Research Experience
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2019.04-Now
Waseda University Faculty of Science and Engineering
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2012.04-2019.03
東北大学 教授
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2006.04-2012.03
埼玉大学大学院理工学研究科 教授
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1992.04-2002.03
埼玉大学理学部 助教授
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1989.10-1992.03
東京都立大学理学部 助手
Updated on 2025/04/04
Personnel Information
Research Activity
Education Activity
Contribution to Society
Updated on 2025/04/04
Waseda University Faculty of Science and Engineering
東北大学 教授
埼玉大学大学院理工学研究科 教授
埼玉大学理学部 助教授
東京都立大学理学部 助手
早稲田大学理工学部 助手
Waseda University Graduate School, Division of Science and Engineering
Waseda University Faculty of Science and Engineering 物理学科
日本数学会 評議員
函数方程式論分科会 解析学賞選考委員
函数方程式論分科会 解析学賞選考委員
函数方程式論分科会 解析学賞選考委員長
日本数学会
粘性解理論
Fully Nonlinear Partial Differential Equations
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
Rate of Convergence for Approximate Solutions in Obstacle Problems for Nonlinear Operators
Shigeaki Koike, Takahiro Kosugi
Springer Proceedings in Mathematics & Statistics 63 - 93 2024.07
Aleksandrov-Bakelman-Pucci maximum principle for Lp-viscosity solutions of equations with unbounded terms
Shigeaki Koike, Andrzej Swiech
Journal de Mathematiques Pures et Applquees 168 ( 9 ) 192 - 212 2022.12 [Refereed]
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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]
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
ABP maximum principle with upper contact sets for fully nonlinear elliptic PDEs
Shigeaki Koike [Invited]
OIST PDE seminar
Presentation date: 2022.12
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
粘性解理論とその応用
Project Year :
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 :
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (S)
Project Year :
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 :
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 :
Ishii Hitoshi
View Summary
This research builds on previous work to develop new developments in the theory and applications of viscosity solutions. The research has solved various problems in the theory and applications of viscosity solution theory, focusing on the basic theory of the comparison principle of solutions for differential and integral equations, unique existence and continuity of solutions, and applications to various asymptotic problems, optimal control, differential games, and geometric problems such as curvature flow, and has advanced the research on viscosity solution theory from both theoretical and applied perspectives. The results obtained have been used in natural science, engineering, society, and society. The results obtained are important as a fundamental theory for natural science, engineering, and social science.
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 :
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 :
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 :
ISHII Hitoshi, OTANI Mitsuharu, NAGAI Hideo, GIGA Yoshikazu, KOIKE Shigeaki, MIKAMI Toshio, MITAKE Hiroyoshi, YAMADA Naoki, ISHII Katsuyuki, ICHIHARA Naoyuki, FUJITA Yasuhiro
View Summary
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 :
Koike Shigeaki
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We obtained comparison principle for unbounded viscosity solutions of degenerate elliptic PDE with superlinear gradient terms. We presented a representation formula for viscosity solutions of integro-differential equations of Isaacs type. We established the local maximum principle fro Lp-viscosity solutions of fully nonlinear uniformly elliptic PDE with unbounded coefficients to the first derivatives. We discussed regularity and large time behavior of viscosity solutions of integro-differential equations with coercive first derivative terms. We obtained existence and uniqueness of entire solutions of fully nonlinear elliptic equations with superlinear growth in the first derivatives. We showed the Lipschitz continuity of viscosity solutions of mean curvature flow equations with bilateral obstacles.
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 :
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 :
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 :
OZAWA Tohru, KOIKE Shigeaki, TANAKA Kazunaga
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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 :
NAGASAWA Takeyuki, KOIKE Shigeaki
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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 :
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
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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 :
NAGAI Hideo, KOHATSU-HIGA Arturo, SEKINE Jun, MIYAHARA Yoshio, KOIKE Shigeaki, ISHII Hitoshi, HATA Hiroaki, AIDA Shigeki, NAGAHATA Yukio, TAMURA Takashi, KAISE Hidehiro
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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 :
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 :
KOIKE Shigeaki, ISHII Hitoshi, MIKAMI Toshio, ISHII Katsuyuki, NAGAI Hideo, MORIMOTO Hiroaki
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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 :
MIKAMI Toshio, TAKEDA Masayoshi, KOIKE Shigeaki, KAISE Hidehiro
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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.
Differential Geometry and Partial Differential Equations as an application of Singularity theory
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research
Project Year :
IZUMIYA Shyuichi, ISHIKAWA Goo, ONO Kaoru, YAMAGUCHI Keizo, OHMOTO Toru, TONEGAWA Yoshihiro, UMEHARA Masaaki, KOIKE Shigeaki, UMEHAPR Masaaki, KOIKE Shigeaki
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In this research project, we applied Singularity Theory to some areas in Mathematics such as Differential Geometry, Symplectic Geometry, non-linear Partial Differential Equations etc, so that we have obtained several results. Moreover, we have obtained related results on some boundary areas such as Astrophysics etc. Especially, we applied Singularity Theory to Differential Geometry of submanifolds in several kinds of space forms. Then we constructed new geometries (Horospherical Geometry, Slant Geometry) and induced some new invariants. We also clarified the geometric meanings of these invariants. As a result, we have a geometric characterization of the singularities andthe shape of event horizons
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 :
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 :
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 :
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 :
NAGASAWA Takeyuki, KOIKE Shigeaki, OHTA Masahito, SAKAMOTO Kunio, KOHSAKA Yoshihito, TACHIKAWA Atsushi
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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 :
NAGAI HIideo, KOHATSU-HIGA Artuto, KOTANI Shinichi, MATSUMOTO Hiroyuki, ISHII Hitoshi, KOIKE Shigeaki
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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 :
KOIKE Shigeaki, MORIMOTO Hiroaki, ISHII Hitoshi, NAGAI Hideo, MIKAMI Toshio, ISHII Katsuynki
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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 :
FUKUI Toshizumi, KOIKE Satoshi, IZUMI Shuzo, SAKAI Fumio, MIZUTANI Tadayoshi, KOIKE Shigeaki
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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 :
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 :
ISHII Hitoshi, GIGA Yoshikazu, KOIKE Shigeaki, NAGAI Hideo, ISHII Katusyuki, MIKAMI Toshio
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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 :
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 :
NAGAI Hideo, KIOKE Shigeaki, SEKINE Jun, AIDA Shigeki, FUNAKI Tadahisa, ISHII Hitoshi
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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 :
ISHII Hitoshi, SAKAI Makoto, MOCHIZUKI Kiyoshi, GIGA Yoshikazu, ISHII Katsuyuki, KOIKE Shigeaki
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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 :
KOIKE Shigeaki, NII Shunsaku, SAKURAI Tsutomu, TSUJIOKA Kunio, ISHII Hitoshi
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(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 :
TSUJIOKA Kunio, KOIKE Shigeaki, NAGASE Masayoshi, YANO Tamaki, SAKURAI Tsutomu
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(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 :
SAKURAI Tsutomu, FUKUI Toshizumi, KOIKE Shigeaki, YANO Tamaki
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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 :
ISHII Hitoshi, TOMITA Yoshihito, GIGA Yoshikazu, MOCHIZUKI Kiyoshi, ISHII Katsuyuki, KOIKE Shigeaki
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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 :
KOIKE Shigeaki, NII Shunsaku, FUKUI Toshizumi, SAKURAI Tsutomu, TSUJIOKA Kunio, ISHII Hitoshi
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(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 :
YANO Tamaki, FUKUI Toshizumi, OKUMURA Masafumi, SATOH Takakazu, NAGASE Masayoshi, SAKAI Fumio
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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 :
櫻井 力, 柳井 久江, 福井 敏純, 小池 茂昭, 奥村 正文, 辻岡 邦夫
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(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 :
SAKAI Fumio, EGASHIRA Shinji, KOIKE Shigeaki, MIZUTANI Tadayoshi, TAKEUCHI Kisao, OKUMURA Masafumi
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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 :
ISHII Hitoshi, LIONS P.L., SONER H.M., SOUGANIDIS P.E., GRANDALL M.G., EVANS L.C., GIGA Y., KOIKE S.
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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 :
小池 茂昭
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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 :
辻岡 邦夫, 小池 茂昭, 長瀬 正義, 桜井 力, 酒井 文雄, 佐藤 祐吉
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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.
二階非線形退化楕円型方程式及び方程式系の粘性解の理論と応用
日本学術振興会 科学研究費助成事業
Project Year :
小池 茂昭
多様体の大域的構造の研究
日本学術振興会 科学研究費助成事業
Project Year :
長瀬 正義, 酒井 文雄, 奥村 正文, 木村 真琴, 小池 茂昭, 金銅 誠之
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長瀬“Gauss-Bonnet operator on singular algebraic curves"において特異点を持つ曲線上のGauss-Bonnet作用素D=d+δの(L^2-)指数の考察がなされた。この研究はAtiyah-Singerの指数定理の一つの拡張であった。この研究ののち,その指数定理の出発点となったスピン構造論の拡張に取り組んだ。投稿中のため11.研究発表には記入しなかったが,現在そのスピン構造の四元数ケーラー多様体に適した変形物の導入に成功している(Nagase,Spin^q structures(preprint,1992),q=quaternionic)。AtiyahとSingerはSpin構造の導入と同時に複素多様体に適したそれの変形物,Spin^c構造(c=complex),をも導入している。今回,研究代表者は,最近の四元数ケーラー多様体への関心の高まりに刺激されてそれに適した変形に取り組んだ。Spin^q群の表現,概四元数構造の可く標準的Spin^q構造,Spin^qベクトル束,Dirac作用素,その指数,等,について上述preprintにおいて論じている。いうなればSpin,Spin^Cにおいて論じられた代表的テーマを,Spin^qについても考察してみた。
このSpin^q構造の研究はこれで終わりか?そのことについて,今後の展開をも含めて少々述べて起きたい。一見(Spin^c構造がある意味でそうであるように)Spin^q構造はSpin,Spin^cの類似物でしかないように見える。著者自身の感触も初期の段階ではその程度であった。ところがその後の考察によるとSpin^cとSpin^qはtwistor理論を介して深いつながりを持つことがわかってきている(Spin,Spin^cの間にはない関係)。この点については現在研究が進行中の段階でもあり詳しくは述べられないが,標語だけでもSpin^c,Spin^q,twistor,Diracといったつながりは何かを予感させるものがある。
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)
Koike Shigeaki
RIMS Kokyuroku 1695 139 - 147 2010.07
Perron-Ishii method for viscosity solutions(Potential Theory and its Related Fields)
Koike Shigeaki
RIMS Kokyuroku 1553 44 - 58 2007.05
Koike Shigeaki
RIMS Kokyuroku 1545 1 - 12 2007.04
Koike Shigeaki
RIMS Kokyuroku 1428 1 - 8 2005.04
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)
Koike Shigeaki
RIMS Kokyuroku 1287 1 - 11 2002.09
Ishibashi Toshihiro, Koike Shigeaki
RIMS Kokyuroku 1197 84 - 94 2001.04
Koike Shigeaki
RIMS Kokyuroku 1135 110 - 119 2000.04
Koike Shigeaki
RIMS Kokyuroku 1111 107 - 116 1999.08
Semicontinuous solutions of Hamilton-Jacobi equations with degeneracy
KOIKE Shigeaki
RIMS Kokyuroku 973 120 - 130 1996.11
Koike Shigeaki
RIMS Kokyuroku 911 20 - 28 1995.05
小池茂昭, 山田直記
数学 47 ( 2 ) 20 - 28 1995
Article, review, commentary, editorial, etc. (other)
Koike Shigeaki
RIMS Kokyuroku 785 62 - 75 1992.05
Koike Shigeaki
RIMS Kokyuroku 679 163 - 173 1989.02
粘性解の基礎理論の新展開
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
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
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上記研究集会の開催
粘性解理論とその先端的応用
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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
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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
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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
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粘性解理論による数理ファイナンスの基礎理論
Ordinary Differential Equation [S Grade]
School of Advanced Science and Engineering
2025 spring semester
Rectation in Applied Mathematics [S Grade]
School of Advanced Science and Engineering
2025 full year
Introduction to Mathematics A [S Grade]
School of Advanced Science and Engineering
2025 spring semester
Rectation in Applied Mathematics [S Grade]
School of Advanced Science and Engineering
2025 full year
Ordinary Differential Equation [S Grade]
School of Advanced Science and Engineering
2025 spring semester
Introduction to Mathematics A [S Grade]
School of Advanced Science and Engineering
2025 spring semester
Master's Thesis (Department of Pure and Applied Mathematics)
Graduate School of Fundamental Science and Engineering
2025 full year
Master's Thesis (Department of Pure and Applied Mathematics)
Graduate School of Fundamental Science and Engineering
2025 full year
Seminar on Applied Analysis and Nonlinear Partial Differential Equations D
Graduate School of Fundamental Science and Engineering
2025 fall semester
Seminar on Applied Analysis and Nonlinear Partial Differential Equations C
Graduate School of Fundamental Science and Engineering
2025 spring semester
Seminar on Applied Analysis and Nonlinear Partial Differential Equations B
Graduate School of Fundamental Science and Engineering
2025 fall semester
Seminar on Applied Analysis and Nonlinear Partial Differential Equations A
Graduate School of Fundamental Science and Engineering
2025 spring semester
Research on Applied Analysis and Nonlinear Partial Differential Equations
Graduate School of Fundamental Science and Engineering
2025 full year
Seminar on Applied Analysis and Nonlinear Partial Differential Equations D
Graduate School of Fundamental Science and Engineering
2025 fall semester
Seminar on Applied Analysis and Nonlinear Partial Differential Equations C
Graduate School of Fundamental Science and Engineering
2025 spring semester
Seminar on Applied Analysis and Nonlinear Partial Differential Equations B
Graduate School of Fundamental Science and Engineering
2025 fall semester
Seminar on Applied Analysis and Nonlinear Partial Differential Equations A
Graduate School of Fundamental Science and Engineering
2025 spring semester
Research on Applied Analysis and Nonlinear Partial Differential Equations
Graduate School of Fundamental Science and Engineering
2025 full year
Research on Applied Analysis and Nonlinear Partial Differential Equations
Graduate School of Fundamental Science and Engineering
2025 full year
Seminar on Mathematical Physics B
Graduate School of Advanced Science and Engineering
2025 fall semester
Seminar on Mathematical Physics A
Graduate School of Advanced Science and Engineering
2025 spring semester
Master's Thesis (Department of Pure and Applied Physics)
Graduate School of Advanced Science and Engineering
2025 full year
Master's Thesis (Department of Pure and Applied Physics)
Graduate School of Advanced Science and Engineering
2025 full year
Seminar on Mathematical Physics B
Graduate School of Advanced Science and Engineering
2025 fall semester
Seminar on Mathematical Physics A
Graduate School of Advanced Science and Engineering
2025 spring semester
Advanced Theory of Partial Differential Equations
Graduate School of Advanced Science and Engineering
2025 fall semester
Advanced Theory of Partial Differential Equations
Graduate School of Advanced Science and Engineering
2025 fall semester
Study Abroad in Physics and Applied Physics D
Graduate School of Advanced Science and Engineering
2025 full year
Study Abroad in Physics and Applied Physics C
Graduate School of Advanced Science and Engineering
2025 full year
Study Abroad in Physics and Applied Physics A
Graduate School of Advanced Science and Engineering
2025 full year
Study Abroad in Physics and Applied Physics B
Graduate School of Advanced Science and Engineering
2025 full year
Fourier Analysis
Waseda University
Introduction of Mathematics B
Waseda University
Topics in Mathematical Physics A
Waseda University
Functional Analysis
Waseda University
Ordinary Differential Equations
Waseda University
Introduction of Mathematics A
Waseda University
Topics in Partial Differential Equations
Waseda University
仙台数学セミナー
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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
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仙台一高で高校生に講義を行った。
Waseda Research Institute for Science and Engineering Concurrent Researcher
2020
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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
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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。
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