Details of the Researcher

PHOTO

Shinya Okabe
Section
Graduate School of Science
Job title
Professor
Degree

  • 博士（理学）（東北大学）

  • 修士（理学）（東北大学）

Research History 3

  • 2008/10 - 2011/03
    岩手大学人文社会科学部　准教授

  • 2007/04 - 2008/09
    東北大学大学院理学研究科　助教

  • 2006/04 - 2007/03
    北海道大学理学研究院数学部門　COE学術研究員

Education 3

  • Tohoku University　Graduate School, Division of Natural Science　数学専攻

    - 2006/03/31

  • Tohoku University　Graduate School, Division of Natural Science　数学専攻

    - 2003/03/31

  • Tohoku University　Faculty of Science　数学専攻

    - 2001/03/31

Professional Memberships 1

  • 日本数学会

Research Interests 2

  • 変分法

  • 偏微分方程式論

Research Areas 1

  • Natural sciences / Mathematical analysis /

Awards 1

  1. 第七回函数方程式論分科会 福原賞

    2015/12/19　日本数学会 函数方程式論分科会

Papers 34

  1. Optimality of smallness conditions in Willmore obstacle problems under Dirichlet boundary conditions Peer-reviewed

    Hans-Christoph Grunau, Shinya Okabe

    Nonlinear Analysis: Real World Applications　85　104363-104363　2025/10

    Publisher: Elsevier BV

    DOI： 10.1016/j.nonrwa.2025.104363 　

    ISSN： 1468-1218

  2. Curve Diffusion Flow with Boundary on Skew Lines Peer-reviewed

    Fuya Hiroi, Shinya Okabe

    The Journal of Geometric Analysis　35　(7)　2025/05/10

    Publisher: Springer Science and Business Media LLC

    DOI： 10.1007/s12220-025-02021-4 　

    ISSN： 1050-6926

    eISSN： 1559-002X

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    Abstract We consider the curve diffusion flow for open planar curves with boundary on two skew lines. For each angle $$\theta \in (0, \pi )$$ of the two skew lines, we prove the existence of global-in-time solutions under a suitable initial condition. Furthermore, we show the full limit convergence of solutions to the arc of the sector with the central angle $$\theta $$ and the same area as that of the initial curve.

  3. An obstacle problem for the p-elastic energy

    Anna Dall’Acqua, Marius Müller, Shinya Okabe, Kensuke Yoshizawa

    Calculus of Variations and Partial Differential Equations　63　(6)　2024/06/24

    Publisher: Springer Science and Business Media LLC

    DOI： 10.1007/s00526-024-02752-2 　

    ISSN： 0944-2669

    eISSN： 1432-0835

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    Abstract In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler–Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers.

  4. Willmore Obstacle Problems under Dirichlet Boundary Conditions Peer-reviewed

    Hans-Christoph Grunau, Shinya Okabe

    ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE　1415-1462　2023/09/29

    Publisher: Scuola Normale Superiore - Edizioni della Normale

    DOI： 10.2422/2036-2145.202105_064 　

    ISSN： 0391-173X

    eISSN： 2036-2145

  5. The p-elastic flow for planar closed curves with constant parametrization Peer-reviewed

    Shinya Okabe, Glen Wheeler

    Journal de Mathématiques Pures et Appliquées　173　1-42　2023/05

    Publisher: Elsevier BV

    DOI： 10.1016/j.matpur.2023.02.001 　

    ISSN： 0021-7824

  6. Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity Peer-reviewed

    Kazuhiro Ishige, Shinya Okabe, Tokushi Sato

    Advanced Nonlinear Studies　23　(1)　2023

    Publisher: Walter de Gruyter GmbH

    DOI： 10.1515/ans-2022-0073 　

    eISSN： 2169-0375

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    Abstract In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ∗>0{\kappa }^{\ast }\gt 0 such that the elliptic problem for an inhomogeneous elliptic equation −Δu+u=up+κμ-\Delta u+u={u}^{p}+\kappa \mu in RN{ { \bf{R } } }^{N} possesses a positive minimal solution decaying at the space infinity if and only if 0<κ≤κ∗0\lt \kappa \le {\kappa }^{\ast }. Here, N≥2N\ge 2, μ\mu is a nontrivial nonnegative Radon measure in RN{ { \bf{R } } }^{N} with a compact support, and p>1p\gt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.

  7. Existence of solutions to nonlinear parabolic equations via majorant integral kernel Peer-reviewed

    Kazuhiro Ishige, Tatsuki Kawakami, Shinya Okabe

    Nonlinear Analysis　223　113025-113025　2022/10

    Publisher: Elsevier BV

    DOI： 10.1016/j.na.2022.113025 　

    ISSN： 0362-546X

  8. Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity Peer-reviewed

    Kazuhiro Ishige, Shinya Okabe, Tokushi Sato

    Advances in Nonlinear Analysis　11　(1)　968-992　2022/02/25

    Publisher: Walter de Gruyter GmbH

    DOI： 10.1515/anona-2021-0220 　

    ISSN： 2191-9496

    eISSN： 2191-950X

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    Abstract In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P) −Δu+u=F(u)+κμ  in  RN, u>0  in  RN, u(x)→0  as  |x|→∞,- \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad { { \bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad { { \bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty ,where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN)\{0}\mu \in L_{\rm{c } }^1({ { \bf R}^N})\backslash \{ 0\} is nonnegative. Then, under a suitable integrability condition on μ, there exists a threshold parameter κ* > 0 such that problem (P) possesses a solution if 0 < κ < κ* and it does not possess no solutions if κ > κ*. Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ*.

  9. On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

    Tatsuya Miura, Shinya Okabe

    Archive for Rational Mechanics and Analysis　239　(2)　1111-1129　2021/02

    Publisher: Springer Science and Business Media LLC

    DOI： 10.1007/s00205-020-01591-7 　

    ISSN： 0003-9527

    eISSN： 1432-0673

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    <title>Abstract</title>In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application, we obtain a global existence result for the surface diffusion flow, providing that an initial curve is <inline-formula><alternatives><tex-math>$$H^2$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math></alternatives></inline-formula>-close to a multiply covered circle and is sufficiently rotationally symmetric.

  10. Asymptotic Behavior of Solutions for a Fourth Order Parabolic Equation with Gradient Nonlinearity via the Galerkin Method

    Nobuhito Miyake, Shinya Okabe

    Geometric Properties for Parabolic and Elliptic PDE's　247-271　2021

    Publisher: Springer International Publishing

    DOI： 10.1007/978-3-030-73363-6_12 　

    ISSN： 2281-518X

    eISSN： 2281-5198

  11. A dynamical approach to the variational inequality on modified elastic graphs Peer-reviewed

    Shinya Okabe, Kensuke Yoshizawa

    Geometric Flows　5　(1)　78-101　2020/10/31

    Publisher: Walter de Gruyter GmbH

    DOI： 10.1515/geofl-2020-0100 　

    eISSN： 2353-3382

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    <title>Abstract</title>We consider the variational inequality on modified elastic graphs. Since the variational inequality is derived from the minimization problem for the modified elastic energy defined on graphs with the unilateral constraint, a solution to the variational inequality can be constructed by the direct method of calculus of variations. In this paper we prove the existence of solutions to the variational inequality via a dynamical approach. More precisely, we construct an <italic>L</italic>2-type gradient flow corresponding to the variational inequality and prove the existence of solutions to the variational inequality via the study on the limit of the flow.

  12. A gradient flow for the p-elastic energy defined on closed planar curves Peer-reviewed

    Shinya Okabe, Paola Pozzi, Glen Wheeler

    Mathematische Annalen　378　(1-2)　777-828　2020/10

    Publisher: Springer Science and Business Media LLC

    DOI： 10.1007/s00208-019-01885-6 　

    ISSN： 0025-5831

    eISSN： 1432-1807

  13. Existence of solutions for a higher-order semilinear parabolic equation with singular initial data Peer-reviewed

    Kazuhiro Ishige, Tatsuki Kawakami, Shinya Okabe

    Annales de l'Institut Henri Poincaré C, Analyse non linéaire　37　(5)　1185-1209　2020/09

    Publisher: Elsevier BV

    DOI： 10.1016/j.anihpc.2020.04.002 　

    ISSN： 0294-1449

  14. The obstacle problem for a fourth order semilinear parabolic equation Peer-reviewed

    Shinya Okabe, Kensuke Yoshizawa

    Nonlinear Analysis　198　111902-111902　2020/09

    Publisher: Elsevier BV

    DOI： 10.1016/j.na.2020.111902 　

    ISSN： 0362-546X

  15. Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

    Hans-Christoph Grunau, Nobuhito Miyake, Shinya Okabe

    Advances in Nonlinear Analysis　10　(1)　353-370　2020/08/02

    Publisher: Walter de Gruyter GmbH

    DOI： 10.1515/anona-2020-0138 　

    ISSN： 2191-9496

    eISSN： 2191-950X

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    Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

  16. A remark on the first p-buckling eigenvalue with an adhesive constraint

    Yoshihisa Kaga, Shinya Okabe

    Mathematics in Engineering　3　(4)　1-15　2020

    Publisher: American Institute of Mathematical Sciences (AIMS)

    DOI： 10.3934/mine.2021035 　

    ISSN： 2640-3501

  17. Blowup for a Fourth-Order Parabolic Equation with Gradient Nonlinearity Peer-reviewed

    Kazuhiro Ishige, Nobuhito Miyake, Shinya Okabe

    SIAM Journal on Mathematical Analysis　52　(1)　927-953　2020/01

    Publisher: Society for Industrial & Applied Mathematics (SIAM)

    DOI： 10.1137/19m1253654 　

    ISSN： 0036-1410

    eISSN： 1095-7154

  18. A supercritical scalar field equation with a forcing term Peer-reviewed

    Kazuhiro Ishige, Shinya Okabe, Tokushi Sato

    Journal de Mathématiques Pures et Appliquées　128　183-212　2019/08

    Publisher: Elsevier BV

    DOI： 10.1016/j.matpur.2019.04.003 　

    ISSN： 0021-7824

  19. Controllability of hybrid PDE‐ODE systems with structural instability and applications to mathematical models on intermittent hormonal therapy for prostate cancer Peer-reviewed

    Kurumi Hiruko, Shinya Okabe

    Mathematical Methods in the Applied Sciences　41　(17)　8229-8247　2018/11

    DOI： 10.1002/mma.5284 　

  20. Stability analysis on a hybrid PDE-ODE system describing intermittent hormonal therapy of prostate cancer Peer-reviewed

    Kurumi Hiruko, Shinya Okabe

    Mathematical Models and Methods in Applied Sciences　28　(3)　487-523　2018/03/01

    Publisher: World Scientific Publishing Co. Pte Ltd

    DOI： 10.1142/S0218202518500136 　

    ISSN： 0218-2025

  21. Shape memory wires in R^3 Peer-reviewed

    Shinya Okabe, Takashi Suzuki, Shuji Yoshikawa

    Shape Memory Alloys - Fundamentals and Applications　2017

    DOI： 10.5772/66914 　

  22. The two-obstacle problem for the parabolic biharmonic equation Peer-reviewed

    Matteo Novaga, Shinya Okabe

    Nonlinear Analysis, Theory, Methods and Applications　136　215-233　2016/05/01

    DOI： 10.1016/j.na.2016.02.004 　

    ISSN： 0362-546X

    eISSN： 1873-5215

  23. Dynamical Aspects of a Hybrid System Describing Intermittent Androgen Suppression Therapy of Prostate Cancer Peer-reviewed

    Kurumi Hiruko, Shinya Okabe

    GEOMETRIC PROPERTIES FOR PARABOLIC AND ELLIPTIC PDE'S　176　191-230　2016

    DOI： 10.1007/978-3-319-41538-3_12 　

    ISSN： 2194-1009

  24. Regularity of the obstacle problem for the parabolic biharmonic equation Peer-reviewed

    Matteo Novaga, Shinya Okabe

    Mathematische Annalen　363　(3-4)　1147-1186　2015/12/01

    DOI： 10.1007/s00208-015-1200-5 　

    ISSN： 0025-5831

    eISSN： 1432-1807

  25. Convergence to equilibrium of gradient flows defined on planar curves Peer-reviewed

    Matteo Novaga, Shinya Okabe

    Journal fur die Reine und Angewandte Mathematik　2015　(733)　2015

    DOI： 10.1515/crelle-2015-0001 　

    ISSN： 1435-5345 0075-4102

    eISSN： 1435-5345

  26. The gradient flow for the modified one-dimensional Willmore functional defined on planar curves with infinite length Peer-reviewed

    Shinya Okabe

    NONLINEAR DYNAMICS IN PARTIAL DIFFERENTIAL EQUATIONS　64　193-200　2015

  27. Curve shortening-straightening flow for non-closed planar curves with infinite length Peer-reviewed

    Matteo Novaga, Shinya Okabe

    Journal of Differential Equations　256　(3)　1093-1132　2014/02/01

    DOI： 10.1016/j.jde.2013.10.009 　

    ISSN： 0022-0396 1090-2732

    eISSN： 1090-2732

  28. Remarks on a dynamical aspect of shortening-straightening flow for non-closed planar curves with fixed boundary Peer-reviewed

    Shinya Okabe

    RIMS Kokyuroku Bessatsu　B35　41-64　2012/12

    Publisher: Kyoto University

    ISSN： 1881-6193

  29. The variational problem for a certain space-time functional defined on planar closed curves Peer-reviewed

    Shinya Okabe

    Journal of Differential Equations　252　(10)　5155-5184　2012/05/15

    DOI： 10.1016/j.jde.2012.01.020 　

    ISSN： 0022-0396 1090-2732

  30. The existence and convergence of the shortening-straightening flow for non-closed planar curves with fixed boundary Peer-reviewed

    Shinya Okabe

    Gakuto International Series Mathematical Sciences and Applications－International Symposium on Computational Science 2011－　34　2011

  31. The dynamics of elastic closed curves under uniform high pressure Peer-reviewed

    Shinya Okabe

    Calculus of Variations and Partial Differential Equations　33　(4)　493-521　2008/12

    DOI： 10.1007/s00526-008-0179-0 　

    ISSN： 0944-2669

    eISSN： 1432-0835

  32. Winter School on Nonlinear Mathematical Sciences 2006

    Okabe Shinya

    Bulletin of the Japan Society for Industrial and Applied Mathematics　17　(2)　183-184　2007

    Publisher: The Japan Society for Industrial and Applied Mathematics

    DOI： 10.11540/bjsiam.17.2_183 　

    ISSN： 0917-2270

  33. The motion of elastic planar closed curves under the area-preserving condition Peer-reviewed

    Shinya Okabe

    Indiana University Mathematics Journal　56　(4)　1871-1912　2007

    DOI： 10.1512/iumj.2007.56.3015 　

    ISSN： 0022-2518

  34. Asymptotic form of solutions of the Tadjbakhsh-Odeh variational problem Peer-reviewed

    Shinya Okabe

    Asymptotic Analysis and Singularities — Elliptic and parabolic PDEs and related problems　47　(2)　709-728　2007

    Publisher: Mathematical Society of Japan

    DOI： 10.2969/aspm/04720709 　

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Misc. 9

  1. $H^2$(ds)-Sobolev gradient flow for the modified elastic energy Invited

    Shinya Okabe

    RIMS Kokyuroku Innovation of the theory for evolution equations: developments via cross-disciplinary studies　2277　79-89　2024/02

  2. 高階放物型問題に対する変分的時間離散近似解法

    岡部真也

    応用数学勉強会 レクチャーノート　1-81　2021/09

  3. Construction of solutions to a fourth order parabolic obstacle problem

    Shinya Okabe

    RIMS Kokyuroku　1896　12-25　2014/05

    Publisher: Kyoto University

    ISSN： 1880-2818

  4. Remarks on the motion of non-closed planar curves governed by shortening-straightening flow

    Shinya Okabe

    数理解析研究所講究録－現象の数理解析へ向けた非線形発展方程式とその周辺－　1746　34-48　2011/06

    Publisher: Kyoto University

    ISSN： 1880-2818

  5. 形状記憶合金ワイヤーの運動を記述する熱弾性方程式の導出

    Shinya Okabe, Takashi Suzuki, Syuji Yoshikawa

    数理解析研究所講究録－非線形発展方程式と現象の数理－　1693　1-10　2010/06

    Publisher: 京都大学

    ISSN： 1880-2818

  6. ある束縛条件下における平面弾性閉曲線のダイナミクス

    Shinya Okabe

    北海道大学数学講究録　116　1-31　2006/12

  7. 十分大きな圧力差をもつTadjbakhsh-Odeh変分問題の解の漸近形

    岡部 真也, OKABE Shinya

    盛岡応用数学小研究集会報告集　2005　13-18　2006/01/01

    Publisher: 岩手大学人文社会科学部

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    平面内に弾性でできた閉曲線があり,外側から一様な圧力がかかっている状況を考える.Thdjbdkhsh-0deh(1967)によってそのような閉曲線の形状を記述するモデルとしてある変分問題が提唱された.(問題1参照･)この変分問題のEuler-Lagrange方程式は周期境界条件をもつ曲率に関する二階常微分方程式で与えられる.そして,その解の漸近形は圧力が限りなく大きくなる場合に現れる.本稿では,Euler-Lagrange方程式に関する特異摂動問題を考察することにより,その漸近形の正確な表示を与える.

  8. 一様な圧力を受ける弾性閉曲線の運動

    岡部 真也, OKABE Shinya

    盛岡応用数学小研究集会報告集　2003　38-42　2004/01/01

    Publisher: 岩手大学人文社会科学部

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    平面内にある弾性体でできた閉曲線を考える.この閉曲線に,その内側と外側からそれぞれ一様な(例えば水圧のような)圧力pi＾＾＾→,po＾＾＾→がかかっているとする.ただし,圧力pi＾＾＾→,po＾＾＾→の大きさpi,Poは定数であり,その向きは閉曲線に対して垂直であるとする.また,P=PO-pi>0,つまり,外側からの圧力のほうが大きいとする.このとき,閉曲線は以下で述べるあるエネルギーを減らすように変形していく.従って,このような閉曲線のダイナミクスは,非伸縮であるという束縛条件に従う閉曲線のエネルギー汎函数に対する勾配流方程式に支配されることになる.

  9. The motion of an elastic closed curve with constant enclosed area

    OKABE Shinya

    数理解析研究所講究録　1405　197-213　2004

    Publisher: Kyoto University

    ISSN： 1880-2818

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Presentations 104

  1. Ideal flow for planar closed curves with local length constraint Invited

    Shinya Okabe

    2025/05/31

  2. 弾性曲線に対する障害物問題 Invited

    岡部真也

    東北大学数学教室談話会　2025/04/14

  3. A gradient flow for the ideal energy under a length constraint Invited

    Shinya Okabe

    Joint Meeting of the AMS, AustMS and NZMS, Mathematics of nonlinear diffusion processes　2024/12/10

  4. A gradient flow for the ideal functional under a length constraint Invited

    Shinya Okabe

    Integration of theory and application for a deeper understanding of nonlinear phenomena　2024/10/30

  5. A gradient flow for the ideal energy under a length constraint Invited

    Shinya Okabe

    China-Japan Workshop on Nonlinear Elliptic and Parabolic Equations　2024/10/09

  6. A gradient flow for the ideal energy under a length constraint Invited

    Shinya Okabe

    Mini-Workshop on Geometric Variational Problems, Geometric Measure Theory, and Related Topics　2024/09/25

  7. Ideal curve flow with constraints on length Invited

    Shinya Okabe

    The 81st Fujihara seminar ``Mathematical Aspects for Interfaces and Free Boundaries"　2024/06/05

  8. Dynamical approach to a generalized isoperimetric inequality Invited

    Shinya Okabe

    2024 OIST Workshop, Geometric Aspects of Partial Differential Equations　2024/01/16

  9. Recent advances in Sobolev gradient flows for curves Invited

    Shinya Okabe

    Euro-Japan Conference on Nonlinear Diffusions　2023/10/19

  10. 数学と医療の交叉点〜癌治療と関わるある数理モデル〜 Invited

    岡部真也

    日本数学会2023年度秋季総合分科会 市民講演会　2023/09/23

  11. Convergence of Sobolev gradient trajectories to elastica Invited

    Shinya Okabe

    Frontiers of gradient flows: well-posedness, asymptotics, singular limits　2023/08/21

  12. Convergence of Sobolev gradient trajectories to elastica Invited

    Shinya Okabe

    2022/12/25

  13. Convergence of Sobolev gradient trajectories to elastica Invited

    Shinya Okabe

    RIMS Workshop ``Innovation of the theory for evolution equations: developments via cross-disciplinary studies"　2022/10/18

  14. Convergence of Sobolev gradient trajectories to elastica Invited

    Shinya Okabe

    2022/10/08

  15. Sobolev gradient flow for the elastic energy defined on closed curves Invited

    Shinya Okabe

    2022/08/29

  16. Convergence of Sobolev gradient trajectories to elastica Invited

    Shinya Okabe

    Tuesday Seminar of Analysis　2022/05/31

  17. Convergence of Sobolev gradient trajectories to elastica Invited

    Shinya Okabe

    2022/03/16

  18. Convergence of Sobolev gradient trajectories to elastica Invited

    Okabe, Shinya

    2021/08/18

  19. The p-elastic flow for planar closed curves with constant parametrization Invited

    岡部真也

    第72回東工大数理解析セミナー　2021/06/11

  20. 高階放物型問題に対する変分的時間離散近似解法 Invited

    岡部真也

    応用数学勉強会2020　2020/11/15

  21. The modified p-elastic flow on planar closed curves Invited

    岡部真也

    九州関数方程式セミナー　2020/11/06

  22. On the isoperimetric inequality and surface diffusion flow for multiply winding curves Invited

    岡部真也

    応用数理解析セミナー　2020/07/09

  23. Relaxation to equilibrium in a Cahn--Hilliard system Invited

    岡部真也

    反応拡散方程式と非線形分散型方程式の解の挙動　2020/02/21

  24. Global existence of the surface diffusion flow for multiply winding curves with rotational symmetry Invited

    Okabe Shinya

    Nonlinear Geometric Partial Differential Equations　2020/02/05

  25. Variational approaches to higher order parabolic problems Invited

    2019/10/04

  26. Stability analysis on a hybrid PDE-ODE system describing intermittent hormonal therapy of prostate cancer Invited

    Okabe Shinya

    Mathematical Biology and Computational Biology　2019/06/22

  27. The p-elastic flow of curves in the plane Invited

    Okabe Shinya

    6th Italian-Japanese Workshop ``Geometric properties for parabolic and elliptic PDEs"　2019/05/23

  28. A gradient flow for the p-elastic energy defined on inextensible closed curves in the plane Invited

    Okabe Shinya

    AMS Sectional Meeting at the University of Hawaii, New trends on variational calculus and non-linear partial differential equations, II　2019/03/22

  29. Remarks on the asymptotic behavior of planar closed curves governed by the curve diffusion flow Invited

    岡部 真也

    楕円型・放物型微分方程式研究集会　2018/11/16

  30. Remarks on the asymptotic behavior of planar closed curves governed by the curve diffusion flow Invited

    岡部 真也

    応用解析研究会　2018/07/14

  31. Weak solutions to p-elastic flow defined on planar closed curves International-presentation

    Workshop on the geometric PDE and related topics　2017/08/11

  32. Convergence to equilibrium of gradient flow defined on planar curves International-presentation

    International Conference on PDEs, Geometric Analysis and Functional Inequalities　2017/03/07

  33. The gradient flow for elastic energy defined on planar curves International-presentation

    Workshop on interface motions and free boundary problems: mathematical analysis, numerical analysis, modellings and experiments　2016/07/08

  34. Convergence to equilibrium of gradient flow defined on planar curves

    Geometric flows and related problems　2016/03/03

  35. The two-obstacle problem for parabolic biharmonic equation International-presentation

    The 17th Northeastern Symposium on Mathematical Analysis　2016/02/15

  36. Convergence to equilibrium of gradient flow defined on planar curves International-presentation

    The 33th Kyushu Symposium on Partial Differential Equations　2016/01/27

  37. The two obstacle problem for the parabolic biharmonic equation

    愛媛大学 解析セミナー　2015/10/31

  38. Convergence to equilibrium of gradient flow defined on planar curves

    応用数学セミナー@芝浦工大　2015/08/26

  39. Convergence to equilibrium of gradient flow defined on planar curves

    小山高専数学談話会　2015/07/09

  40. Convergence to equilibrium of gradient flow defined on planar curves International-presentation

    Geometric Properties for Parabolic and Elliptic PDE's - 4th Italian-Japanese Workshop -　2015/05/25

  41. Convergence to equilibria of steepest descent flows defined on planar curves

    南大阪応用数学セミナー　2014/12/13

  42. The obstacle problem for a fourth order parabolic equation International-presentation

    Mathematical Approaches to Pattern Formation　2014/10/28

  43. The obstacle problem for a fourth order parabolic equation International-presentation

    表面・界面のダイナミクスの数理 VIII　2014/10/22

  44. 四階放物型方程式に対する障害物問題

    日本数学会秋期総合分科会　2014/09/25

  45. The obstacle problem for the parabolic biharmonic equation

    京都大学 NLPDE セミナー　2014/06/27

  46. Regularity of solutions to the obstacle problem for the biharmonic equation International-presentation

    The 6th Nagoya Workshop on Differential Equations　2014/03/10

  47. Regularity of solutions to the obstacle problem for the parabolic biharmonic equation International-presentation

    Joint Research Program on Nonlinear PDE's Universita di Firenze and Tohoku University　2014/03/03

  48. Asymptotic behavior of curve shortening-straightening flow for planar curves with infinite length

    Dynamics and Special Functions　2014/02/22

  49. Convergence to equilibrium of gradient flows defined on planar curves International-presentation

    The 15th Northeastern Symposium on Mathematical Analysis　2014/02/17

  50. Construction of solutions to a fourth order parabolic obstacle problem via minimizing movements International-presentation

    Geometry of solutions of partial differential equations　2013/11/20

  51. The variational problem for a certain space-time functional defined on planar closed curves International-presentation

    Differential Equations and Applications　2013/02/20

  52. Existence and convergence of solutions to the shortening-straightening flow for non-closed planar curves with infinite length International-presentation

    Calculus of Variations and Geometric Measure Theory　2012/11/21

  53. Long time existence of shortening-straightening flow for non-closed planar curves with infinite length International-presentation

    The 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications　2012/07/01

  54. Long time existence of shortening-straightening flow for non-closed planar curves with infinite length International-presentation

    Tohoku-Fudan Workshop on the Occasion of the Centennial of the Faculty of Science　2012/05/17

  55. 幾何学的発展方程式に従う平面曲線のダイナミクス

    第一回室蘭連続講演会　2012/01/11

  56. Long time existence of shortening-straightening flow for non-closed planar curves with infinite length International-presentation

    2011 NCTS Taiwan-Japan Workshop on PDEs and Geometric Analysis　2011/12/19

  57. The gradient flow for the modified one-dimensional Willmore functional defined on planar curves with infinite length International-presentation

    The 4th MSJ-SI Nonlinear Dynamics in Partial Differential Equations　2011/09/12

  58. The existence of shortening-straightening flow for non-closed planar curves with infinite length International-presentation

    ＲＩＭＳ研究集会「非線形現象に現れる界面運動の数理解析・数値解析　2011/07/12

  59. The existence of shortening-straightening flow for non-closed planar curves with infinite length International-presentation

    International Symposium on Computational Science 2011　2011/02/15

  60. The motion of non-closed planar curves governed by shortening-straightening flow International-presentation

    ＲＩＭＳ研究集会「現象の数理解析に向けた非線形発展方程式とその周辺」　2010/10/13

  61. The motion of non-closed planar curves governed by shortening-straightening flow

    洞爺解析セミナー　2010/09/28

  62. The variational problem for a certain space-time functional defined on closed curves

    神戸大学解析セミナー　2009/11/05

  63. 閉曲線上で定義された時空汎函数に対する変分問題

    東北大学応用数学セミナー　2009/07/16

  64. The variational problem for a certain action functional defined on closed curves International-presentation

    Variational problems for curves and surfaces and related topics　2009/06/30

  65. The variational problem for a certain space-time functional defined on closed curves International-presentation

    保存則と幾何学的偏微分方程式およびその解析　2009/06/10

  66. The variational problem for a certain action functional defined on closed curves

    北海道大学偏微分方程式セミナー　2009/04/23

  67. The variational problem for a certain action functional defined on closed curves

    福島応用数学小研究集会　2009/02/06

  68. The variational problem for a certain action functional defined on closed curves International-presentation

    Mathematical Science and Nonlinear Partial Differential Equations　2009/02

  69. The variational problem for a certain action functional defined on closed curves

    曲線と曲面の非線型解析　2008/12/16

  70. 閉曲線上で定義されたある action 汎函数に対する変分問題

    2008年度日本数学会秋季総合分科会　2008/09/24

  71. The variational problem for a certain action functional defined on closed curves

    第4回非線型の諸問題　2008/09/21

  72. The variational problem for a certain action functional defined on closed curves International-presentation

    The 33rd Sapporo Symposium on Partial Differential Equations　2008/08

  73. The variational problem for a certain action functional defined on closed curves

    室蘭工業大学談話会　2008/07/14

  74. 閉曲線上で定義されたある action 汎函数に対する変分問題

    NLPDE セミナー　2008/06/06

  75. The variational problem for a certain action functional defined on closed curves

    熊本大学応用解析セミナー　2008/05/10

  76. The dynamical aspects of elastic planar closed curves under uniform high pressure

    神楽坂解析セミナー　2008/04/26

  77. The dynamics of elastic closed curves under uniform high pressure

    第７回偏微分方程式ワークショップ　2008/03/13

  78. The dynamical aspects of elastic closed curves under uniform high pressure International-presentation

    The 9th Northeastern Symposium on Mathematical Analysis　2008/02/21

  79. The dynamics of elastic closed curves under uniform high pressure

    首都大学東京 変分問題セミナー　2008/01/18

  80. The stability and instability of elastic closed curves under uniform high pressure

    第５回浜松偏微分方程式研究集会　2007/12/17

  81. The structure of solutions of Tadjbakhsh-Odeh variational problem with large pressure term International-presentation

    Asis PDE mini-workshop at Sendai　2007/09/13

  82. あるエネルギー汎函数の勾配に支配される曲線のダイナミクス

    東北大学数学教室談話会　2007/06/04

  83. The dynamics of elastic closed curves under uniform high pressure

    九州関数方程式セミナー　2007/05/25

  84. 一様に強い圧力をうける平面弾性閉曲線のダイナミクス

    月曜解析セミナー　2007/05/21

  85. 一様に強い圧力をうける平面弾性閉曲線のダイナミクス

    埼玉大学解析ゼミ　2007/02/28

  86. 一様に強い圧力をうける平面弾性閉曲線の安定性および不安定性

    日本数学会2006年度秋季総合分科会函数方程式分科会　2006/09/19

  87. On the motion of an elastic closed curve with constant enclosed area International-presentation

    Workshop on Singularities in PDE and the Calculus of Variations　2006/07

  88. ある束縛条件に従う平面弾性閉曲線のダイナミクス

    COE研究員連続講演会　2006/06/13

  89. On the motion of closed curves under some constraints

    偏微分方程式セミナー　2006/04/17

  90. Asymptotic form of solutions of the Tadjbakhsh-Odeh variational problem International-presentation

    The 7th Northeasten Symposium on Mathematical Analysis　2006/02/20

  91. The dynamics of an elastic closed curve under external high pressure International-presentation

    Asymptotic Methods for Partial Differential Equations　2006/02/08

  92. The dynamics of elastic closed curves in the plane under some constraints International-presentation

    The 3rd The 21st COE Symposium -Exploring New Science by Bridging Particle-Matter Hierarchy-　2006/02

  93. 一つまたは二つの束縛条件をみたす平面閉曲線の運動

    微分方程式の総合的研究　2005/12/16

  94. 十分大きな圧力差をもつ Tadjbakhsh-Odeh 変分問題の解の漸近形

    盛岡応用数学小研究集会「遷移過程に現れるパターン形成」- モデリング, シミュレーション, そして解析 -　2005/10/08

  95. The asymptotic form of solutions of Tadjbakhsh-Odeh variational problem International-presentation

    MSJ-IRI 2005「漸近解析と特異点」　2005/07/18

  96. 囲む面積が一定な平面内の弾性曲線の運動 International-presentation

    平成16年度21世紀COEシンポジウム「物質階層融合科学の構築」　2005/03

  97. 囲む面積が一定な平面内の弾性閉曲線の運動

    曲線と曲面の変分問題と発展方程式　2005/02/16

  98. 囲む面積が一定な平面内の弾性閉曲線の運動

    曲線と曲面の非線型解析　2004/12/17

  99. 囲む面積が一定な平面内の弾性閉曲線の運動

    日本数学会2004年度秋季総合分科会函数方程式分科会　2004/09/19

  100. 囲む面積が一定な平面内の弾性閉曲線の運動

    応用数学セミナー　2004/07/15

  101. The motion of an elastic closed curve with constant enclosed area Invited

    Shinya Okabe

    2004/06/25

  102. The motion of an elastic closed curve under uniform pressure International-presentation

    The 5th Northeastern Sympsium on Mathematical Analysis　2004/02

  103. 一様な圧力をうける弾性閉曲線の運動

    盛岡応用数学小研究集会　2003/10/06

  104. 一様な圧力をうける弾性閉曲線の運動

    日本数学会2003年度秋季総合分科会函数方程式分科会　2003/09/24

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Research Projects 15

  1. 偏微分方程式論における解の変容過程の究明と新展開

    石毛 和弘, 川上 竜樹, 石渡 通徳, 石渡 哲哉, 岡部 真也, 宮本 安人, 高津 飛鳥

    Offer Organization: 日本学術振興会

    System: 科学研究費助成事業

    Category: 基盤研究(A)

    Institution: 東京大学

    2025/04/01 - 2030/03/31

  2. Mathematical and numerical analysis of Sobolev gradient flows appearing in interface and materials science

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (C)

    Institution: Okayama University of Science

    2022/04/01 - 2026/03/31

  3. Analysis on singularities of higher order geometric gradient flows

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Tohoku University

    2021/04/01 - 2026/03/31

  4. New development on higher order elliptic and parabolic PDEs -- cooperation between harmonic analysis and geometric analysis

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research (B))

    Institution: Tohoku University

    2020/10 - 2025/03

  5. Systematical geometric analysis and asymptotic analysis for evolution equations

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)

    Category: Grant-in-Aid for Scientific Research (S)

    Institution: The University of Tokyo

    2019/06/26 - 2024/03/31

  6. 曲線がなす距離空間におけるエネルギー勾配流の構成とその応用

    岡部 真也, SCHRADER PHILIP

    Offer Organization: 日本学術振興会

    System: 科学研究費助成事業 特別研究員奨励費

    Category: 特別研究員奨励費

    Institution: 東北大学

    2019/11/08 - 2022/03/31

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    本年度は弾性エネルギーに対するH2勾配流の定常解への完全収束について研究を実施した。L2勾配流の場合には、パラメータの変換など何らかの修正を加えた上でないと定常解への完全収束を示すことができない。その要因の一つとして、完全収束を示す際に必要となる勾配不等式をそういった変換を加えることなく示すことが困難であることが挙げられる。本研究において考察したH2勾配流の場合にはH2(ds)に適当な距離を定義した距離空間が完備となることを利用して、何の変換も加えることなく、勾配不等式を示すことに成功した。この結果を基に定常解への完全収束を証明し、論文として纏め学術誌に投稿中である。 弾性エネルギーに対するH2勾配流の研究は幾何学的汎関数に対する高階Sobolev勾配流を構成せんとする目的の第一歩と位置付けることができる。実際、上記の研究を基盤として、様々な応用を展開している。まず、閉曲線の長さ汎関数に対するH1勾配流に曲線が囲む面積を一定に保つという束縛を付した幾何学的発展方程式を考案した。現在、我々とG. Wheeler氏、V. Wheeler氏との共同研究として研究を継続しているところである。また、弾性エネルギーにメビウス汎関数を加えた汎関数に対するH2勾配流についても研究を展開している。この汎関数に対するL2勾配流については幾つかの研究が既になされているが、3/2階放物型と分類される型の方程式となるため、その解析は容易ではない。本研究は、新しい観点による勾配流を構成することによって、弾性結び目について動的な考察を与えることも目指すものである。 以上のように、当該年度に行なった研究は、幾何学的汎関数に対する高階Sobolev勾配流の研究のきっかけを作るに至った、学術的に価値のあるものであるといえる。

  7. Evolution equations describing non-standard irreversible processes

    Akagi Goro

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Tohoku University

    2016/04/01 - 2020/03/31

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    Irreversible phenomena represented by diffusion are major factors of important phenomena closely related to our life such as unidirectionality of time, aging of lives and fracture. Classical theories for irreversible phenomena have already been established in the last century. However, many important irreversible phenomena beyond the scope of the classical theories have been observed, and therefore, studies of mathematical analysis have been developed in order to analyze and understand those new phenomena. In this research project, we have developed a new framework on Evolution Equations for covering new mathematical models which describe various non-standard irreversible phenomena. In particular, principal methods of analysis have been established for phase-field equations with strong irreversibility arising from fracture and damage models as well as nonlocal evolution equations involving fractional Laplacians, and moreover, systematic research has been done for those equations.

  8. Fusion and evolution of asymptotic analysis and geometric analysis in partial differential equations

    Ishige Kazuhiro

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Scientific Research (A)

    2015/04/01 - 2020/03/31

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    We developed the arguments in geometric analysis and asymptotic analysis, and studied power concavity properties of solutions and singular phenomena such as blow-up phenomena. Furthermore, we established a new method to study asymptotic analysis which is applicable to fractional heat equations. More precisely, we studied the following topics: (1) Power concavity of solutions; (2) Solvability of nonlinear elliptic equations with dynamical boundary conditions; (3) Initial trace of solutions to nonlinear diffusion equations; (4) Blow-up set for systems of nonlinear heat equations; (5) Asymptotic analysis for the heat equation with a potential and its applications; (6) Higher order asymptotic analysis for fractional heat equations.

  9. Dynamical aspects of geometric evolution equations defined on curves or surfaces

    Okabe Shinya

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

    Category: Grant-in-Aid for Young Scientists (B)

    Institution: Tohoku University

    2012/04/01 - 2016/03/31

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    We consider the following problems: (A) Dynamical aspects of geometric evolution equations defined on planar curves; (B) Obstacle problem for parabolic biharmonic equation. Regarding the theme (A), we proved that (A1) a planar open curve with infinite length converges to the borderline elastica as time goes to infinity; (A2) if a solution of geometric evolution equation converges to an equilibrium along a sequence of time, then the curve converges to an equilibrium as time goes to infinity, provided that the set of the equilibria has a discrete structure. Concerning the theme (B), we considered (B1) obstacle problem for the parabolic biharmonic equation, and (B2) two-obstacle problem for the parabolic biharmonic equation. For each problems (B1) and (B2), we proved the long time existence of solutions and investigated the regularity of the solutions.

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

    Koike Shigeaki

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Tohoku University

    2011/04/01 - 2016/03/31

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    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.

  11. Theory of Differential Equations Applied to Biological Pattern Formation--from Analysis to Synthesis

    TAKAGI Izumi, IKEDA Hideo, OGAWA Takayoshi, NAGASAWA Takeyuki, YANAGIDA Eiji, UEYAMA Daishin, OKABE Shinya, NAKASHIMA Kimie, YAMADA Sumio

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

    Category: Grant-in-Aid for Scientific Research (A)

    Institution: Tohoku University

    2010/04/01 - 2015/03/31

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    The purpose of this project is to build mathematical theories on reaction-diffusion systems and on the deformation of curves and surfaces, which are necessary to understand, through mathematical models, the dynamic process of morphogenesis in embryonic stages. In particular, we have succeeded in building a theory on pattern formation in strongly heterogeneous environments, and this will help us devise more biologically realistic models of pattern formation.

  12. Development of mathematical analysis via phase field method

    TONEGAWA Yoshihiro, NISHIURA Yasumasa, OKABE Shinya, MAEKAWA Yasunori

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Category: Grant-in-Aid for Scientific Research (B)

    Institution: Hokkaido University

    2009 - 2012

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    A family of smooth surfaces parameterized by time is called mean curvature flow if the velocity of motion at each point and time is equal to its mean curvature vector. We have made fundamental advance of knowledge on the general existence and regularity theory of mean curvature flow which may have singularities, and moreover, on those of geometric time evolution problems in large.

  13. Dynamics of curves and surfaces governed by various geometric evolution equations

    OKABE Shinya

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

    Category: Grant-in-Aid for Young Scientists (B)

    2009 - 2011

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    The summary of our results are stated as follows :(1)We prove that there exists a critical point of the variational problem for a certain space-time functional defined on planar curves which is concerned with stochastically perturbed mean curvature flow for curves ; (2)we show that, for a planar smooth non-closed curve with finite or infinite length, there exists a smooth solution of the steepest descent flow for the modified total squared curvature for any finite time. Moreover, we prove that, for an initial curve perturbed from line segment, the solution smoothly converges to a stationary solution along a certain sequence of time.

  14. Qualitative Properties of Solutions of Differential Equations Modeling Biological Pattern Formation

    TAKAGI Izumi, YANAGIDA Eiji, IKEDA Hideo, NAGASAWA Takeyuki, IIDA Masato, ISHIGE Kazuhiro, UEYAMA Daishin, OGAWA Takayoshi, MOCHIZUKI Atsuhi, YAMADA Sumio

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

    Category: Grant-in-Aid for Scientific Research (A)

    Institution: Tohoku University

    2006 - 2009

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    Collapse of patterns is a newly found phenomenon characteristic to some reaction-diffusion systems possessing singular nonlinearities, where patterns are formed at first but eventually converge to a nonregular steady state. We have given sufficient conditions for patterns to collapse and also for solutions to blow-up in finite time. In addition, qualitative properties of solutions such as the dynamics of maximum points and/or asymptotic forms of solutions have been studied in detail. Moreover, movement of planar closed curves driven by bending energy is considered as a lower dimensional analogue for the geometric variational problem which determines the shape of red blood cells. All the critical points of the energy functional under some constraints are found and the gradient flow of the constraint minimization problem has been constructed.

  15. The dynamics of curves and surfaces governed by gradient of energy functionals

    OKABE Shinya

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

    Category: Grant-in-Aid for Young Scientists (B)

    2007 - 2008

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Social Activities 4

  1. 第14回仙台数学セミナー

    2007/08/09 - 2007/08/11

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    東北地方の高校生を対象とした「第１４回仙台数学セミナー」（財団法人 川井数理科学財団主催）において演習を担当した。

  2. 現代数学講演会

    2014/07/15 -

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    公益財団法人川井数理科学財団による「現代数学講演会」の講師として秋田県立秋田高等学校に赴き、講演「数学と臨床医療の関わり」を3年生向けに行った。

  3. 第18回仙台数学セミナー

    2011/08 -

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    東北地方の高校生を対象とした「第１８回仙台数学セミナー」（財団法人 川井数理科学財団主催）において講義・演習を担当した。

  4. 第15回仙台数学セミナー

    2008/08 -

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    東北地方の高校生を対象とした「第１５回仙台数学セミナー」（財団法人 川井数理科学財団主催）において演習を担当した。