropic mean curvature flow (New developments of the theory of evolution equations in the analysis of non-equilibria)

  Ishii Katsuyuki

  Kyoto University, Oct. 2013, RIMS Kokyuroku, 1856, 102 - 112, English


• Rate of convergence of the Bence-Merriman-Osher algorithm for motion by mean curvature(Viscosity Solution Theory of Differential Equations and its Developments)

  Ishii Katsuyuki

  Kyoto University, Apr. 2006, RIMS Kokyuroku, 1481, 11 - 19, English


• Nonlinear second order elliptic equations with subdifferential terms (Viscosity Solutions of Differential Equations and Related Topics)

  Ishii Katsuyuki

  Kyoto University, Sep. 2002, RIMS Kokyuroku, 1287, 155 - 163, English


• Interior regularity of viscosity solutions for nonlinear second order ellitic partial differential equations (Studies on qualitative estimates on regularity and singularity of solutions of PDEs)

  Ishii Katsuyuki

  Kyoto University, Jan. 2002, RIMS Kokyuroku, 1242, 40 - 49, Japanese


• On $L^p$ theory of viscosity solutions (Studies on structure of solutions of nonlinear PDEs and its analytical methods)

  Ishii Katsuyuki

  Kyoto University, Apr. 2001, RIMS Kokyuroku, 1204, 114 - 127


• Singular limit of solutions of Ginzburg-Landau equation (Related topics on regularity of solutions to nonlinear evolution equations)

  Ishii Katsuyuki

  Kyoto University, May 1998, RIMS Kokyuroku, 1045, 103 - 118, Japanese


• On unbounded viscosity solutions of nonlinear second order partial differential equations(Nonlinear Evolutions Equations and Their Applications)

  ISHII Katsuyuki, TOMITA Yoshihito

  Kyoto University, Jun. 1995, RIMS Kokyuroku, 913, 54 - 67, English


• On the nonlinear degenerate elliptic PDEs with obstacles(Nonlinear Evolution Equations and Their Applications)

  ISHII KATSUYUKI

  Kyoto University, Feb. 1995, RIMS Kokyuroku, 898, 132 - 148, English


• An Estimate on the Rate of Convergence of Viscosity Solutions for the Singular Perturbation Problems(Evolution Equations and Applications to Nonlinear Problems)

  Ishii Katsuyuki, Yamada Naoki

  京都大学数理解析研究所, Jun. 1991, 数理解析研究所講究録, 755, 1 - 9, English

• Image processing and curvature flow equations

  Katsuyuki Ishii

  Joint work, Saiensu=sha Co., Ltd. Publishers, Apr. 2008, Japanese

  General book

• Approximate problems of a threshold-type to mean curvature flow

  ISHII KATSUYUKI

  Workshop on applied mathematics 2018, Dec. 2018, Japanese, Domestic conference

  [Invited]

  Invited oral presentation


• Convergence of a threshold-type algorithm to mean curvature flow

  ISHII KATSUYUKI

  Conference on elliptic/parabolic differential equations, Nov. 2018, Japanese, Osaka Prefectural University, Domestic conference

  [Invited]

  Invited oral presentation


• Convergence of a threshold-type algorithm to curvature-dependent motions

  ISHII KATSUYUKI

  Viscosity solutions and Related Topics, Nov. 2018, English, Tohoku University, International conference

  [Invited]

  Invited oral presentation


• Convergence of a threshold-type algorithm for mean curvature flow

  ISHII KATSUYUKI

  The 11th Mathematical Society of Japan (MSJ) Seasonal Institute (SI) The Role of Metrics in the Theory of Partial Differential Equations, Jul. 2018, English, 北海道大学, International conference

  [Invited]

  Invited oral presentation


• Convergence of a threshold-type algorithm for mean curvature flow

  ISHII KATSUYUKI

  12th AIMS on Dynamical Systems, Differential Equations and Applications, Jul. 2018, English, National Taiwan University, International conference

  [Invited]

  Invited oral presentation


• 平均曲率流の近似問題について

  ISHII KATSUYUKI

  九州関数方程式セミナー, 2018, Japanese, 福岡大学セミナーハウス, 1992 年に Bence, Merriman, Osher によって提案された 平均曲率流に対する閾値型近似アルゴリズムはその簡明さから 多くの研究者の興味を引き、現在でも研究が進められている。 本講演ではこのアルゴリズムに対する最近の講演者の研究につ いて紹介する。, Domestic conference

  [Invited]

  Invited oral presentation


• 空間曲線に対する曲率流の近似問題について

  ISHII KATSUYUKI

  第 8 回北海道・東北コンソーシアムセミナー, 2018, Japanese, 弘前大学, Domestic conference

  Oral presentation


• Convergence of a threshold-type algorithm for curvature-dependent motion of hypersurface

  ISHII KATSUYUKI

  Matsuyama Analysis Seminar 2017, Feb. 2017, Japanese, Ehime University, In this talk I present the rate of convergence of a threshold-type algorithm for curvature-dependent motions of hypersurface. The optimality of this rate is also shown., Domestic conference

  [Invited]

  Invited oral presentation


• 空間曲線に対する曲率流の近似問題について

  ISHII KATSUYUKI

  第 7 回北海道・東北コンソーシアムセミナー, 2017, Japanese, 公立はこだて未来大学, Domestic conference

  Oral presentation


• Convergence of a threshold-type algorithm for curvature-dependent motion of hypersurface

  ISHII KATSUYUKI

  Workshop on Hamilton-Jacobi Equations, Jul. 2016, English, Fudan University, Shanghai, China, In this talk I present the rate of convergence of a thershold-type algorithm for curvature-dependent motions of hypersurface. the optimality of our estimate is also shown., International conference

  [Invited]

  Invited oral presentation


• Rate of convergence of an algorithm for curvature-dependent motions of hypersurfaces

  ISHII KATSUYUKI

  抽象発展方程式理論から見た偏微分方程式に関する評価方法の再考, Oct. 2014, English, 仙葉 隆, 京都大学数理解析研究所, International conference

  [Invited]

  Invited oral presentation


• Convergence of a threshold-type algorithm for curvature-dependent motions of hypersurfaces

  ISHII KATSUYUKI

  東北大学応用数学セミナー, Oct. 2014, Japanese, 東北大学理学研究科, Domestic conference

  [Invited]

  Invited oral presentation


• Convergence of an approximation scheme for the mean curvature flow

  ISHII KATSUYUKI

  Workshop on nolinear PDEs --PDE approach to network and related topics--, Jun. 2013, English, Tohoku University, International conference

  [Invited]

  Invited oral presentation


• An approximate problem for mean curvature flow based a variatinoal problem

  ISHII KATSUYUKI

  Kanazawa analysis seminar, May 2013, Japanese, Sxchool of Mathematics and Physics, Kanazawa University, Domestic conference

  [Invited]

  Invited oral presentation


• On the convergence of an algorithm for the anisotropic mean curvature flow and its application to the crystalline curvature flow in the plane

  ISHII KATSUYUKI

  Murorna Applied Analysis Seminar, May 2013, Japanese, Muroran Institute of Technology, Domestic conference

  Oral presentation


• An approximate problem for mean curvature flow based a variatinoal problem

  ISHII KATSUYUKI

  Kobe analysis seminar, Apr. 2013, Japanese, Department of Mathematics, Kobe University, Domestic conference

  [Invited]

  Invited oral presentation


• An approximation scheme for the anisotropic and the planar crystalline curvature flow

  ISHII KATSUYUKI

  PDE and Real analysis seminar, Apr. 2013, Japanese, Graduate school of Mathematical Sciences, University of Tokyo, Domestic conference

  [Invited]

  Invited oral presentation


• An approximation scheme for the planar motion by crystalline curvature

  ISHII KATSUYUKI

  南大阪応用解析セミナー, Feb. 2013, Japanese, Osaka City University, Domestic conference

  Keynote oral presentation


• An approximation scheme for the planar motion by crystalline curvature

  ISHII KATSUYUKI

  Functional Analysis and Applications, Feb. 2013, English, Kobe University, International conference

  [Invited]

  Invited oral presentation


• On the convergence of an area minimizing scheme for anisotropic mean curvature flow,

  ISHII KATSUYUKI

  第 10 回浜松偏微分方程式研究集会, Dec. 2012, Japanese, Shizuoka University, Domestic conference

  [Invited]

  Invited oral presentation


• 変分法に基づく平均曲率流の近似アルゴリズムについて

  ISHII KATSUYUKI

  語ろう数理解析 セミナー, Nov. 2012, Japanese, Osaka University, Domestic conference

  [Invited]

  Invited oral presentation


• On the convergence of an area minimizing scheme for anisotropic mean curvature flow

  ISHII KATSUYUKI

  富山大学基礎解析セミナー, Oct. 2012, Japanese, The University of Toyama, Domestic conference

  Oral presentation


• On the convergence of an area minimizing scheme for anisotropic mean curvature flow

  ISHII KATSUYUKI

  非平衡現象の解析における発展方程式理論の新展開, Oct. 2012, English, RIMS, Kyoto University, International conference

  [Invited]

  Invited oral presentation


• 平均曲率流に対する近似アルゴリズムの数学解析

  ISHII KATSUYUKI

  日本数学会秋季総合 分科会 関数方程式論分科会特別講演, Sep. 2012, Japanese, Kyushu University, Domestic conference

  [Invited]

  Invited oral presentation


• On the convergence of an area minimizing scheme for anisotropic mean curvature flow

  ISHII KATSUYUKI

  第 8 回非線型の諸問題, Sep. 2012, Japanese, 宮崎県婦人会館, Domestic conference

  [Invited]

  Invited oral presentation


• On the convergence of an area minimizing scheme for anisotropic mean curvature flow

  ISHII KATSUYUKI

  Seminar on Partial Differential Equations in Osaka 2012, Aug. 2012, English, Osaka University, International conference

  Oral presentation


• An area minimizing scheme for anisotropic mean curvature flow.

  石井 克幸

  The 29th J\Kyushu Symposium on Partial Differential Equations, Jan. 2012, English, 川島秀一, 栄伸一郎, 隠居良行, Kyushu University Nishjin Plaza, International conference

  [Invited]

  Invited oral presentation


• An area minimizing scheme for anisotropic mean curvature flow.

  石井 克幸

  Analysis seminar, Department of Mathematics, Saitama University, Dec. 2011, Japanese, 小池茂昭, 長澤壮之, Department of Mathematics, Saitama University, Domestic conference

  [Invited]

  Invited oral presentation


• An area minimizing scheme for anisotropic mean curvature flow.

  石井 克幸

  Nagoya seminar on Differential Equations, Nov. 2011, Japanese, 杉本充, 菱田俊明, 津川光太郎, 加藤淳, Graduate School of Mathematics, Nagoya University, Domestic conference

  [Invited]

  Invited oral presentation


• An area minimizing scheme for anisotropic mean curvature flow.

  石井 克幸

  Front propagation, biological problems and related topics: viscosity solution methods for asymptotic analysis, Sep. 2011, English, P.E. Souganidis, 石井仁司, 儀我美一, 神保秀一, 利根川吉廣, Department of Mathematics, Hokkaido University, International conference

  [Invited]

  Invited oral presentation


• A time discretization to the American option pricing

  Katsuyuki Ishii

  偏微分方程式セミナー, May 2010, Japanese, 北海道大学数学教室, 北海道大学数学教室, Domestic conference

  [Invited]

  Invited oral presentation


• A time discretization to the American option pricing

  Katsuyuki Ishii

  松山解析セミナー, Feb. 2010, Japanese, 愛媛大学理工学研究科 内藤雄基, 愛媛大学理工学研究科, Domestic conference

  [Invited]

  Invited oral presentation


• A time discretization to the American option pricing

  Katsuyuki Ishii

  第8 回広島応用解析セミナー, Jan. 2010, English, 広島大学工学研究科 坂口茂, 広島大学工学研究科, Domestic conference

  [Invited]

  Invited oral presentation


• An approximate problem to the American option pricung

  Katsuyuki Ishii, Katsuyuki Ishii

  神戸大学解析セミナー, Oct. 2009, Japanese, 神戸大学数学教室, 神戸大学数学教室, Domestic conference

  Oral presentation


• A time discretization to the American option pricing

  Katsuyuki Ishii

  2nd International conference on Reaction-Diffusion equations and Viscosity solutions, Jul. 2009, English, Providence University, Taichung, International conference

  [Invited]

  Invited oral presentation


• An approximate problem to the free boundary of the American option pricing

  Ishii Katsuyuki

  Fukuoka Mini Workshop on Evolution Equations and Related Topics, Aug. 2006, English, Yamada, Naoki, 福岡大学セミナーハウス, International conference

  Oral presentation


• An approximate problem of the free boundary to the American option pricing

  Ishii Katsuyuki

  International conference on Mathematical finance and its applications, Aug. 2006, English, AKAHORI, Jiro, DELBAEN, Freddy, ISHIMURA, Naoyuki, KUSUOKA, Shigeo, NAKAGAWA, Hidetoshi, OMATA, Seiro, SEKINE, Jun, TAKAOKA, Koichiro, Graduate School of Natural Science & Technology, Kanazawa University, International conference

  [Invited]

  Invited oral presentation


• Optimal rate of convergence of the Allen-Cahn equation to the motion by mean curvature

  Ishii Katsuyuki

  Nonlinear Partial Differential Equations and Applications, Jun. 2006, English, 石井 仁司, Loretti Paola, Cortona, Italy, International conference

  [Invited]

  Invited oral presentation


• アメリカンオプションに現れる自由境界問題の近似

  Ishii Katsuyuki

  数理ファイナンス金沢研究集会, Feb. 2006, Japanese, 竹原 均, 小俣 正朗, 金沢大学サテライト・プラザ, Domestic conference

  [Invited]

  Invited oral presentation


• Rate of convergenc of the Bece-Merriman-Osher algorithm for motion by mean curvature

  Ishii Katsuyuki

  九州関数方程式セミナー, Dec. 2005, Japanese, 中尾 愼宏, 川島 秀一, 栄 伸一郎, 隠居 良行, 山田 直記, 内藤 幸一郎, 小林 孝行, 水町 徹, 九州大学 六本松キャンパス, Domestic conference

  Oral presentation


• Rate of convergence of the Bence-Merriman-Osher algorithm for motion

  Ishii Katsuyuki

  確率論とPDE, Aug. 2005, English, 三上 敏夫, 竹田 雅好, 北海道大学理学部, International conference

  [Invited]

  Invited oral presentation


• Rate of convergence of the Bence-Merriman-Osher algorithm for motion by mean curvature

  Ishii Katsuyuki

  MSJ-IRI Asymptotic Analysis and Singularity, Jul. 2005, English, Yoshio Tsutsumi, Hideo Kozono, Eiji Yanagida, Kazunaga Tanaka, Sendai International Center, International conference

  Oral presentation


• Rate of convergence of the Bence-Merriman-Osher algorithm for motion by mean curvature

  Ishii Katsuyuki

  Viscosity Solution Theory of Differential Equations and its Developments, Jun. 2005, English, 小池 茂昭, 石井 仁司, 儀我 美一, 京都大学数理解析研究所, International conference

  [Invited]

  Invited oral presentation


• Optimal rate of convergence of the Bence-Merriamn-Osher algoritm for motion by mean curvature

  Ishii Katsuyuki

  神楽坂解析セミナー, May 2005, Japanese, 東京理科大学神楽坂校舎, Domestic conference

  [Invited]

  Invited oral presentation


• Optimal rate of convergence of the Bence-Merriman-Osher algorithm for motion by mean curvature.

  Ishii Katsuyuki

  第二回非線形偏微分方程式研究集会, Mar. 2005, Japanese, 三沢 正史, 小川 卓克, 妙見ホテル, Domestic conference

  [Invited]

  Invited oral presentation

• 日本数学会

• Canonical mean curvature flow and its application to evolution problems

  利根川 吉廣, 高坂 良史, 石井 克幸, 三浦 達哉, 高棹 圭介, 可香谷 隆, 小野寺 有紹, 水野 将司

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Tokyo Institute of Technology, 01 Apr. 2023 - 31 Mar. 2028


• 曲率流に対する閾値型近似アルゴリズムとそれを用いた広義解の性質の研究

  石井 克幸, 高坂 良史, 上田 好寛

  日本学術振興会, 科学研究費助成事業, 基盤研究(C), 神戸大学, 01 Apr. 2023 - 31 Mar. 2026


• 表面拡散方程式によって時間発展する曲線・曲面の形状と特異性の解析

  高坂 良史, 石井 克幸

  日本学術振興会, 科学研究費助成事業, 基盤研究(C), 神戸大学, 01 Apr. 2019 - 31 Mar. 2024

  Willmore流によって運動が記述される曲面の閾値型近似アルゴリズムの研究を行った。Willmore流の場合は4階熱方程式に対する基本解のTaylor展開をもとに閾値型近似アルゴリズムが導出できることを2020年度に示したので、2021年度はその近似アルゴリズムにおいて基本解の導関数の展開を導くことによって、閾値関数から得られる閾値集合の境界での閾値関数の勾配評価を得た。また、Willmore汎関数に表面積汎関数を加えたエネルギー汎関数の勾配流によって運動が記述される曲面についても閾値型近似アルゴリズムを研究し、4階熱方程式に空間変数に関する2階導関数の項を加えた線形4階放物型偏微分方程式の基本解のTaylor展開をもとにすれば、この場合も閾値型近似アルゴリズムが構成できることを示した。さらにWillmore流の場合と同様にしてその閾値関数から得られる閾値集合の境界での閾値関数の勾配評価を導いた。今回得た閾値型近似アルゴリズムは変分的時間離散近似との関係付けが期待できることに気づいたので、現在は曲げエネルギーに長さ汎関数を加えたエネルギー汎関数の勾配流によって運動が記述される曲線の場合に変分的時間離散近似との関係性をもとにした収束証明を検討している。また、上記の閾値型近似アルゴリズムはHelfrich流(Willmore汎関数に表面積汎関数と体積汎関数を加えたエネルギー汎関数の勾配流)によって運動が記述される曲面への応用も期待できるので、それについても現在検討している。


• 曲面・曲線からなる曲率流に対する近似アルゴリズムとそれを用いた広義解の性質の研究

  石井 克幸, 高坂 良史, 上田 好寛

  日本学術振興会, 科学研究費助成事業, 基盤研究(C), 神戸大学, 01 Apr. 2020 - 31 Mar. 2023

  石井は Willmore 流に対する閾値型近似アルゴリズムの研究を行った。4 階熱方程式や 2 階の項をもつ線形 4 階放物型偏微分方程式について、その基本解を Taylor 展開することによって閾値型近似アルゴリズムを構成した。更に 4 階熱方程式の場合には、解の挙動を詳しく調べることにより、その解を使って定義される閾値集合の性質やその境界での解の勾配評価を得た。この証明には半空間に対する定義関数を初期値とする 4 階熱方程式の解のある種の正値性が鍵となっている。2 階の項をもつ線形 4 階放物型偏微分方程式の解を用いて定義される閾値集合の場合にも同様の性質が得られると考えており、研究を進めている。
  高坂はWillmore汎関数に表面積 (曲線の場合は長さ) 汎関数を加えたエネルギー汎関数の勾配流について、閾値型近似アルゴリズムの研究を行った。低階項をもつ線形 4 階放物型偏微分方程式の基本解の Taylor 展開をもとに、上記の勾配流に対する閾値型近似アルゴリズムを構成した。解を用いて定義される閾値集合について、その境界における解の勾配評価について研究を進めている。
  上田はこれまでに研究を進めてきた対称双曲型偏微分方程式系の安定性解析に関する研究に着手し、既知の結果の拡張に成功した。また、一般論の拡張にあたってその最適性に関する問題が浮上したが、京都大学の前川泰則氏と共にその最適性に関する条件の導入にも着手し、ある一定の結果を得ることに成功している。


• Multifaceted studies on dynamical problems in the calculus of variations using geometric measure theory

  利根川 吉廣, 高坂 良史, 石井 克幸, 三浦 達哉, 高棹 圭介, 可香谷 隆, 小野寺 有紹

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Tokyo Institute of Technology, 01 Apr. 2018 - 31 Mar. 2023

  主な研究成果として以下の２つを挙げる． （１）Salvatore Stuvard（ミラノ大学）と共同で前年度から引き続き研究を行っている余次元１の閉修正可能集合を初期値としたブラッケ流の存在定理について，特にブラッケ流の囲む領域の体積変化について，当初の我々の予想を凌ぐシャープな結果を得ることに成功した．論文はAdvances in Calculus of Variationsに出版受理済である．存在を示したブラッケ流は有界変動関数の枠組みにおける平均曲率流にもなっており，そのためブラッケ流特有の非一意性の問題を解決している．有界変動関数の枠組みの時間大域解の存在定理自体が知られていなかった中，この存在定理は一般化された平均曲率流に対して新しい概念を発見した，とも言える結果である． （２）曲面のブラッケの意味での法線方向速度が，平均曲率と外力項の和で表せる問題を考える．葛西－利根川(2014)による正則性定理により，ほとんどの点において動く曲面はパラボリックの意味で局所的にC^1級グラフになることがわかっていたが，外力項がヘルダー連続ではない場合，界面運動を表す2階放物型方程式の強解になるかどうかは未解決だった．この問題に対し，森龍之介（東工大特別研究員）と富松瑛太（D２）と共同で，もしグラフの時間微分がラドン測度であれば，グラフは期待される２階偏微分方程式の強解になっていることが解明された．興味深いことに，Allen-Cahn方程式から得られる解はこの条件を満たしている．これはブラッケの意味での界面速度の繊細な性質を表すものであり，未知であった現象である．論文はIndiana University Math Journalに出版受理済みである．


• 曲率によって動く曲線・曲面に対する数値計算アルゴリズムとその正則性・特異性

  石井 克幸

  学術研究助成基金助成金／基盤研究(C), Apr. 2017 - Mar. 2020, Principal investigator

  Competitive research funding


• Deepening and application of a theory for the logarithmic Sobolev inequality

  Fujita Yasuhiro, Ishii Hitoshi, Ishii Katsuyuki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), University of Toyama, 01 Apr. 2015 - 31 Mar. 2018

  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.


• Critical exponent and the behavior of solutions to nonlinear parabolic partial differential equations

  Yuki Naito, Yanagida Eiji, Ishwata Michinori, Senba Takasi, Kajikiya Ryuji, Yoshikawa Syuji, Ioku Norisuke

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Ehime University, 01 Apr. 2014 - 31 Mar. 2018

  We consider the semilinear elliptic equation and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures. We show that, under the suitable conditions, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across the critical exponent which is determined by the space dimension and the order of the behavior of the coefficient function. We consider the behavior of solutions to the Cauchy problem for a semilinear heat equation with supercritical nonlinearity. We study the convergence of solutions to steady states in a weighted norm, and show the global attractivity property of steady states. We also give its convergence rate for a class of initial data. Proofs are given by a comparison method based on matched asymptotic expansion.


• Variational analysis on dynamic geometric problems

  TONEGAWA Yoshihiro, KIM Lami, WICKRAMASEKERA Neshan

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), 31 May 2013 - 31 Mar. 2018

  We proved the basic existence and regularity theorems for the mean curvature flow considered in the framework of geometric measure theory which is called Brakke's mean curvature flow. As for the existence theorem, when given an arbitrary n-dimensional closed set in an n+1-dimensional Euclidean space, we proved the time global existence of the Brakke's mean curvature flow that evolves from the given initial data. For the analysis of the singular set, we proved the regularity theory around triple junction in the one-dimensional case, and showed the strong stability property of the triple junction within the weak topology of measure.


• Well-posedness for nonautonomous differential systems with dissipativity structure described by metric-like functionals

  TANAKA Naoki, SHIMIZU Senjo, ISHII Katsuyuki, MATSUMOTO Toshitaka

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Shizuoka University, 01 Apr. 2013 - 31 Mar. 2016

  We characterize the well-posedness for nonautonomous differential equations governed by continuous operators, using dissipativity conditions with respect to metric-like functionals, subtangential conditions and connectedness conditions. Toward to the non-continuous case, we generalize the cerebrated well-posedness result on autonomous differential equations governed by maximal monotone operators due to Komura and Brezis. Moreover, we discuss the well-posedness for functional differential equations and the solvability of abstract Cauchy problems for weakly continuous operators.


• Fundamental theory for viscosity solutions of fully nonlinear equations and its applications

  Koike Shigeaki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Tohoku University, 01 Apr. 2011 - 31 Mar. 2016

  We obtained comparison principle for unbounded viscosity solutions of degenerate elliptic PDE with superlinear gradient terms. We presented a representation formula for viscosity solutions of integro-differential equations of Isaacs type. We established the local maximum principle fro Lp-viscosity solutions of fully nonlinear uniformly elliptic PDE with unbounded coefficients to the first derivatives. We discussed regularity and large time behavior of viscosity solutions of integro-differential equations with coercive first derivative terms. We obtained existence and uniqueness of entire solutions of fully nonlinear elliptic equations with superlinear growth in the first derivatives. We showed the Lipschitz continuity of viscosity solutions of mean curvature flow equations with bilateral obstacles.


• Deepening of the theory of viscosity solutions and its applications

  ISHII Hitoshi, OTANI Mitsuharu, NAGAI Hideo, GIGA Yoshikazu, KOIKE Shigeaki, MIKAMI Toshio, MITAKE Hiroyoshi, YAMADA Naoki, ISHII Katsuyuki, ICHIHARA Naoyuki, FUJITA Yasuhiro

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Waseda University, 01 Apr. 2011 - 31 Mar. 2016

  We investigated the asymptotic problems of partial differential equations such as the long-time asymptotic behavior of solutions of Hamilton-Jacobi equations and viscous Hamilton-Jacobi equation, the vanishing discount problem, and obtained many important new pieces of knowledge regarding these asymptotic problems as well as the theory of viscosity solutions. We developed the basic theory of the existence and uniqueness of solutions for singular diffusion equations and for integral-differential equations. Based on the analysis of solutions of Hamilton-Jacobi-Bellman equations, we established certain estimates on the large-time asymptotic behavior of the minimizing large deviation probabilities, the verification theorem for optimal consumption-investment in a non-complete market model, a new approach to the stochastic optimal transportation problem.


• Study for Hamilton-Jacobi equations and logarithmic Sobolev inequality

  FUJITA Yasuhiro, ISHII Hitoshi, ISHII Katsuyuki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), University of Toyama, 01 Apr. 2012 - 31 Mar. 2015

  Throughout this study, I wrote three papers. First, I gave a new and complete proof of Lp logarithmic Sobolev inequality by using the hypercontractivity of the solution of some Hamilton-Jacobi equation. Next, I clarified many properties of the inequality which is obtained by letting p→∞ in this Lp logarithmic Sobolev inequality. This was done by using Laplace transforms and the theory of regular variations. Finally, I investigated the set which determines the structure of solutions of Hamilton-Jacobi equations. This set plays the same role as the uniqueness set for boundary value problems.


• Singularity of solutions for nonlinear partial differential equations of parabolic type and structure of solutions for the stationary problems

  NAITO Yuki, KAJIKIYA Ryuji, ISHII Katsuyuki, YANAGIDA Eiji, SENBA Takasi, YOSHIKAWA Syuji, IOKU Norisuke

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Ehime University, 28 Apr. 2011 - 31 Mar. 2015

  We study the singular behavior of solutions for nonlinear partial differential equations of parabolic type, and investigate the relations between the singularity and the solution structure of the stationary problems. We verify the roles of self-similar solutions in the Cauchy problems for semilinear heat equations in the case where the problem has multiple self-similar solutions. We consider the Cauchy problem for semilinear heat equations, and show the optimal spatial decay condition of initial functions at infinity for the blow-up in finite time. We consider the elliptic partial differential equations involving p-Laplace operator, and show the geometrical properties of radially symmetric solutions which has singular behavior near the boundary.


• 平均曲率流に対する近似問題と正則性・特異性に関する研究

  石井 克幸

  科学研究費補助金／基盤研究(C), Apr. 2012 - Mar. 2015, Principal investigator

  Competitive research funding


• New development and its application of Aubry-Mather theory for Hamilton-Jacobi equations

  YASUHIRO Fujita, HITOSHI Ishii, KATSUSHI Ohmori, KATSUYUKI Ishii

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), University of Toyama, 2009 - 2011

  About the study of this Kakenhi, I have obtained several results related with the Aubry-Mather theory and talked about these results in several conferences and seminars. These results are published in some journals. The first result is to clarify the relation between the quotient Aubry sets and uniqueness sets for minimization formula for Hamilton-Jacobi equations. The second one is to provide a new proof of classical inequalities by using a comparison theorem for the Aubry set of Hamilton-Jacobi equations. The third one is to derive an optimal logarithmic Sobolev inequality with Lipschitz constant. In the proof of this inequality, an asymptotic solution of the Aubry-Mather theory for a Hamilton-Jacobi equation is used. The fourth one is to investigate a rate of convergence appearing in the asymptotic behavior of a viscosity solution to the Cauchy problem for the Hamilton-Jacobi equation with quadratic gradient term. I showed that the semiconvexity property of this Hamiltonian is an important factor which determines this rate. Here, the Aubry set is closely related with the semiconvexity property of this Hamiltonian. As a conclusion, I think that I have done a complete job about the study of this Kakenhi by using the Aubry-Mather theory.


• Self-similarity and singularity of solutions for nonlinear parabolic PDEs

  NALTO Yuki, ISHII Katsuyuki, KUWAMURA Masataka, SUZUKI Takashi, SENBA Takashi, ISHIGE Kazuhiro

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 2008 - 2010

  The structure of self-similar solutions for semilinear heat equations is studied, and the role of self-similar solutions in the problem of asymptotic behavior of time-glabal solutions is investigated. In particular, new results on the structure of self-similar solutions for semilinear heat equations with Sobolev critical nonlinarity are obtained. Furthermore, the applications to the nonlinear problem involving chemotaxis system are studied.


• Viscosity solution theory for fully nonlinear equations and its applications

  KOIKE Shigeaki, ISHII Hitoshi, MIKAMI Toshio, ISHII Katsuyuki, NAGAI Hideo, MORIMOTO Hiroaki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Saitama University, 2008 - 2010

  The basic theory for viscosity solutions of fully nonlinear second order elliptic partial differential equations is studied. In case when uniformly elliptic equations contain unbounded coefficients to the first derivatives, it is proved that the weak Harnack inequality holds for Lp-viscosity solutions. As applications, it turns out that qualitative properties such as the strong maximum principle, the maximum principle for unbounded domains, the Phragmen-Lindelov theorem etc. are shown. In case when degenerate elliptic equations contain the first derivative terms with supearlinear growth, by setting appropriate function spaces, to which viscosity solutions belong, the comparison principle for them is proved.


• Analysis on anisotropic curvature flow equations in phenomena

  GIGA Mi-Ho, ISHII Katsuyuki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), The University of Tokyo, 2006 - 2009

  To analyze crystal growth phenomena and image processing technology, various differential equations governing motion of curves or surfaces under a particular law are proposed. We focused on anisotropic curvature flow equations with very singular interfacial energy. We analyzed one of typical equations, so-called a crystalline curvature flow equation. For any initial polygon, we proved the well-posedness of the initial value problem of the corresponding system of ordinary differential equations by using self-similar solutions and a geometric method. On the other hand, we showed that the idea of singular interfacial energy is effective to mathematical analysis of shock waves. Meanwhile anisotropic curvature flow equations with inhomogeneous external force are important to understand crystal growth phenomena. We derived fundamental comparison principle of them.


• RESEARCH ON THE THEORY OF VISCOSITY SOLUTIONS OF DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS

  ISHII Hitoshi, KOBAYASI Kazuo, OTANI Mitsuharu, GIGA Yoshikazu, NAGI Hideo, KOIKE Shigeaki, MIKAMI Toshio, YAMADA Naoki, GOTO Syun'ichi, ISHII Katsuyuki, FUJITA Yasuhiro, OHNUMA Masaki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Waseda University, 2006 - 2009

  On the theme of researching the theory of viscosity solutions of differential equations and its applications, we investigated viscosity solutions of boundary value problems, weak KAM theory, regularity of viscosity solutions, optimizations problems, several kinds of asymptotic problems in differential equations, curvature flows and motions of phase boundaries, mass transportation problems, problems in engineering and economics. Based on the investigations done before, we have succeeded to obtain many, new observations on each of subjects listed above. Our contributions to research on Aubry sets in weak KAM theory and its application to asymptotic problems are significant.


• Study on asymptotic solutions of Hamilton-Jacobi equations based on the theory of viscosity solutions

  YASUHIRO Fujita, ISHII Hitoshi, YOSHIDA Norio, IKEDA Hideo, ISHII Katsuyuki, ISHII Hitoshi, YOSHIDA Norio, IKEDA Hideo, ISHII Katsuyuki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), University of Toyama, 2006 - 2008

  まず,領域がn次元ユークリッド空間全体のとき, Ornstein-Uhlenbeck作用素を含むHamilton-Jacobi方程式の解の時間無限大での漸近解への収束の様子を明らかにした. 続いて,このHamilton-Jacobi方程式からOrnstein-Uhlenbeck作用素の粘性項が消去された場合に,漸近解への収束の様子を明らかにした. 前者は確率制御理論への応用を持っている. 後者は, Aubry-Mather理論に基づく解の表現を新たに与えることを可能にした. 次に,領域がn次元ユークリッド空間全体のとき, Hamilton-Jacobi方程式の解の時間無限大での漸近解への収束率を遅くする要因がAubry集合の幾何学的な性質と初期値の下限との関係であることを明らかにした. これは,従来の研究が漸近解への収束のみを考えるものであった点から,収束率を遅くする要因を明らかにしたと言う点で一歩踏み込んだ研究と考えられる. その他として, 上記研究に関連して2つの研究成果が得られた. ひとつは, 相加相乗の不等式, ヘルダーの不等式,ヒルベルトの不等式の証明をAubry-Mather理論により導くというものである. もうひとつは, Ornstein-Uhlenbeck作用素がポアンカレの不等式においてどのような役割を果たしているかについて明らかにしたものである.


• Mathematical analysis for phenomena in the living body and its simulation

  HOSHINO Hiroki, HOSONO Yuzo, IIDA Masato, ISHII Katsuyuki, KUBO Akisato, HOSONO Yuzo, IIDA Masato, ISHII Katsuyuki, NAITO Morihiro, KUBO Akisato

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 2006 - 2008

  生体現象、特に悪性腫瘍細胞の結合組織への浸潤を記述した数学モデルに対する理論解析および数値シミュレーションを行った。これまで数値計算的にさまざまな進行波解が得られていたが、今回の研究において、滑らかな進行波解の存在に対する解析的な証明を与えるとともに漸近的性質を調べた。また広い意味での生命現象の数理解析を目指し、侵入現象に対する進行波解や競争系に対する解の遷移層などのパターン形成の解明を行った。


• 平均曲率流の近似問題と正則性・特異性に関する研究

  石井 克幸

  科学研究費補助金／基盤研究(C), 2008, Principal investigator

  Competitive research funding


• On the study of the theory of viscosity solutions and its new developments

  KOIKE Shigeaki, MORIMOTO Hiroaki, ISHII Hitoshi, NAGAI Hideo, MIKAMI Toshio, ISHII Katsuynki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Saitama University, 2004 - 2007

  The Aleksandrov-Bakelman-Pucci maximum principle for Lp-viscosity solutions of fully nonlinear second order uniformly elliptic/parabolic partial differential equations with possibly superllinear growth terms of the first derivatives, unbounded coefficients, unbounded inhomogeneous terms has been established under appropriate hypotheses in two research papers with A. Swiech. Some counter-examples have been presented when there are no hypotheses. Perron's method has been first applied to Lp-viscosity solutions of fully nonlinear elliptic partial differential equations by introducing semicontinuous Lp-visosity solutions. For several nonlinear variational inequalities arising in Mathematical Finance, optimal controls have been constructed by showing that associated value functions admit enough regularity in research papers with H. Morimoto, and H. Morimoto and S. Sakaguchi.


• 画像処理の数理における実解析的手法の探索

  小川 卓克, 服部 哲弥, 木村 正人, 後藤 俊一, 石井 克幸, 松本 敏隆, 長藤 かおり

  日本学術振興会, 科学研究費助成事業, 萌芽研究, 2003 - 2005

  研究実績は以下のとおり. 研究代表者の小川は研究分担者の石井克幸と協力者の後藤陽子と共同で平均曲率流方程式を等高面の方法で考え、そのBence-Merrimen-Osher型の数値解析アルゴリズムの解への収束を、半線形熱方程式の解に対する特異摂動の観点から考え、粘性解の方法により証明した。平均曲率流方程式は特に画像処理の際のノイズ消去に有効に用いられるがその場合のBMOアルゴリズムの有効性が示せた. また,小川は単独で,鉄磁性体の2次元ising型spinモデル(シグマ模型)に対する連続体近似を考え、その半線形化方程式のエネルギー空間における可解性を新しいゲージ変換を考えることにより与えた。また関連して、粘性が入る場合に鉄磁性体モデルとSchrodinger写像、調和写像熱流との相関を議論し、それぞれ係数が極限と成る場合の状況について、ゲージ変換による議論、単調性公式による議論により特異性の発生について考察した。 分担者の服部はプレシルピンスキーガウケットと呼ばれる無限フラクタル格子上の単純ランダムウォークと自己回避確率連鎖を連続的に内挿する自己抑制・吸引的確率連鎖の族を構成し,変位の指数を与えた. 分担者の木村はパラメータを含む移流項を持つ楕円型方程式の第一固有値の特異摂動問題を考察した.移流の代表速度を表すパラメータが無限大に近づくとき起こる固有値の指数減衰現象について,空間1次元の場合に精密な漸近挙動評価を得た. 分担者の松本は生成作用素の定義域が稠密でないanalytic semigroupおよび、integrated semigroupの時間に依存しない非線形摂動を考察し、汎関数を用いた一般的な増大条件の下で、弱解を与える発展作用素が存在するための必要十分条件を、方程式に対する陰的差分近似の存在によって与えた。


• Mathematical modeling and analysis for phenomena in medical science

  HOSHINO Hiroki, KUBO Akisato, ISHII Katsuyuki, NAITO Morihiro

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Fujita Health University College, 2003 - 2005

  H.Hoshino studied reaction-diffusion systems for the model of tumor angiogenesis presented by Othmer and Stevens in 1997 and Anderson and Chaplain in 1998 with A.Kubo and T.Suzuki (Osaka University). Hoshino showed the well-posedness in the Holder spaces of the solutions to the systems and the existence of Lyapunovfunctions. Moreover, by numerical simulations, he obtained the animations for the discrete models of angiogenesis by the Monte Carlo method and those for the continuous models by the finite element method. A.Kubo constructed global solutions to the degenerate hyperbolic equations derived from the model of tumorangiogenesis and investigated their asymptotic behavior. Especially, he considered the cases where initial functions have small perturbations from constant states and applied Galerkin's method to get the results. K.Ishii investigated the convergence of the algorithm for computing the motion of a hypersurface by mean curvature flow given by Bence, Merriman and Osher. He obtained the convergence rate of the algorithm for the motion of a smooth and compact hypersurface by mean curvature. Furthermore, he considered the specialcase of a circle evolving by curvature and showed that the rate is optimal. With Proteus mirabilis having active motility, M.Naito observed that how its motility depends on conditions of culture mediums, and he investigated whether the difference of the states of mediums (e.g., solid or semi-solid mediums) or of the temperature of culture would give that of the motility or not. By this observation, he considered the validity of mathematical models with discontinuous nonlinear term describing the growth of bacteria.


• Research on the theory of viscosity solutions and its applications

  ISHII Hitoshi, GIGA Yoshikazu, KOIKE Shigeaki, NAGAI Hideo, ISHII Katusyuki, MIKAMI Toshio

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Waseda University, 2003 - 2005

  We proposed and proved the effectiveness of singular diffusions in the vertical direction in the level set approach to first-order partial differential equations (pde for short). We established the strong maximum principle to viscosity solutions of fully nonlinar elliptic pde including the minimal surface eqaution. We builded an example of fully nonlinear uniformly elliptic pde for which the maximum principle does not holds, and established the maximum principle, Holder regularity, and the solvability of the Dirichlet problem for such nonlinear pde under suitable hypotheses. We introduced the convexified Gauss curvature flow, formulated the level set approach to its generalizations, and established existence and uniqueness of solutions of the pde which appears in the level set approach. We also introduced a stochastic approaximation scheme to the generalized convexified Gauss flow and proved its convergence. We proved on a mathematical basis the occurrence of Berg's effect when the crystal shape is a cylinder. For the BMO (Bence-Merrima-Osher) scheme, we gave a new proof of its convergence to the mean curvature flow and the optimal estimate on the rate of convergence. We proved the convergence the asymptotic solutions as time goes to infinity of solutions of parabolic pde with the Ornstein-Uhlenbeck operator. We analized the simultaneous effects of homogenization and vanishing viscosity in periodic homogenization of uniformly elliptic pde. We proved existence and uniqueness of the limit in the zero-noise of certain h-path processes and established existence and uniqueness of the Monge-Kantorovich problem with a quadratic cost. Regarding mathematical finance, we studied optimal stopping time problems and risk-sensitive portfolio optimization problems for general factor models and constructed their optimal strategies. We analized the asymptotic behavior of solutions of p-Laplace equations as p goes to infinity in a fairly general setting.


• Ｒでの非線形退化楕円型偏微分方程式の非有界な粘性解の構造の研究

  丸尾 健二

  科学研究費補助金／基盤研究(C), 2005

  Competitive research funding


• 非線形放物型偏微分方程式の自己相似構造と解の特異性

  内藤 雄基

  科学研究費補助金／基盤研究(C), 2005

  Competitive research funding


• 非線形発展方程式の最適制御と逆問題解析の研究

  中桐 信一

  科学研究費補助金／基盤研究(C), 2005

  Competitive research funding


• 数値計算アルゴリズムを用いた平均曲率流の解析的研究

  石井 克幸

  科学研究費補助金／基盤研究(C), 2005, Principal investigator

  Competitive research funding


• Studies on the applications of the theory of viscosity solutions to some singular perturbation problems

  ISHII Katsuyuki, MARUO Kenji, KAGEYAMA Yasuo

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 2002 - 2004

  Katsuyuki Ishii studied a numerical algorithm for motion by mean curvature, which is proposed by Bence, Merriman and Osher and obtained the following results 1.I gave a proof of convergence showing how the mean curvature flow equation is derived from this algorithm. (This is a joint work with Yoko Goto and Takayoshi Ogawa.) 2.I obtained the rate of convergence of this algorithm in the case of smoth motion by mean curvature. I also showed the optimality in the case of a circle evolving by curvature. Kenji Maruo studied semilinear degenerate elliptic partial differential equations in the plane. He obtained the following 3.Assuming that the coefficients of the equation are radially symmetric, he proved that, under some growth conditions at infinity, the continuous viscosity solutions are radially symmetric Yasuo Kageyama obtained the rate of convergence and some properties of modified Bernstein polynomials


• Study of viscosity solutions for partial differential equations with subdifferential

  TANAKA Naoto, MARUO Kenji, KUROKIBA Masaki, YAMADA Naoki, ISHII Katsuyuki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Fukuoka University, 2001 - 2003

  The aim of present research is to extend the notion of viscosity solutions to multi-valued nonlinear partial differential equations (PDE's) with subdifferential and to investigate the existence, uniqueness and other properties of solutions. The theory of viscosity solutions has been applied to wide class of frilly nonlinear PDE's. Among them, the obstacle problem is an important class of applications of viscosity solutions. We have put emphasis of the research on following three points: 1. We formulate various PDE's by subdifferential and apply the theory of viscosity solutions for it. 2. We compare the formulation by viscosity solution with known ones and study an advantage for it. 3. We prove solvability of various nonlinear PDE's and discuss the relation between the methods by viscosity solution. It is shown that the notion of viscosity solutions are extended for nonlinear second order PDE's with subdifferential whose typical example is obstacle problem, and unique, existence theorem and stability of the solution are proved. Moreover, we investigate convergence of Yoshida approximation for subdiffrential term.


• Research on viscosity solutions of differential equations and their applications

  ISHII Hitoshi, SAKAI Makoto, MOCHIZUKI Kiyoshi, GIGA Yoshikazu, ISHII Katsuyuki, KOIKE Shigeaki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), 2000 - 2002

  The results obtained in our project are : (1) In 1974, W. Firey proposed the Gauss curvature flow as a mathematical model of the wearing process of a stone rolling on the beach by wave. This model assumes that the stone has a convex shape. In our research we considered the case when a stone is not convex. We introduced the convexified Gauss curvature flow which models the wearing process of a nonconvex stone rolling on the beach and established the level set approach based on viscosity solutions method. (2) We introduced weak solutions to the integral equation which describes the convexified Gauss curvature flow, proved that the uniqueness of the weak solution for the Cauchy problem, and proved its existence by a discrete stochastic approximation. (3) We studied a general stochastic optimal control problem with state constraint and proved, under relatively weak assumptions, the Lipschitz continuity and Holder continuity of the associated value function, that the value function satisfy the corresponding Bellman equation in the viscosity sense and that the state constraint problem for the Bellman equation has a unique viscosity solution. (4) We introduced the notion of proper viscosity solutions of a wide class of first-order partial differential equations including the Burgers equation, proved the unique existence of proper viscosity solutions for the class of equations, which may not have divergence form, and established the convergence of the approximation by the vanishing viscosity method to proper viscosity solutions in the sense of convergence of graphs with respect to the Hausdorff distance.


• viscosity solutions of nonlinear partial differential equations with singularities

  ISHII Katsuyuki, MARUO Kenji

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Kobe University of Mercantile Marine, 1998 - 2001

  In this project, I considered the existence, uniqueness and stability of viscosity solutions of nonlinear partial differential equations (PDE 's in short) with singularities and their applications of some approximate problems. I had some results on the motion of planar polygons with singular curvature and its application to an approximation for the planar motion of a simple closed curve by its curvature. I also showed that a version of an algorithm, which was proposed by Bence, Merryman and Osher in 1992, can be applied to approximate the motion by mean curvature with right-angle boundary condition in a bounded domain. I studied elliptic/parabolic PDE's with nonlinear terms of the spatial gradient. I classifed completely the interaction between the growth properties of nonliner terms and the uniqueness classes for viscosity solutions and proved the existence of viscosity solutions in such classes. I also treated nonlinear second order ellitpic PDE's with subdifferential. Using the definition of the subdifferential, we modified the notion of the usual viscosity solutions and obtained the uniqueness, existence and stability. Maruo mainly studied the radially symmetry of continuous viscosity solutions of Dirichlet problem for nonlinear degenerate elliptic PDE's. He gave the necessary and sufficient condition which assures that the continuous viscosity solutions are radially symmetric. It seems that this condition is optimal. He also obtained the existence and uniqueness of bounded radial viscosity solutions and those of unbounded ones in the whole space.


• On symmetric and radial viscosity solutions for elliptic partial differential equation.

  MARUO Kenji, INOUE Tetuo, ISHII Katsuyuki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Kobe University of Mercantile Marine, 1997 - 2000

  We consider the Dirichelet problem for a semilinear degenerte elliptic equation (DP) : -g(|x|)Δu+f(|x|, u(x))=0, and Boundary Condition where N【greater than or equal】2 and g (|x|), f(|x|, u) are continuous and the domain is a bounded ball in N-dimensional space. We discuss the problem (DP) under the following assumptions : 1)g is nonnegative. 2) f is strictly monotone for u. We frist define a standard viscosity solution by the viscosity solution such that f (|x|, u(x))=0 if g(|x|)=0. Then we can prove that the any continuous standard viscosity solution is the radial solution and unique. We add an assumption : 3)∫^g^<-1> (s) ds=∞ or ∫_ g^<-1> (s) ds=∞ for any a : g (a)=0. Then We obtain that any continuous viscosity solution is the radial solution and uniqne. If the assumption 3) is not satisfied there exist examples such that the continuous viscosity solutions are not uniqne. We next state the existence and uniqueness of the continuous unbounded viscosity solution in R^N. We use the order of the infinite neiborhood of the solution as the boundary condition. We know that the existence or nonexistece of the solution are dependent on a kind of the order of the solution. Moreover, we get the results which the uniqueness or non-uniqueness are also dependent on a kind of the order of the solution. In this case, we assume that g, f is sufficiently smooth. We now show the existence and uniquness of the continuous viscosity solution to quasi-semilinear degenrate elliptic problem. Here, g (|x|, u), f (|x|, u) are continuous and f is strictly monotone for u. Moreover, we assume there exists an implicite function of f=0 and the implicite function holds some smootheness. Then we can prove the existence of the continuous viscosity solution. We next state the uniquenss of the continuous viscosity solution. Assume that g (|x|, u) and f (|x|, u) hold the some relations such that f (|x|, u)/g (|x|, u) is monotone for u. Then we have the uniquness theorem and get the result this viscosity solution is the radial solution.


• Research in viscosity solutions using the method of Functional Analysis.

  MARUO Kenji, INOUE Tetuo, ISHII Katsuyuki, TOMITA Yoshihito, MIYAKODA Tuyako, KAGEYAMA Yasuo

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Kobe University of Mercantile Marine, 1998 - 1999

  We consider the Dirichelet problem for a semilinear degenerate elliptic equation (DP) : -g(|x|)Δu+f(|x|, u(x)) = 0, and Boundary Condition where N【greater than or equal】2 and g(|x|), f(|x|, u) are continuous. We discuss the problem (DP) under the following assumption : 1)g is nonnegative. 2)f is strictly monotone for u. We first define a standard viscosity solution by the viscosity solution such that if g(|x|) = 0 then f(|x|, u(x)) = 0. Then we can prove that the any continuous standard viscosity solution is the radial solution and it is unique. We add an assumption : 3)∫ィイD1a-0ィエD1gィイD1-1ィエD1(s)ds = ∞ or ∫ィイD2a+0ィエD2gィイD1-1ィエD1(s)ds = ∞ for any a : g(a) = 0. Then We obtain that any continuous viscosity solution is the radial solution and it is unique. If the assumption 3) is not satisfied there exist examples such that the continuous viscosity solutions are not unique. Here, the domain is a bounded boall in n-dimension space. We next state the existence and uniqueness of the continuous unbounded viscosity solution in RィイD12ィエD1. We use the order of the infinite neighborhood of the solution as the boundary condition. We know that the existence or nonexistence of the solution are dependent on a kind of the order of the solution. Moreover, we get the results which the uniqueness or non-uniqueness are also dependent on a kind of the order of the solution. In case, we assume that g, f is sufficiently smooth. We now show the existence of a continuous viscosity solution to quasi-semilinear degenerate elliptic problem. Here, g(|x|, u), f(|x|, u) are continuous and f is strictly monotone for u. Moreover, we assume there exists an implicite function of f = 0 and the implicite function holds some smoothness. Then we can prove the existence of the continuous viscosity solution. But it is difficult to prove the uniqueness of the solution.


• Asymptotics of the solutions of nonlinear partial differential equations and their applications to phase transition

  YAMADA Naoki, ISHII Katsuyuki, KUROKIBBA Masaki, KUSANO Takashi

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Fukuoka University, 1997 - 1999

  We formulated obstacle problems especially bilateral obstacle problems by the framework of viscosity solutions, and obtained an estimate of the size of touching subdomains of the solution. This estimate is an extension of the previous works that estimated the size of touching subdomains for unilateral obstacle problems. The motion of polygons governed by crystalline curvature is a model of crystal growth under the super cooling. We consider a two-dimensional problem with a crystalline energy whose level sets are regular n-polygons and got an result of the convergence of these solutions to the unique smooth solution of the mean curvature flow. We also investigate a system of partial differential equations associated with the free energy of surface tension. This system is proposed as a model of phase separation phenomena in alloy of two metals. We proved the global existence and the uniqueness of the solution. We considered various nonlinear ordinary differential equations and got conditions that the solutions to be oscillatory or nonoscillatory. These investigations are not only interesting in itself but also useful to construct approximate solutions in partial differential equations treated in our project.


• Theory and applications of viscosity solutions

  ISHII Hitoshi, TOMITA Yoshihito, GIGA Yoshikazu, MOCHIZUKI Kiyoshi, ISHII Katsuyuki, KOIKE Shigeaki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), TOKYO METROPOLITAN UNIVERSITY, 1997 - 1999

  The results obtained are summarized as follows. 1. We introduced a method of constructing an approximate feedback control for state-constraint control problems via viscosity solutions of the corresponding Hamilton-Jacobi equations. 2. The uniqueness and existence theorem due to Barron-Jensen on semicontinuous viscosity solutions of Hamilton-Jacobi equations is a fundamental tool in characterizing value functions in optimal control when the value functions are semicontinuous. We established a theorem similar to the Barron-Jensen theorem in Hilbert spaces. 3. We considered the Hamilton-Jacobi equation in ergodic control and gave a characterization of the existence of viscosity solutions of the Hamilton-Jacobi equation through a kind of value function of the corresponding ergodic optimal control. 4. In the Barron-Jensen theory of semicontinuous viscosity solutions the convexity of Hamiltonians is a key assumption. We introduced a notion of semicontinuous viscosity solution for Hamilton-Jacobi equations with non-convex Hamiltonian for which nice uniqueness and existence properties hold. 5. We studied the solvability, uniqueness, smoothness of solutions of Bellman equations in risk-sensitive stochastic control as well as the relation between its singular limit and a differential game. 6. We introduced a geometric approximation scheme for Gauss curvature flow of a convex body and proved its convergence. 7. We proved the equivalence between the invariance of a controlled stochastic differential equation with respect to a compact set and the restriction property to the compact set of viscosity solutions of the corresponding Bellman equation. 8. We studied the waiting time phenomena for Gauss curvature flow of a convex set and proved that if two principal curvatures vanish at a point on the initial surface then the waiting time of the point is positive.


• 関数解析的方法方法による非線型波動方程式の研究

  丸尾 健二, 都田 艶子, 石井 克幸, 井上 哲夫, 村上 隆彦, 富田 義人

  日本学術振興会, 科学研究費助成事業, 一般研究(C), 神戸商船大学, 1994 - 1994

  時間遅れを持つ非線形波動方程式(1)d^2u(t)/dt^2+∂Φu(t)+∂Ψu(t-h)+∫^0_<-h>a(-s)∂Φu(s+t)ds∋fを実ヒルベルト空間H(L^2(Ω))で考える。ここでΦ、ΨはHから(-∞,∞)への下半連続な凸関数でその劣微分を∂Φ、∂Ψと表わしている。しかしこのままの非線型で考えるのは大変難しいので、まず線型の場合を考えた。すなはちAを自己共役な正値線形作用素、A_1を線形閉作用素(A⊂A_1)とし、∂Φ=A、∂Ψ=A_1としたとき上記の解の存在と一意性を調べた。遅れを持つ作用素や外力f、初期値u(0)=a,(du/dt)(0)=bがA_nのn乗巾の領域に入っていることと(1)の解の存在域との関係がはっきりしてきた。また非斉次項fの微分可能性と解の存在域との関係も同様にわかってきた。これらから線型方程式の場合の解の存在と一意性についての十分条件が(ほぼ必要条件)見つかったと思う。今原稿の作成中である。また非線型方程式については∂Ψ(X)のnormがΦ(X)で押えられるなら解の存在はいうことができた。しかしいまだ満足するものではなくいましばらく研究を要すると思う。また遅れを含む方程式という観点から、遅れを含む非線型拡散方程式du(t)/dt+∂Φu(t)+∂Φu(t-h)+∫a(-s)∂Φu(s+t)ds∋fにつても調べた。fがL_1(0,T;H)に含まれているとき、今までにわかっていた連続解よりは滑らかな無条件連続な解を見つけることができた。ただしΦには適当な仮定が入る。しかしこの仮定は線型の場合は無条件に成るような仮定である。またこの場合の方程式は生物モデルも含んでいるので、生物モデルに現われる楕円形方程式についても調べた。すなはち一点で特異性を持つ楕円型方程式の解の一意性についてである。


• 二階非線形偏微分方程式に対する粘性解の研究

  冨田 義人, 石井 克幸, 丸尾 健二, 井上 哲男, 村上 隆彦

  日本学術振興会, 科学研究費助成事業, 一般研究(C), 神戸商船大学, 1993 - 1993

  1.全空間における非線形楕円型方程式(*)F(x,u,Du,D^2u)=0 in R^nの粘性解が一意に存在するための解のクラスをFの構造と関連させて決定した.主結果を粗く述べると、F(x,u,p,X)がpに関してm(m≧1)次の多項式、xに関してμ(μ≧1)次の多項式のようなふるまいをし、mとμとの間に1<μ1,N≧2,LAMBDA(x)=(1-|×|)^λ(|×|<1のとき);=(|×|-1)^λ(1<|×|


• 関数解析的方法による発展方程式の研究

  丸尾 健二, 都田 艶子, 石井 克幸, 冨田 義人

  日本学術振興会, 科学研究費助成事業, 一般研究(C), 神戸商船大学, 1992 - 1992

  (ア)制御システムに表れる方程式を含む、時間遅れhを持つ方物型方程式をヒルベルト空間で発展方程式として表現し、その方程式系が解を一意的に持つ為の外力項の研究をした。その結果は外力項がt=0における変格積分は可能でありかつL^2loc(O,T:H)に入っているが、L^1(O,T:H)には入るとはかぎらないという大前提の元、t=nh(n=0,1,2,…,N)では連続になるとはかぎらないがt≠nhでは方程式を満す弱解の存在と一意性を示す事ができた。又弱解であって、解が連続になる外力項の必要十分条件を示す事にも成功した。これらを一つにまとめO.J.Mに投稿中である。 投稿の時点においてはまだ未解決であった、弱解で中t=0で不連続になる解が存在する方程式系の外力項を、具体的に構成する事に成功した。その上に弱解でnohまでは連続であるがt=nohでは不連続となる解を持つ方程式系の外力項の構成もほぼ完成しており、現在論文作成中である。次に劣微分作用素を非線型項に持ちかつ時間遅れhを持つ非線型方物型方程式を考え、線型と同様な問題を研究した。線型と同様な議論ができず、弱解の定義すらはっきりしないが、連続な強解を得る為の外力項の十分条件を見つける事ができた。この外力項は当然我々の大前提は満足している関数であって真にL^1(O,T:H)に入らない物である。これらをまとめて論文とし投稿中である。 (イ)劣微分項を非線型項に持つ非線型双曲型方程式の解の存在について、空間次元一次元で障害物の上に弦の張った時の振動方程式を含む様な条件を劣微分作用素に付けかつその障害物が時間と共に動く様な場合で、解の存在を示した論文はF.Eに授理された。又障害物は動かないが、方程式系に時間遅れを入れた方程式系の解の存在については、O.J.Mに発表した。