Updated on 2022/12/02

写真a

 
KOIKE, Shigeaki
 
Scopus Paper Info  
Paper Count: 0  Citation Count: 0  h-index: 6

Citation count denotes the number of citations in papers published for a particular year.

Affiliation
Faculty of Science and Engineering, School of Advanced Science and Engineering
Job title
Professor

Research Institute

  • 2020
    -
    2022

    理工学術院総合研究所   兼任研究員

Education

  •  
    -
    1988.03

    Waseda University   Graduate School, Division of Science and Engineering  

  •  
    -
    1981.03

    Waseda University   Faculty of Science and Engineering   物理学科  

Degree

  • Waseda University   理学(数学)

Research Experience

  • 2019.04
    -
    Now

    Waseda University   Faculty of Science and Engineering

  • 2012.04
    -
    2019.03

    東北大学   教授

  • 2006.04
    -
    2012.03

    埼玉大学大学院理工学研究科   教授

  • 1992.04
    -
    2002.03

    埼玉大学理学部   助教授

  • 1989.10
    -
    1992.03

    東京都立大学理学部   助手

  • 1988.04
    -
    1989.09

    早稲田大学理工学部   助手

▼display all

Professional Memberships

  •  
     
     

    日本数学会

 

Research Areas

  • Basic analysis

Research Interests

  • 粘性解理論

  • Nonlinear Partial Differential Equations

Papers

  • Regularity of solutions of obstacle problems -old &new-

    Shigeaki Koike

    Springer Proceedings in Mathematics & Statistics   346   205 - 243  2021.05  [Refereed]  [Invited]

  • On Lp-viscosity solutions of bilateral obstacles with unbounded ingredients

    Mathematische Annalen   377 ( 3-4 ) 833 - 910  2020.08  [Refereed]

  • Weak Harnack inequality for fully nonlinear uniformly parabolic equations with unbounded ingredients and applications

    KOIKE Shigeaki, SWIECH Andrzej & TATEYAMA Shota

    Nonlinear Analysis   185   264 - 289  2019.08  [Refereed]

  • On the rate of convergence of solutions in free boundary problems via penalization

    Shigeaki Koike, Takahiro Kosugi, Makoto Naito

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   457 ( 1 ) 436 - 460  2018.01  [Refereed]

     View Summary

    The rate of convergence of approximate solutions via penalization for free boundary problems are concerned. A key observation is to obtain global bounds of penalized terms which give necessary estimates on integrations by the nonlinear adjoint method by L.C. Evans. (c) 2017 Elsevier Inc. All rights reserved.

    DOI

    Scopus

    2
    Citation
    (Scopus)
  • Maximum principle for Pucci equations with sublinear growth in Du and its applications

    Shigeaki Koike, Takahiro Kosugi

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   160   1 - 15  2017.09  [Refereed]

     View Summary

    It is obtained that there exist strong solutions of Pucci extremal equations with sublinear growth in Du and measurable ingredients. It is proved that a strong maximum principle holds in a local sense in Lemma 4.1 although even the (weak) maximum principle fails. By using this existence result, it is shown that the ABP type maximum principle and the weak Harnack inequality for viscosity solutions hold true. As an application, the Holder continuity for viscosity solutions of possibly singular, quasilinear equations is established. (C) 2017 Elsevier Ltd. All rights reserved.

    DOI

    Scopus

    1
    Citation
    (Scopus)
  • Remarks on viscosity solutions for mean curvature flow with obstacles

    K. Ishii, H. Kamata, S. Koike

    Springer Proceedings in Mathematics and Statistics   215   83 - 103  2017  [Refereed]

     View Summary

    Obstacle problems for mean curvature flow equations are concerned. Existence of Lipschitz continuous viscosity solutions are obtained under several hypotheses. Comparison principle globally in time is also discussed.

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    2
    Citation
    (Scopus)
  • Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term

    G. Galise, S. Koike, O. Ley, A. Vitolo

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   441 ( 1 ) 194 - 210  2016.09  [Refereed]

     View Summary

    In this paper we consider second order fully nonlinear operators with an additive superlinear gradient term. Like in the pioneering paper of Brezis for the semilinear case, we obtain the existence of entire viscosity solutions, defined in all the space, without assuming global bounds. A uniqueness result is also obtained for special gradient terms, subject to a convexity/concavity type assumption where superlinearity is essential and has to be handled in a different way from the linear case. (C) 2016 Elsevier Inc. All rights reserved.

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    13
    Citation
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  • Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians

    Guy Barles, Shigeaki Koike, Olivier Ley, Erwin Topp

    CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   54 ( 1 ) 539 - 572  2015.09  [Refereed]

     View Summary

    In this paper we obtain regularity results for elliptic integro-differential equations driven by the stronger effect of coercive gradient terms. This feature allows us to construct suitable strict supersolutions from which we conclude Holder estimates for bounded subsolutions. In many interesting situations, this gives way to a priori estimates for subsolutions. We apply this regularity results to obtain the ergodic asymptotic behavior of the associated evolution problem in the case of superlinear equations. One of the surprising features in our proof is that it avoids the key ingredient which are usually necessary to use the strong maximum principle: linearization based on the Lipschitz regularity of the solution of the ergodic problem. The proof entirely relies on the Holder regularity.

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    Scopus

    17
    Citation
    (Scopus)
  • REMARKS ON THE COMPARISON PRINCIPLE FOR QUASILINEAR PDE WITH NO ZEROTH ORDER TERMS

    Shigeaki Koike, Takahiro Kosugi

    COMMUNICATIONS ON PURE AND APPLIED ANALYSIS   14 ( 1 ) 133 - 142  2015.01  [Refereed]

     View Summary

    A comparison principle for viscosity solutions of second-order quasilinear elliptic partial differential equations with no zeroth order terms is shown. A different transformation from that of Barles and Busca in [3] is adapted to enable us to deal with slightly more general equations.

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    6
    Citation
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  • On the ABP maximum principle for L-p-viscosity solutions of fully nonlinear PDE

    Shigeaki Koike

    NONLINEAR DYNAMICS IN PARTIAL DIFFERENTIAL EQUATIONS   64   113 - 124  2015  [Refereed]

     View Summary

    Fully nonlinear second-order uniformly elliptic partial differential equations (PDE for short) with unbounded ingredietns are considered. The Aleksandrov-Bakelman-Pucci (ABP for short) maximum principle for LP-viscosity solutions of fully nonlinear, second-order uniformly elliptic PDE are shown.
    The results here are joint works with A. Swiech in [12], [13], [14], [15].

  • Representation formulas for solutions of Isaacs integro-PDE

    Shigeaki Koike, Andrzej Świȩch

    Indiana University Mathematics Journal   62 ( 5 ) 1473 - 1502  2013  [Refereed]

     View Summary

    We prove sub-and super-optimality inequalities of dynamic programming for viscosity solutions of Isaacs integro-PDE associated with two-player, zero-sum stochastic differential game driven by a Lévy-type noise. This implies that the lower and upper value functions of the game satisfy the dynamic programming principle and that they are the unique viscosity solutions of the lower and upper Isaacs integro-PDE. We show how to regularize viscosity sub-and super-solutions of Isaacs equations to smooth sub-and supersolutions of slightly perturbed equations.

    DOI

    Scopus

    9
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  • On the ABP maximum principle and applications

    S. Koike

    RIMS Kokyuroku   1845   107 - 120  2013

  • LOCAL MAXIMUM PRINCIPLE FOR L-p-VISCOSITY SOLUTIONS OF FULLY NONLINEAR ELLIPTIC PDES WITH UNBOUNDED COEFFICIENTS

    Shigeaki Koike, Andrzej Swiech

    COMMUNICATIONS ON PURE AND APPLIED ANALYSIS   11 ( 5 ) 1897 - 1910  2012.09  [Refereed]

     View Summary

    We establish local maximum principle for L-p-viscosity solutions of fully nonlinear elliptic partial differential equations with unbounded ingredients.

    DOI

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    16
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  • Comparison principle for unbounded viscosity solutions of degenerate elliptic PDEs with gradient superlinear terms

    Shigeaki Koike, Olivier Ley

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   381 ( 1 ) 110 - 120  2011.09  [Refereed]

     View Summary

    We are concerned with fully nonlinear possibly degenerate elliptic partial differential equations (PDEs) with superlinear terms with respect to Du. We prove several comparison principles among viscosity solutions which may be unbounded under some polynomial-type growth conditions. Our main result applies to PDEs with convex superlinear terms but we also obtain some results in nonconvex cases. Applications to monotone systems of PDEs are given. (C) 2011 Elsevier Inc. All rights reserved.

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    7
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  • REMARKS ON THE PHRAGMEN-LINDELOF THEOREM FOR L-p-VISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES WITH UNBOUNDED INGREDIENTS

    Shigeaki Koike, Kazushige Nakagawa

    ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS   2009 ( 146 ) 1 - 14  2009.11  [Refereed]

     View Summary

    The Phragmen-Lindelof theorem for L-p-viscosity solutions of fully nonlinear second order elliptic partial differential equations with unbounded coefficients and inhomogeneous terms is established.

  • Existence of strong solutions of Pucci extremal equations with superlinear growth in Du

    Shigeaki Koike, Andrzej Swiech

    JOURNAL OF FIXED POINT THEORY AND APPLICATIONS   5 ( 2 ) 291 - 304  2009.08  [Refereed]

     View Summary

    We prove existence of strong solutions of Pucci extremal equations with superlinear growth in Du and unbounded coefficients. We apply this result to establish the weak Harnack inequality for L(p)-viscosity supersolutions of fully nonlinear uniformly elliptic PDEs with superlinear growth terms with respect to Du.

    DOI

    Scopus

    17
    Citation
    (Scopus)
  • Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients

    Shigeaki Koike, Andrzej Swiech

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   61 ( 3 ) 723 - 755  2009.07  [Refereed]

     View Summary

    The weak Harnack inequality for L-p-viscosity solutions is shown for fully nonlinear, second order uniformly elliptic partial differential equations with unbounded coefficients and inhomogeneous terms. This result extends those of Trudinger for strong solutions [21] and Fok for L-p-viscosity solutions [13]. The proof is a modification of that of Caffarelli [5], [6]. We apply the weak Harnack inequality to obtain the strong maximum principle, boundary weak Harnack inequality, global C-alpha estimates for solutions of fully nonlinear equations, strong solvability of extremal equations with unbounded coefficients, and Aleksandrov-Bakehnan-Pucci maximum principle in unbounded domains.

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    Scopus

    32
    Citation
    (Scopus)
  • Recent developments on maximum principle for Lp -viscosity solutions of fully nonlinear elliptic/parabolic PDES

    Shigeaki Koike

    Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions     131 - 153  2009.01  [Refereed]

     View Summary

    The ABP type maximum principle for 'formula presented'-viscosity solutions of fully nonlinear second order elliptic/parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is exhibited.

    DOI

  • Maximum principle for fully nonlinear equations via the iterated comparison function method

    Shigeaki Koike, Andrzej Swiech

    MATHEMATISCHE ANNALEN   339 ( 2 ) 461 - 484  2007.10  [Refereed]

     View Summary

    We present various versions of generalized Aleksandrov-Bakelman-Pucci (ABP) maximum principle for L-p-viscosity solutions of fully nonlinear second-order elliptic and parabolic equations with possibly superlinear-growth gradient terms and unbounded coefficients. We derive the results via the "iterated" comparison function method, which was introduced in our previous paper (Koike and Swiech in Nonlin. Diff. Eq. Appl. 11, 491-509, 2004) for fully nonlinear elliptic equations. Our results extend those of (Koike and Swiech in Nonlin. Diff. Eq. Appl. 11, 491-509, 2004) and (Fok in Comm. Partial Diff. Eq. 23(5-6), 967-983) in the elliptic case, and of (Crandall et al. in Indiana Univ. Math. J. 47(4), 1293-1326, 1998; Comm. Partial Diff. Eq. 25, 1997-2053, 2000; Wang in Comm. Pure Appl. Math. 45, 27-76, 1992) and (Crandall and Swiech in Lecture Notes in Pure and Applied Mathematics, vol. 234. Dekker, New York, 2003) in the parabolic case.

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    31
    Citation
    (Scopus)
  • A linear-quadratic control problem with discretionary stopping

    Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B   8 ( 2 ) 261 - 277  2007.09  [Refereed]

     View Summary

    We study a the variational inequality for a 1-dimensional linear-quadratic control problem with discretionary stopping. We establish the existence of a unique strong solution via stochastic analysis and the viscosity solution technique. Finally, the optimal policy is shown to exist from the optimality conditions.

  • Optimal Consumption and Portfolio Choice with Stopping

    Shigeaki Koike, Hiroaki Morimoto

    FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA   48 ( 2 ) 183 - 202  2005.08  [Refereed]

     View Summary

    We study the Bellman equation associated with the optimal consumption and portfolio choice problem with stopping times in a complete market. We establish the existence of a strong solution by using the viscosity solutions technique. The optimal policy is shown to exist from the optimality conditions in the variational inequality.

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    Scopus

    2
    Citation
    (Scopus)
  • Perron's method for Lp-viscosity solutions

    S. Koike

    Saitama Mathematical Journal   23   9 - 28  2005  [Refereed]

  • Maximum principle and existence of L-p-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

    S Koike, A Swiech

    NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS   11 ( 4 ) 491 - 509  2004  [Refereed]

     View Summary

    We study L-p-viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for L-p-viscosity solution. We also prove stability and existence results for the equations under consideration.

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    Scopus

    19
    Citation
    (Scopus)
  • Variational inequalities for leavable bounded-velocity control

    S Koike, H Morimoto

    APPLIED MATHEMATICS AND OPTIMIZATION   48 ( 1 ) 1 - 20  2003.07  [Refereed]

     View Summary

    We study the variational inequality associated with a bounded-velocity control problem when discretionary stopping is allowed. We establish the existence, of a strong solution by using the viscosity solution techniques. The optimal policy is shown to exist from the optimality conditions in the variational inequality.

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    Scopus

    9
    Citation
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  • Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients

    S. Koike, T. Takahashi

    Advances in Differential Equations   7 ( 4 ) 493 - 512  2002  [Refereed]

  • On fully nonlinear PDEs derived from variational problems of Lp norms

    Toshihiro Ishibashi, Shigeaki Koike

    SIAM Journal on Mathematical Analysis   33 ( 3 ) 545 - 569  2001  [Refereed]

     View Summary

    The p-Laplace operator arises in the Euler-Lagrange equation associated with a minimizing problem which contains the Lpnorm of the gradient of functions. However, when we adapt a different Lpnorm equivalent to the standard one in the minimizing problem, a different p-Laplace-type operator appears in the corresponding Euler-Lagrange equation. First, we derive the limit PDE which the limit function of minimizers of those, as p → ∞, satisfies in the viscosity sense. Then we investigate the uniqueness and existence of viscosity solutions of the limit PDE. © 2001 Society for Industrial and Applied Mathematics.

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    27
    Citation
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  • Pursuit-evasion games with state constraints: Dynamic programming and discrete-time approximations

    M Bardi, S Koike, P Soravia

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   6 ( 2 ) 361 - 380  2000.04  [Refereed]

     View Summary

    In this paper we study the boundary value problem for the Hamilton-Jacobi-Isaacs equation of pursuit-evasion differential games with state constraints. We prove existence of a continuous viscosity solution and a comparison theorem that we apply to establish uniqueness of such a solution and its uniform approximation by solutions of discretized equations.

  • Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem

    O Alvarez, S Koike, Nakayama, I

    SIAM JOURNAL ON CONTROL AND OPTIMIZATION   38 ( 2 ) 470 - 481  2000.02  [Refereed]

     View Summary

    We obtain the uniqueness of lower semicontinuous (LSC) viscosity solutions of the transformed minimum time problem assuming that they converge to zero on a "reachable" part of the target in appropriate directions. We present a counter-example which shows that the uniqueness does not hold without this convergence assumption.
    It was shown by Soravia that the uniqueness of LSC viscosity solutions having a "subsolution property" on the target holds. In order to verify this subsolution property, we show that the dynamic programming principle (DPP) holds inside for any LSC viscosity solutions.
    In order to obtain the DPP, we prepare appropriate approximate PDEs derived through Barles' inf-convolution and its variant.

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    9
    Citation
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  • On ε-optimal controls for state constraint problem

    H. Ishii, S. Koike

    Annales de l'Institut Henri Poincar\'{e}, Analyse Non lin\'{e}aire   17 ( 4 ) 473 - 502  2000  [Refereed]

  • Semicontinuous viscosity solutions for Hamilton-Jacobi equations with a degenerate coefficient

    S. Koike

    Differential and Integral Equations   10 ( 3 ) 455 - 472  1997  [Refereed]

  • A comparison result for the state constraint problem of differential games

    S. Koike

    Proceedings of Korean-Japan Partial Differential Equations Conference     1 - 8  1997

  • A new formulation of state constraint problems for first-order PDES

    H Ishii, S Koike

    SIAM JOURNAL ON CONTROL AND OPTIMIZATION   34 ( 2 ) 554 - 571  1996.03  [Refereed]

     View Summary

    The first-order Hamilton-Jacobi-Bellman equation associated with the state constraint problem for optimal control is studied. Instead of the boundary condition which Soner introduced, a new and appropriate boundary condition for the PDE is proposed. The uniqueness and Lipschitz continuity of viscosity solutions for the boundary value problem are obtained.

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    52
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  • On the bellman equations with varying control

    S Koike

    BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY   53 ( 1 ) 51 - 62  1996.02  [Refereed]

     View Summary

    The value function is presented by minimisation of a cost functional over admissible controls. The associated first order Bellman equations with varying control are treated. It turns out that the value function is a viscosity solution of the Bellman equation and the comparison principle holds, which is an essential tool in obtaining the uniqueness of the viscosity solutions.

  • The state constraint problem for differential games

    S. Koike

    Indiana University Mathematics Journal   44 ( 2 ) 467 - 487  1995  [Refereed]

  • UNIQUENESS OF VISCOSITY SOLUTIONS FOR MONOTONE SYSTEMS OF FULLY NONLINEAR PDES UNDER DIRICHLET CONDITION

    S KOIKE

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   22 ( 4 ) 519 - 532  1994.02  [Refereed]

  • Viscosity solutions of monotone systems for Dirichlet problems

    M. Katsoulakis, S. Koike

    Differential and Integral Equations   7 ( 2 ) 367 - 382  1994  [Refereed]

  • Viscosity solutions of functional differential equations

    H. Ishii, S. Koike

    Advances in Mathematical Sciences and Applications   3   191 - 218  1993  [Refereed]

  • A viscosity solution approach to functional differential equations

    S. Koike

    Proceedings of the Second GARC SYMPOSIUM on Pure and Applied Mathematics   17   213 - 219  1993

  • ON THE RATE OF CONVERGENCE OF SOLUTIONS IN SINGULAR PERTURBATION PROBLEMS

    S KOIKE

    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   157 ( 1 ) 243 - 253  1991.05  [Refereed]

  • Viscosity solutions for monotone systems of second-order elliptic PDEs

    H. Ishii, S. Koike

    Communications in Partial Differential Equations   16 ( 6-7 ) 1095 - 1128  1991  [Refereed]

  • Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games

    H. Ishii, S. Koike

    Funkcialaj Ekvacioj   34 ( 1 ) 143 - 155  1991  [Refereed]

  • Remarks on elliptic singular perturbation problems

    Hitoshi Ishii, Shigeaki Koike

    Applied Mathematics & Optimization   23 ( 1 ) 1 - 15  1991.01  [Refereed]

     View Summary

    We show the effectiveness of viscosity-solution methods in asymptotic problems for second-order elliptic partial differential equations (PDEs) with a small parameter. Our stress here is on the point that the methods, based on stability results [3], [16], apply without hard PDE calculations. We treat two examples from [11] and [23]. Moreover, we generalize the results to those for Hamilton-Jacobi-Bellman equations with a small parameter. © 1991 Springer-Verlag New York Inc.

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  • AN ASYMPTOTIC FORMULA FOR SOLUTIONS OF HAMILTON-JACOBI-BELLMAN EQUATIONS

    S KOIKE

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   11 ( 3 ) 429 - 436  1987.03  [Refereed]

  • On the regularity of solutions of a degenerate parabolic Bellman equation

    S. Koike

    Hiroshima Mathematical Journal   16 ( 2 ) 251 - 267  1986  [Refereed]

  • Boundary regularity and uniqueness for an elliptic equation with gradient constraint

    Hitoshi Ishii

    Communications in Partial Differential Equations   8 ( 4 ) 317 - 346  1983.01  [Refereed]

    DOI

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Books and Other Publications

  • Nonlinear Partial Differential Equations for Future Applications

    Shigeaki Koike, Hideo Kozono, Takayoshi Ogawa, Shigeru Sakaguchi( Part: Edit)

    2021.05

  • 粘性解 -比較原理を中心に-

    小池 茂昭( Part: Sole author)

    2016.12

  • リメディアル数学

    泉屋周一他( Part: Joint author, 1章)

    数学書房  2011

  • 微分積分

    小池茂昭( Part: Sole author)

    数学書房  2010

  • International Conference for the 25th Anniversary of Viscosity Solutions

    Yoshikazu Giga 他( Part: Joint editor)

    学校図書  2008

  • これからの非線型偏微分方程式

    小薗英雄他( Part: Joint author, 151~168)

    日本評論社  2007.05

  • A Beginner's Guide to the Theory of Viscosity Solutions

    Shigeaki Koike( Part: Sole author)

    Mathematical Society of Japan  2007

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Misc

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Other

  • 粘性解の基礎理論の新展開

    2010.04
     
     

     View Summary

    Lp粘性解理論の研究

  • 完全非線形方程式の粘性解の基礎理論

    2008.04
     
     

     View Summary

    完全非線形方程式のLp粘性解

  • 粘性解生誕25周年国際研究集会

    2007.04
     
     

     View Summary

    上記研究集会の開催

  • 粘性解理論とその先端的応用

    2007.04
     
     

     View Summary

    粘性解理論の数理ファイナンスへの応用

  • 粘性解の微分可能性と最適制御に関する基礎研究

    2006.04
     
     

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    粘性解の微分可能性を高めることで最適制御を決定する

  • 粘性解理論とその応用

    2004.04
     
     

     View Summary

    粘性解の最適制御理論への応用

  • 粘性解理論による数理ファイナンスの基礎理論

    1998.04
     
     

     View Summary

    粘性解理論による数理ファイナンスの基礎理論

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Awards

  • 解析学賞

    2016.09   日本数学会   完全非線形楕円型・放物型偏微分方程式のLp粘性解理論

    Winner: 小池茂昭

  • JMSJ Outstanding Paper Prize

    2010.03   日本数学会  

    Winner: 小池茂昭, Andrzej Swiech

Research Projects

  • 粘性解理論とその応用

    Project Year :

    1988.04
    -
    Now
     

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

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

    Project Year :

    2020.04
    -
    2025.03
     

  • New development of mathematical theory of turbulence by collaboration of the nonlinear analysis and computational fluid dynamics

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)

    Project Year :

    2016.05
    -
    2021.03
     

  • Evolution equations describing non-standard irreversible processes

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

    Project Year :

    2016.04
    -
    2020.03
     

    Akagi Goro

     View Summary

    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.

  • New developments of the theory of viscosity solutions and its applications

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

    Project Year :

    2016.04
    -
    2020.03
     

  • Fusion and evolution of asymptotic analysis and geometric analysis in partial differential equations

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

    Project Year :

    2015.04
    -
    2020.03
     

    Ishige Kazuhiro

     View Summary

    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.

  • Geometric studies on singularity of non-linear phenomena

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

    Project Year :

    2014.04
    -
    2018.03
     

    Izumiya Shyuichi, TONEGAWA Yoshihiro, ONO Kaoru, UMEHARA Masaki, KOIKE Shigeaki

     View Summary

    In this research project we have shown that Lagrangian equivalence among Lagrangian submanifold germs and a certain equivalence relation among the corresponding graph-like wave fronts are the same. Applications of this result include the classical differential geometry, the geometry of the space-time, the differential geometry of singular surfaces and mappings, and so on. On the other hand, a new application of singularity theory to quantum mechanics is also discovered in this research project.

  • Deepening of the theory of viscosity solutions and its applications

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

    Project Year :

    2011.04
    -
    2016.03
     

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

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    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.

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

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

    Project Year :

    2011.04
    -
    2016.03
     

    Koike Shigeaki

     View Summary

    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.

  • Geometric properties and asymptotic behavior of solutions of diffusion equations

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

    Project Year :

    2011.04
    -
    2016.03
     

    Ishige Kazuhiro, KOZONO Hideo, OGAWA Takayoshi, KOIKE Shigeaki, YAMADA Sumio, YANAGIDA Eiji, KABEYA Yoshitsugu

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    We developed the method for studying geometric properties and asymptotic behavior of solutions of parabolic equations, and obtained the asymptotic behavior of hot spots and the optimal decay rates of the Lebesgue norms for the heat equation with a potential. Furthermore, we established a method of obtaining the higher order asymptotic expansions of the solutions behaving like the heat kernel. In addition, we study the location of the blow-up set for a semilinear heat equation by the profile of the solution just before the blow-up time. In particular, we gave a sufficient condition for no boundary blow-up.

  • A research on the geometric singularities of non-linear phenomena

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

    Project Year :

    2010.04
    -
    2014.03
     

    IZUMIYA Shyuichi, ISHIKAWA Goo, TERAO Hiroaki, TONEGAWA Yoshihiro, OHMOTO Toru, ONO Kaoru, UMEHARA Masaaki, KOIKE Shigeaki

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    In this research project we constructed the notion of 'lightlike curvature' for spacelike submanifolds of Lorentzian space forms by using the notion of 'lightcone Gauss maps'.As an application of the theory of Legendrian singularities, we described the singularities of the lighhtlike hypersurface along a spacelike submanifold. Moreover, we constructed a geometric framework to describe the caustics of world sheets which is an important notion in the theory of general relativity and the brane world scenario. We clarified the relation of the caustics and the wave front propagations iby using the theory of graph-like Legendrian unfoldings.

  • Studies on a unified point of view on global theories of nonlinear elliptic equations

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

    Project Year :

    2009.04
    -
    2014.03
     

    OZAWA Tohru, KOIKE Shigeaki, TANAKA Kazunaga

     View Summary

    We studied nonlinear elliptic equations arising in various fields of mathematical physics by means of variational analysis, ordinary differential equations, and viscosity techniques. We studied orbital stability of standing waves, explicit blow-up solutions, and exponential decay of ground states for systems of nonlinear Schr\"odinger type equations.

  • Analysis of gradient flow for the bending energy of plane curves under multiple constraints

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

    Project Year :

    2010
    -
    2012
     

    NAGASAWA Takeyuki, KOIKE Shigeaki

     View Summary

    The Helfrich variational problem is one for the bending energy of closed plane curves under constrains of length and enclosed volume. The problem can be formulated not only for curves but also for surfaces and hypersurface, i.e., higher dimensional case. We study the behavior of the corresponding gradient flow, called the “Helfrich flow". Firstly, we construct the Helfrich flow of general dimension, and discuss the uniqueness. Furthermore we get a new fact on the global existence of generalized rotational hypersurface which is useful for the analysis of behavior for the Helfrich flow.

  • Synthetic study of nonlinear evolution equation and its related topics

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

    Project Year :

    2009
    -
    2012
     

    OTANI Mitsuharu, YAMADA Yoshio, TANAKA Kazunaga, NISHIHARA Kenji, ISHII Hitoshi, OZAWA Tohru, OGAWA Takayoshi, KENMOCHI Nobuyuki, KOIKE Shigeaki, SAKAGUCHI Shigeru, SUZUKI Takashi, HAYASHI Nakao, IDOGAWA Tomoyuki, ISHIWATA Michinori, AKAGI Gorou

     View Summary

    Various types of nonlinear PDEs (nonlinear elliptic equations, nonlinear diffusion equations, nonlinear wave equations, nonlinear Schrodinger equations) arising in physics and engineering were synthetically studied from the viewpoint of the theory of nonlinear evolution equations by using the techniques from the theory of nonlinear functional analysis, the theory of functions of a real variable, the theory of ordinary differential equations and the calculus of variations.

  • Development of the methods of stochastic control and filtering in mathematical finance

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

    Project Year :

    2008
    -
    2012
     

    NAGAI Hideo, KOHATSU-HIGA Arturo, SEKINE Jun, MIYAHARA Yoshio, KOIKE Shigeaki, ISHII Hitoshi, HATA Hiroaki, AIDA Shigeki, NAGAHATA Yukio, TAMURA Takashi, KAISE Hidehiro

     View Summary

    We considered the portfolio optimization problems related to expected utility maximization and valuation of the derivatives as certain kinds of stochastic control problems and developed analysis based on filtering theory and the dynamic programming principles. In particular, we obtained notable results e.g. duality theorems etc., on the large deviation control problems by bringing new aspects in considering down side risk minimization.

  • Stability analysis on solitary wave solutions for systems of nonlinear wave equations

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

    Project Year :

    2009
    -
    2011
     

    OHTA Masahito, KOIKE Shigeaki, NAGASAWA Takeyuki, MACHIHARA Shuji

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    We improved the standard theory on instability of standing wave solutions for nonlinear Schrodinger equations. In particular, we proved a new instability result for a borderline case between stability and instability. Moreover, we applied the abstract theory to a system of nonlinear Schrodinger equations related to the Raman amplification in plasma. Furthermore, under a general condition, we proved that linear instability implies nonlinear instability.

  • Viscosity solution theory for fully nonlinear equations and its applications

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

    Project Year :

    2008
    -
    2010
     

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

     View Summary

    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.

  • Stochastic least principle and its applications

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

    Project Year :

    2007
    -
    2010
     

    MIKAMI Toshio, TAKEDA Masayoshi, KOIKE Shigeaki, KAISE Hidehiro

     View Summary

    We gave maxima and maximizer of joint distribution function of maximally dependent random variables and its application. We defined a generalization of the Knothe-Rozenblatt Rearrangement (KRR)and its stochastic version Knothe-Rozenblatt process (KRP), proved the existence and the uniqueness and gave a characterization as the limit of the minimizer of a class of variational problem with a small parameter. We also gave a representation theorem for a random probability density function of a stochastic flow of KRP. We proved that the mean of a logarithm of the random probability density function with space variable replaced by KRP is convex.

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

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

    Project Year :

    2006
    -
    2009
     

    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

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    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.

  • An arithmetic study of modular forms of half integral weight and Siegel modular forms

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

    Project Year :

    2006
    -
    2008
     

    KOJIMA Hisashi, SAKAI Fumio, MIZUTANI Tadayoshi, SAKAMATO Kunio, KOIKE Shigeaki, FUKUI Toshizumi, NAGASAWA Takeyoki, OHTA Masahito, SHIMOKAWA Koya, EBIHARA Madoka

  • STUDIES ON STABILITY OF SOLITARY WAVES AND BLOWUP OF SOLUTIONS FORNONLINEAR WAVE EQUATIONS

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

    Project Year :

    2006
    -
    2008
     

    OHTA Masahito, KOIKE Shigeaki, NAGASAWA Takeyuki, MACHIHARA Shuji

  • Reserch on the stability of solutions of geometric evolution equation using group equivariance

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

    Project Year :

    2005
    -
    2007
     

    NAGASAWA Takeyuki, KOIKE Shigeaki, OHTA Masahito, SAKAMOTO Kunio, KOHSAKA Yoshihito, TACHIKAWA Atsushi

     View Summary

    In this research we investigate the gradient flow with constraints for functionals defined for family of curves and surfaces as geometric evolution equations. The gradient flow, which decreases the value of functionals via deformation, is one of the method for finding critical points. Various shapes in the nature should be stable in some sense. Functionals are the measure of stability, and therefore the limit of gradient flow should be stable in this sense.
    Nagasawa and Kohsaka consider the Willmore functional for surfaces with prescribed area and enclosed volume (the Helfrich variational problem), and construct the associate gradient flow (the Helfrich flow), and analyze the structure of center manifold near sphere. On the stationary problem for the problem, solutions bifurcating from sphere with more 2, 4, 6 and 8 are constructed by Nagasawa. We reduce the bifurcation equation by use of the group equivariance but not the equivariant branching lemma of the bifurcation theory. Furthermore Nagasawa considers the Helfrich flow for plane curves. There an approximate problem such that the constraints are realized as a singular limit is proposed. It is shown that the uniform estimates for solutions for approximate problem and their convergence.
    In many case, equations of gradient flow are parabolic type. Koike investigates the maximum principle and comparison results for fully nonlinear parabolic and elliptic equations. Ohta studies the stability of solutions for evolution equation of hyperbolic type. Sakamoto investigates the CR-structure of manifolds. Kohsaka studies the nonlinear stability of stationary solutions for surface diffusion with boundary conditions. Solutions of geometric variational problem are weak solution of a nonlinear equation. Hence it is important to analyze their regularity. Tachikawa studies the regularity theory of minimal critical points for integral functional with discontinuous coefficients.

  • Expected utility maximiaation problems and stochastic control

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

    Project Year :

    2004
    -
    2007
     

    NAGAI HIideo, KOHATSU-HIGA Artuto, KOTANI Shinichi, MATSUMOTO Hiroyuki, ISHII Hitoshi, KOIKE Shigeaki

     View Summary

    We considered expected power utility maximization problems on infinite time horizon with transaction costs. Related risk-sensitive quasi-variational inequalities were deduced. The solution of the inequality consists of the pair of constant and a function. We constructed an optimal strategy by using the function of the solution and showed the constant give the optimal value of the problem.
    We studied problems on equilibrium price for insider trading and showed that there exists a stable equilibrium price for such models even though insiders affect stock prices.
    We studied asymptotic behavior of the solution to a Hamilton-Jacobi equation and showed that the solution converges to the asymptotic solution on the whole. Euclidean space under relatively general assumptions.
    By introducing the notion of Lp viscosity solutions we proved the existence of Lp viscosity solutions by a modified version of the Perron's method. Moreover we showed his solution turns out to be Holder continuous if the equation is uniform elliptic.
    By proving differentiabilities of viscosity solutions of the obstacle problems arising from mathematical finance we constructed optimal controls.
    By taking up linear Gaussian models which are incomplete market models we considered the problem minimizing a probability that growth rate of the wealth process lies below the prescribed value. The asymptotics of the probability is characterized as the dual of risk-sensitive portfolio optimization problem.

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

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

    Project Year :

    2004
    -
    2007
     

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

     View Summary

    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.

  • Geometry of singularities of mapping II

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

    Project Year :

    2003
    -
    2006
     

    FUKUI Toshizumi, KOIKE Satoshi, IZUMI Shuzo, SAKAI Fumio, MIZUTANI Tadayoshi, KOIKE Shigeaki

     View Summary

    We consider Thom-Bordman manifolds $Sigma^{i, j}$ (and their Zariski closure) in the jet space. We discussed the Cohen-Macaulay property (It is a big problem to decide whether it is Cohen-Macaulay or not from the view point in intersection theory in algebraic geometry.). It is composed by Mr.Ronga's desingularization and construct complecies supported $Sigma^{n^-p+1,1}$. An explicit formula counting the number of cusps which appeared in stable perturbation of the map-germ $(C^n,0) to(C^2,0)$ when n=2,3,4. (The first paper in the next page)
    Next, a classical differential geometry is discussed from the singular view point, significant point theory. We also discuss several differential equation appeared in this context. The notion of Thom-Boardman submanifold in the foregoing paragraph plays key rule to analyse this.. The notion of rounding and flattening are defined in this way. We also discuss to define the index if these points are isolated. We remark that we can bundle exactly the same way if do not the submanifold has singularities.
    We also obtain an analogy of Lowner's conjecture for rank 1 map $g : R^2to R^3$ (The rounding index is 1 or less for such maps number). Moreover, several differential equation of two variables was caught as generalization of the differential equation of principal line, and define the notion of totally real and investigate the fundamental properties of index, classification of their singularities are discussed. (the third paper in the next page t)
    It is interesting problem to consider the restriction of function to the level of function. A certain map can be naturally defined when the levels of the later function are parallelizable It is shown that its mapping degree are differences of Euler characteristics signposts of a positive point locus and negative point locus. The necessary and sufficient condition for parallelizability was also discussed. This is related with quotanion number structure and the Carey structures. (the fourth paper in next page)
    Additionally, we show the inverse-map theorem concerning the arc analysis map (the second paper in the next page) and the recent progress of the theory of the blow analysis map (the sixth paper in the next page).

  • Arithmetic study of Fourier coefficients of modular forms of half integral weight and Siegel modular forms

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

    Project Year :

    2004
    -
    2005
     

    KOJIMA Hisashi, TAKEUCHI Kisao, SAKAI Fumio, MIZUTANI Tadayoshi, SAKAMOTO Kunio, KISHIMOTO Takashi

     View Summary

    (1) We shall constract the Shimura correspondence S from Hilbert-Maass wave forms f of half integral weight over algebraic number field to Hilbert-Maass wave forms S(f) of integral weight over algebraic number fields. Moreover, we establish an explicit connection between the square of Fourier coefficients of f and the central value of quadratic twisted L-series associated with the image S(f) off.
    (2) W.Kohnen reformulated the Ikeda lifting as a linear mapping and he formulated a Maass space of Siegel modular forms of degree 2n. Moreover, he purposed a conjecture that the image of Ikeda lifting is equal to the Maass space of degree 2n. In a joint work with W.Kohnen, we proved that this conjecture is ture in the case where 4|n and 4|n-1.

  • Research on the theory of viscosity solutions and its applications

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

    Project Year :

    2003
    -
    2005
     

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

     View Summary

    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.

  • Research on geometric evolution equations for hypersurfaoes

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

    Project Year :

    2003
    -
    2004
     

    NAGASAWA Takeyuki, KOIKE Shigeaki, SAKAMOTO Kunio, TAKAGI Izumi, YANAGIDA Eiji, TACHIKAWA Atsushi

     View Summary

    In this research, we consider gradient flows associated to functionals defined on some family of hypersurfaces. Gradient flow, which is the deformation of an object to the steepest direction of the gradient of functional, is one of methods for finding critical points of functionals. Therefore it is important for the research to analyze properties of functionals.
    The Helfrich variational problem is the minimizing problem of Willmore functional among the closed suefaces with the prescribed area and enclosed volume. This is one of models for shape transformation theory of human red blood cell. The associated gradient flow is called the Helfrich flow. It is not difficult to see spheres are stationary solutions. Nagasawa and Kohsaka studied this geometric flow, and obtained the flowing facts, (1)The time local existence theorem and the uniqueness theorem for arbitrary initial surfaces. (2)The global existence theorem for initial surfaces that are close to spheres. (3)The existence of the center manifold near spheres and estimates of its dimension. These results have been submitted for an academic journal. Nagasawa and Takagi studied stationary solutions bifurcating from spheres. They had already obtained results of the existence and stability for axially symmetric bifurcating solutions before this research project. To study the existence of not necessarily axially symmetric solutions, the reduced bifurcation equation was derived. Furthermore we deduced the normal form from the reduced bifurcation equation, and determined all of solutions for modes 2 and 4. Perturbing them it might be possible to construct solutions of the bifurcation equation.
    Sakamoto considered the functional defined by the squared integral of normal curvature associated immersions of manifold. He derived the first variation formula and investigated the structure of critical points. The Willmore functional is a special case of his study. Yanagida considered the geometric flow associated with three-face free boundary problem with triple junction, and he got a criterion for stability of steady solutions. Tachikawa researched the regularity of weak solutions to equations from geometric variational problem. In particular he obtained a result on the regularity of harmonic maps into Finsler manifold. Koike and Arisawa considered various equations containing geometric evolution equations by using the theory of viscosity solutions. Ohta and Shimokawa researched on the blowup problem of solutions.

  • BELLMAN EQUATIONS OF RISK-SENRSITIVE STOCHASTIC AND THEIR APPLICATIONS

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

    Project Year :

    2001
    -
    2003
     

    NAGAI Hideo, KIOKE Shigeaki, SEKINE Jun, AIDA Shigeki, FUNAKI Tadahisa, ISHII Hitoshi

     View Summary

    1. We considered risk-sensitive portfolio optimization problems on infinite time horizon for linear Gaussian models and general factormodels. Proving existence of solutions of ergodic type Bellman equations we got the results constructing explicitly the optimal strategies from the solutions. As for linear Gaussian models we got the same results in the case of partial information as well by only using the informations of security prices.
    2. In the case of partial information, using the information of only security prices, we obtained maximum principle as necessary conditions for optimality for the problems on a finite time horizon
    3. In the above case we showed that optimal strategies could be expressed explicitly by using the solution of Bellman equation with degenerate coefficients for conditionally Gaussian models
    4. We showed semi-classical behavior of the minimum eigenvalues of Schrodinger operators on Wiener space can be captured in a similar way to the case of finite dimensions. By using similar idea we proved rough lower estimates holds for the minimum eigenvalues of the operators on path spaces (not pinned) on Riemannian manifolds. We also proved, by considering semi-classical limits on the pinned pathe space on Lie groups, that it implies that harmonic forms vanishes
    5. We studied estimates of log derivatives of the heat kernels on Riemannian manifolds in which curvatures rapidly decrease enough and proved log Sobolev inequalities on path spaces. We also studied relationships between Brownian rough path and weak type poincare inequalities.
    6. We studied optimization problems concerning exponential hedging in mathematical finance. In particular we calculated asymptotic expansion of the backward stochastic differential equations with respect to small parameter and obtained asymptotics of the optimal controls
    7. We constructed optimal portfolio by getting higher order differentiability of the solutions of nonlinear partial differential equations arising from mathematical finance
    8. We got interested in solving optimization problem by the methods of convex duality in mathematical finance and extended known. results in applying the methods to the case of partial information, or super hedging under constraints with respect to delta
    9. We got the results on exsistence and uniqueness of viscosity solutions by deriving Euler equations as singular limits of minimum elements of minimization problems of functionals topologically equivalent. We got the Holder estimates of Lp viscosity solutions of fully nonlinear elliptic partial differential equation with super-linear growth with respect to first order derivatives.
    10. We discussed hydrodynamic limits of critical surface models on walls and derived variational inequalities of evolution type. We also derived Alt-Caffarelli variational problems by proving large deviation principles for equilibrium systems of the critical surfaces with pinning.

  • Research on viscosity solutions of differential equations and their applications

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

    Project Year :

    2000
    -
    2002
     

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

     View Summary

    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.

  • Study on Optimal Controls and Differential Games via the Viscosity Solution Theory

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

    Project Year :

    2000
    -
    2002
     

    KOIKE Shigeaki, NII Shunsaku, SAKURAI Tsutomu, TSUJIOKA Kunio, ISHII Hitoshi

     View Summary

    (1) Showing the equivalence of boundary conditions between Dirichlet and state-constraint types, we characterize the value function of minimum arrival time problems, and give a representation formula of it. We also prove the equivalence between the property of semicontinuous viscosity solution and the dynamic programming principle.
    (2) We set up a typical differential game "pursuit-evasion" problem to characterize the value function. We also obtain the convergence of semi-discretized approximate value functions.
    (3) We construct ε-optimal controls for state constraint problems directly from the Hamilton-Jacobi equations without using semi-discrete approximations.
    (4) We study fully nonlinear second order uniformly elliptic PDEs with superlinear growth for first derivatives, and with possibly discontinuous coefficients and inhomogenious terms. When the growth order is less than quadratic, by the Aleksandrov-Bakelman-Pucci (ABP for short) maximum principle and Caffarelli's argument, we obtain the Harnack inequality to show the Holder continuity. In the quadaratic case, we give a counter-example for the ABP maximum principle while we present a sufficient condition so that the ABP maximum principle holds, and obtain the existence of L^p-viscosity solutions for Dirichlet problems.
    (5) We characterize viscosity solutions for fully discontinuous limit PDEs of variational problems with various energies having equivalent norms.
    (6) We obtain locally W^<2,∞> estimates on solutions of obstacle problems arising in mathematical finance to construct optimal policy via one-dimensional Ito formula.

  • Optimal Control Theory of Enler-BernouIIi Equation

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

    Project Year :

    2000
    -
    2002
     

    TSUJIOKA Kunio, KOIKE Shigeaki, NAGASE Masayoshi, YANO Tamaki, SAKURAI Tsutomu

     View Summary

    (1) In the first year of the term of the project, the head investigator researched controllability of a system coupled by two Euler- Bernoulli beams with control at the coupled point. First he studied asymptotic behavior of eigenvalues of a fourth order differential operator related to the Euler-BernouIIi equation. Then controllability problem is reduced a moment problem in a Hilbert space. The method to solve controllability by reducing to a moment problem is called moment problem method. Our ultimate object of the project is to study controllability of a system coupled several Euler-Bernoulli beams. However corresponding moment problem method is too complicated and does not work to our problem. So this problem is still open. In the second and third year of the term of the project, he turned to controllability of evolution equations with singular boundary condition. A boundary condition in an evolution equation is called singular if its order in spatial derivative is as same as that of the equation. We investigated controllability of wave equation with singular boundary condition using moment problem method. A. similar result will be obtained for Euler-Bernoulli equation with singular boundary condition.
    (2) Investigator Koike contributed greatly to optimal control theory related to viscosity solution. Various problems are proposed and solved by him.

  • On the stracture of solutions of partial differential eauqtions

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

    Project Year :

    1999
    -
    2000
     

    SAKURAI Tsutomu, FUKUI Toshizumi, KOIKE Shigeaki, YANO Tamaki

     View Summary

    The head investigator considered the asymptotic behavior of the surface waves of viscous incomplessible fluids. The suaface waves are governed by the Euler equation with the free boundary condition on the surface. We studied the dispersive behavior of the solutions to the linearlized equation. By using the Airy integral, we gave the asymptotic formula for the solutions at degenerate points. We also studied the coherent states of the light in the quantum mecanics. The coherent states are not mutually orthogonal but make a complete system of the quantum states of the light. Making use of the coherent states expantion, we obtained another derivation for the fundamental solutions to the Harmonic oscilators.
    The head investigator writing a book on the partial differential equations. Some of the above results will be included in this book.

  • Theory and applications of viscosity solutions

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

    Project Year :

    1997
    -
    1999
     

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

     View Summary

    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.

  • Applications to the optimal control and differential game via the viscosity solution theory

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

    Project Year :

    1997
    -
    1999
     

    KOIKE Shigeaki, NII Shunsaku, FUKUI Toshizumi, SAKURAI Tsutomu, TSUJIOKA Kunio, ISHII Hitoshi

     View Summary

    (1) We obtain the uniqueness and representation formula for lower semicontinuous (lsc for short) viscosity solutions of Hamilton-Jacobi (HJ for short) equations with degenerate coefficients.
    This is a "lsc" version of a result of Ishii and Ramaswamy.
    (2) We show the uniqueness of viscosity solutions of pursuit-evasion games of the state constraint (SC for short) problem and the convergence of time-discrete approximations to the value function. These results generalize those of Alziary.
    (3) We show the uniqueness of lsc viscosity solutions for the minimum time problem. To show the equivalence between the Dirichlet type boundary condition and the SC type one, we prove in general that a function is a lsc viscosity solution of the associated PDE if and only if it satisfies the Dynamic Programming Principle. This extends the corresponding result by P.-L. Lions for continuous viscosity solutions.
    (4) We construct ε-optimal feedback controls for the SC problem. To construct them, we use the information from the associated HJ equation. Furthermore, we need the inf-convolution approximations, the monotone formula for super-differentials and the convergence of value functions for SC problems of smaller domains.
    (5) We obtain (a) the comparison principle and (b) interior Holder continuity of viscosity solutions of fully nonlinear, second-order, uniformly elliptic PDEs which involve superlinear growth terms for the gradient. The result (a) extends that of Caffarelli, Crandall, Kocan and Swiech while the result (b) generalizes that of Caffarelli.
    To show (b), we need a generalized Alexandroff-Bakelman-Pucci maximum principle, a precise construction of classical supersolutions of the associated extremal PDEs, and a modified cube decomposition lemma.

  • The study of the pole of the zeta function.

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

    Project Year :

    1996
    -
    1997
     

    YANO Tamaki, FUKUI Toshizumi, OKUMURA Masafumi, SATOH Takakazu, NAGASE Masayoshi, SAKAI Fumio

     View Summary

    The totalty of irredeuchible regular Prehomogenous Vector spaces are classified into 29 types.
    For almost all types, their p-adic zeta functions are determined. But zeta functions for the type SL (5) xGL (4) has not been determined. T.Yano studied in detail for this space, and published a result. For its p-adic zeta functions, we determined several factors in the denominator.
    Zeta functions are defined for algebraic curves. F.Sakai studied them.
    M.Nagase studeied geometric aspects, and T.Fukui made an effort from the singularity point of view.

  • 偏微分方程式の局所および大域解析

    日本学術振興会  科学研究費助成事業 基盤研究(C)

    Project Year :

    1996
     
     
     

    櫻井 力, 柳井 久江, 福井 敏純, 小池 茂昭, 奥村 正文, 辻岡 邦夫

     View Summary

    (1)研究代表者櫻井はSymplecticな特性集合を持つ偏微分方程式の超局所的性質について研究を行なった.Symplecticな特性集合を持つ作用素をHeisenberg群にモデル化し,その上の調和解析の理論を構築することにより方程式の解が持つ特異性について詳細に調べた.具体的には,Sub-elliptic条件をみたす不変作用素のParametricを(1/2,1/2)タイプの擬微分作用素として実現し,そののち,Sub-elliptic条件を満たさない作用素に対して摂動論を展開した.この研究は“Analytic hypoellipticity and local solvability for a class of pseudo-differential operators with symplectic characteristics" (Bnach Center Publications,33,1996,315-335)にまとめられている.
    (2)分担者奥村は主に複素射影空間のある種のCR-部分多様体の上でラプラシアンの固有値とスカラー曲率との関係について研究し,Y.W.Choe-M.Okumura:Scalar curvature of a certain CR-submamifold of complex projective space (to appear in Arkev der Math.)に発表した.
    (3)分担者小池は決定論的な状態拘束条件下での効用関数の,対応する一階Bellman方程式の境界条件を特徴付け,その下で効用関数が一意的な粘性解であることを示した.(A new formulation of state constraint problems for first-order PDEs,SAIM J.,34(1996))
    また,制御集合が状態に依存する場合のBellman方程式の解の一意性のための十分条件を示し,その条件が満たされない場合の例をあげた.

  • On Birational Geometry of Algebraic Varieties

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

    Project Year :

    1995
    -
    1996
     

    SAKAI Fumio, EGASHIRA Shinji, KOIKE Shigeaki, MIZUTANI Tadayoshi, TAKEUCHI Kisao, OKUMURA Masafumi

     View Summary

    Sakai generalizel Zarishi's theorem on cyolic coverings of the projective plane to the cyclic coveings of algebraic surfaces under the hyposhesis that the degree of the covering's a power of a prime number and the Branch euwe could be reducible. He also improve an estimate of the total Milnor member of plane cuwes with simple singulinties.
    Okumura obtained a sufficient condition which guaranties that a CR-submeniforld of a complex projective opere is a product of odd dimensional sphers.
    Takeuchi classified all moduler subgroups G of the modular group SL_2 (TS) which has signatine (o ; e_1, e_2, e_3). Moreour, he gave the matrix forms.
    Koike considored the solution of a Bell monequotion. He obtoineda sufficient condition for the uniqueness of the solution.
    Egashira studied C^2-class codimeusion one foliatiation on a compact monifold.

  • Joint Study on Viscosity Solutions and Their Applications

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for international Scientific Research

    Project Year :

    1995
     
     
     

    ISHII Hitoshi, LIONS P.L., SONER H.M., SOUGANIDIS P.E., GRANDALL M.G., EVANS L.C., GIGA Y., KOIKE S.

     View Summary

    We studied basic properties and their applications of viscosity solutions of first and second order partial differential equations. In particular, we obtained several results in the fundamental theory and applications of level set approach to evolutions of surfaces and deterministic optimal control of differential equations.
    1. We analyzed some of mathematical models of etching in manufacturing of computer chips, showed that the level set approach gives the right way to study these models, and thus justified the former numerical computations. This research was done jointly by H.Ishii and L.C.Evans.
    2. We showed for semilinear parabolic partial differential equations with periodic functions like the sine function as nonlinear term that under appropriate scalings solutions of these equations coverge to those functions of which every level sets evolve by mean curvature motion. This result was obtained by H.Ishii.
    3. We unified the Bence, Merriman, Osher algorithm and the threshold growth dynamics by Griffeath and others to the threshold dynamics and applied this threshold dynamics to yield approximation schemes for anisotropic mean curvature motions and curvature-independent motions. This research was done jointly by H.Ishii, G.E.Pir** and P.E.Souganidis.
    4. We showed that the usual formulation of ergodc problems for Hamilton-Jacobiequations in finite-dimensional spaces is not sufficient to treat the ergodic problems in infinite-dimensional spaces. This research was done jointly by M.Arisawa, H.Ishii, and P.L.Lions.
    5. We studied surface energy-driven motion of curves when the interface energy is not smooth and extended the theory of viscosity solutions to cover the motion by the nonsmooth energy including crystalline energy. This was done by Y.Giga and M.-H.Giga.
    7. We studied differential games with state constraints and showed that the value function is a unique continuous viscosity solution of the Bellman equation when the boundary condition is appropriately interpreted. This was done jointly by M.Bardi, S.Koike, and P.Soravia.

  • 非線形退化楕円型方程式の粘性解の研究

    日本学術振興会  科学研究費助成事業 奨励研究(A)

    Project Year :

    1994
     
     
     

    小池 茂昭

     View Summary

    制御理論に表れる状態拘束問題を研究した.具体的には,対応する値関数(value function)が満たすべき正しい境界値問題を設定し,値関数が唯一の粘性解であることを示し,これを特徴付けた.
    1.状態拘束問題における値関数を特徴付ける境界値問題は,Soner(1996年)により導入され様々な一般化が研究されたが,解の定義に「連続性」の制約が付き一般理論との間にギャップがあった.これを1階Bellman型偏微分方程式に対して,動的計画原理に基づいた正確な定義を与え,解の表現・一意性・微分可能性等を得た.
    2.微分ゲームに対しても状態拘束問題を提唱し,対応する境界値問題を与えた.その問題の値関数は制限付き戦略(strategy)を用いて与えられた始めてのものと思われる。更に,同問題において解の表現・一意性等を示し,値関数の特徴付けをした.
    3.実際の制御問題では制御が空間の位置によって制御を受けることが重要であり,一般論を展開する必要がある.しかし,その様な問題は,本質的に不連続なHamiltonianを考えることであり,困難が予想される.現在は,制御の制約が比較的緩やかな場合の一般論と急激な制御の変化のある特殊な場合について値関数の特徴付けを研究中である。

  • 特異な境界条件を持った偏微分方程式の最適制御理論

    日本学術振興会  科学研究費助成事業 一般研究(C)

    Project Year :

    1993
     
     
     

    辻岡 邦夫, 小池 茂昭, 長瀬 正義, 桜井 力, 酒井 文雄, 佐藤 祐吉

     View Summary

    本研究は,特異な境界条件を持った偏微分方程式の最適制御理論の数学の各分野からの総合的な共同研究である.
    1 ビーム,プレートを支配する偏微分方程式その他いくつかの偏微分方程式に関する最適制御理論については,解の一意存在定理,最適条件,可制御性などが得られており,これらについて特異な境界条件を付して研究することは可能である.
    2 ロボットアームの運動の制御問題に関し,線形システムの場合に解の一意存在定理,近似可制御性については,京都大学数理解析研究所の講究録に研究代表者辻岡の得た結果が発表されている.
    3 非線形制御理論について
    (1)可制御性は不動点定理および,関数解析的なHUM法を用いた研究が盛んで,LIONS学派のLIONS,ZUAZUA,イタリヤのLADIESKA,TRIGGIANI,東大の山本,熊本大の内藤,神戸大の中桐等の研究がある.最適理論については,凸解析の手法が用いられ,中国のYONG,米国のFATTORINIの研究がある.
    (2)粘性解の理論について,研究分担者小池およびその共同研究者によって,いくつかの秀れた成果が得られており,専門学術雑誌に発表もしくは発表が決定している.
    4 代数学的あるいは幾何学側面からは,研究分担者竹内,酒井,長瀬の結果がありそれぞれ発表予定である.

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Presentations

  • Lp viscosity solution theory -revisited-

    KOIKE Shigeaki  [Invited]

    The 20th Northeastern Symposium on Mathematical Analysis 

    Presentation date: 2019.02

  • Obstacle problems for PDE of non-divergence type

    小池 茂昭  [Invited]

    研究集会「第14回 非線型の諸問題」 

    Presentation date: 2018.09

  • Recent development on Lp viscosity solutions for fully nonlinear parabolic PDE

    小池 茂昭  [Invited]

    九州における偏微分方程式研究集会 

    Presentation date: 2018.01

  • 自由境界問題の近似問題

    小池 茂昭  [Invited]

    室蘭非線形解析研究会 

    Presentation date: 2017.12

  • On the rate of convergence in free boundary problems

     [Invited]

    The 5th Italian-Japanese Workshop on Geometrc Properties for Parabolic and Elliptic PDE's  (JAPAN) 

    Presentation date: 2017.05

  • 粘性解のABP最大値原理とその応用

    小池 茂昭  [Invited]

    日本数学会年会 

    Presentation date: 2017.03

  • 自由境界問題の処罰法による近似解の収束レートについて

    小池 茂昭  [Invited]

    福島応用数学研究集会 

    Presentation date: 2017.03

  • Fully nonlinear uniformly elliptic/parabolic PDE with unbounded ingredients

    Hamilton-Jacobi Equations New trends and applications  (France) 

    Presentation date: 2016.05

  • Entire solutions of fully nonlinear elliptic PDE with super linear gradient terms

     [Invited]

    Workshop on nonlinear partial differential equations and related topics  (JAPAN 金沢) 

    Presentation date: 2016.05

  • ABP最大値原理について

     [Invited]

    日本数学会2016年度年会  (JAPAN) 

    Presentation date: 2016.03

  • Holder continuity for subsolutions of integer-differential equations

    偏微分方程式の漸近問題と粘性解  (JAPAN 京都) 

    Presentation date: 2015.12

  • On the ABP maximum principle for Lp-viscosity solutions of fully nonlinear PDE

    小池 茂昭  [Invited]

    The 4th MSJ-SI Nonlinear Dynamics and PDE 国際研究集会 

    Presentation date: 2011.09

  • On the ABP maximum principle and applications

    小池 茂昭  [Invited]

    研究集会「幾何学的偏微分方程式における保存則と正則性の研究」 

    Presentation date: 2011.06

  • On viscosity solutions of fully nonlinear elliptic PDE with measurable and unbounded ingredients

    小池 茂昭  [Invited]

    Nonlinrear PDE's, Valparaiso 

    Presentation date: 2011.01

  • 完全非線形楕円型偏微分方程式の粘性解について

    小池 茂昭  [Invited]

    研究集会「微分方程式の総合的研究」 

    Presentation date: 2010.12

  • On the weak Harnack inequality for fully nonlinear PDEs with unbounded ingredients

    小池 茂昭  [Invited]

    研究集会「Viscosity methods and nonlinear PDE」 

    Presentation date: 2010.07

  • Weak Harnack inequality for fully nonlinear PDEs with unbounded ingredients

    小池 茂昭  [Invited]

    Positivity: A key to fully nonlineart equations Conference 

    Presentation date: 2010.06

  • Weak Harnack inequality for Lp-viscosity solutions of fully nonlinear PDEs with unbounded ingredients

    小池 茂昭  [Invited]

    The Second Chile-Japan Workshop on Elliptic and Parabolic Equations 

    Presentation date: 2009.12

  • Weak Harnack inequality for Lp-viscosity solutions of fully nonlinear elliptic PDEs with unbounded ingredients

    小池 茂昭  [Invited]

    The Second International Conference of Reaction Diffusion Systems and Viscosity Solution 

    Presentation date: 2009.07

  • Weak Harnack inequality for fully nonlinear PDEs with superlinear growth terms in Du

    小池 茂昭

    研究集会「微分方程式の粘性解とその周辺」 

    Presentation date: 2009.06

  • Recent developments on the ABP maximum principle for fully nonlinear elliptic PDEs

    小池 茂昭  [Invited]

    第4回非線型の諸問題 

    Presentation date: 2008.09

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Specific Research

  • 超線形増大項のある完全非線形方程式の理論と応用

    2020  

     View Summary

    二階完全非線形一様楕円型方程式が1階微分項の増大度が1次以上の場合のABP最大値原理が成立するための十分条件は知られている。そこでは、逐次比較関数法を開発することで得られた。これらを利用して、平均場ゲームに現れる方程式系のうち粘性ハミルトン・ヤコビ方程式が低階項が一次以上の増大度がある場合の解の存在を導いた。

  • 臨界係数を持つ完全非線形方程式の粘性解のABP最大値原理とその応用に関する研究

    2019  

     View Summary

    完全非線形一様楕円型方程式が一階微分項に非有界係数μを持つ場合のABP最大値原理において、Lp粘性解に対しては、qが空間次元nより、大きいLq空間にμが属する時には、2007年の研究代表者の研究によって知られていた。しかし、強解に対してはμがLn空間に属していればが成り立つことが古典的な結果として、AleksandrovやBakelman等によって知られている。本研究では、Lp粘性解においてもμがLn空間に属する時に成立することを示すことを研究目的とした。研究成果としては、μがLnに属し、非斉次項fが次元nより大きいpの場合に示した。証明の鍵は、μがLnに属し、非斉次項がLpに属する時の対応する完全非線形方程式の強解の構成にある。

 

Syllabus

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Teaching Experience

  • Fourier Analysis

    Waseda University  

    2020.09
    -
    Now
     

  • Introduction of Mathematics B

    Waseda University  

    2020.09
    -
    Now
     

  • Topics in Mathematical Physics A

    Waseda University  

    2020.04
    -
    Now
     

  • Functional Analysis

    Waseda University  

    2020.04
    -
    Now
     

  • Ordinary Differential Equations

    Waseda University  

    2020.04
    -
    Now
     

  • Introduction of Mathematics A

    Waseda University  

    2020.04
    -
    Now
     

  • Topics in Partial Differential Equations

    Waseda University  

    2019.04
    -
    Now
     

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Committee Memberships

  • 2016.05
    -
    2016.10

    函数方程式論分科会  解析学賞選考委員

  • 2013.05
    -
    2013.10

    函数方程式論分科会  解析学賞選考委員

  • 2012.05
    -
    2012.10

    函数方程式論分科会  解析学賞選考委員長

Social Activities

  • 仙台数学セミナー

    2014.08
     
     

     View Summary

    講演、演習を通して東北地方の高校生に数学の面白さを紹介する。

  • 出張講義

    2013.12
    -
     

     View Summary

    仙台一高で高校生に講義を行った。