Updated on 2022/05/21

写真a

 
KOIKE, Shigeaki
 
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

  • 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

  • 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.

    DOI

  • 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.

    DOI

  • 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.

    DOI

  • 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.

    DOI

  • 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

  • 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

  • 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.

    DOI

  • 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

  • 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.

    DOI

  • 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.

    DOI

  • 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.

    DOI

  • 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.

    DOI

  • 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.

    DOI

  • 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.

    DOI

  • 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.

    DOI

  • 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.

    DOI

  • 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.

    DOI

  • 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

  • On the ABP maximum principle and applications

    KOIKE Shigeaki

    Advanced Studies in Pure Mathematics   64   113 - 124  2015.04  [Refereed]  [Invited]

    Article, review, commentary, editorial, etc. (international conference proceedings)  

  • 粘性解が古典解になる時 -Caffarelliの研究の紹介-

    小池茂昭

    数学   62 ( 3 ) 315 - 328  2010.10

    Article, review, commentary, editorial, etc. (other)  

  • 粘性解による値関数の特徴づけ

    儀我美一, 小池茂昭

      49 ( 1 ) 2 - 7  2005

    Article, review, commentary, editorial, etc. (other)  

    DOI

  • 非線形偏微分方程式の粘性解理論(石井仁司氏の業績)

    小池茂昭, 山田直記

    数学   47   20 - 28  1995

    Article, review, commentary, editorial, etc. (other)  

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
     
     

     View Summary

    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
     

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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Committee Memberships

  • 2013.05
    -
    2013.10

    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

  • 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。