Updated on 2022/07/02

写真a

 
ISHII, Hitoshi
 
Affiliation
Faculty of Education and Integrated Arts and Sciences
Job title
Professor Emeritus

Education

  •  
    -
    1975

    Waseda University   Graduate School, Division of Science and Engineering  

  •  
    -
    1970

    Waseda University   Faculty of Science and Engineering  

Degree

  • Waseda University   Dr. Science

  • Waseda University   Ms. Science

  • 早稲田大学   理学博士

Research Experience

  • 2018.04
    -
    Now

    Tsuda University, Research Fellow

  • 2018.04
    -
    Now

    Waseda University, Professor Emeritus

  • 2019.05
     
     

    Sapienza University of Rome   Department of Mathematics   Visiting Professor

  • 2018.05
     
     

    Sapienza University of Rome   Department of Mathematics   Visiting Professor

  • 2001.04
    -
    2018.03

    Waseda University, Professor

  • 2011.08
    -
    2014.06

    King Abdulaziz University, Adjunct Professor

  • 2011.01
     
     

    College de France, Visiting Professor

  • 2010.09
    -
    2010.11

    University of Chicago、Visiting Professor

  • 1997
    -
    2001

    Tokyo Metropolitan University, Professor

  • 1996
    -
    1997

    Tokyo Metropolitan University, Professor

  • 1989
    -
    1996

    Chuo University, Professor

  • 1981
    -
    1989

    Chuo University, Associate Professor

  • 1976
    -
    1981

    Chuo University, Lecturer

  • 1975
    -
    1976

    Waseda University, Research Associate

▼display all

Professional Memberships

  •  
     
     

    日本応用数理学会

  •  
     
     

    アメリカ数学会

  •  
     
     

    THE MATHEMATICAL SOCIETY OF JAPAN

  •  
     
     

    American Mathematical Society

 

Research Areas

  • Basic mathematics

  • Mathematical analysis

  • Applied mathematics and statistics

  • Basic analysis

Research Interests

  • degenerate elliptic equations

  • asymptotic problems

  • Hamilton-Jacobi equations

  • fully nonlinear elliptic equations

  • curvature flows

  • theory of viscosity solutions

  • Optimal Control

  • Partial Differential Equations

▼display all

Papers

  • Discrete approximation of the viscous HJ equation

    Andrea Davini, Hitoshi Ishii, Renato Iturriaga, Hector Sanchez Morgado

    Stochastics and Partial Differential Equations: Analysis and Computations   9 ( 4 ) 1081 - 1104  2021.12  [Refereed]

    DOI

  • Hamilton–Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown

    Hitoshi Ishii, Kaizhi Wang, Lin Wang, Jun Yan

    Communications in Partial Differential Equations     1 - 36  2021.10  [Refereed]

    DOI

  • Existence through convexity for the truncated Laplacians

    I. Birindelli, G. Galise, H. Ishii

    Mathematische Annalen   379 ( 3-4 ) 909 - 950  2021.04  [Refereed]

    DOI

  • The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling

    Hitoshi Ishii

    Mathematics in Engineering   3 ( 4 ) 1 - 21  2021  [Refereed]

    DOI

  • Averaging of Hamilton-Jacobi equations along divergence-free vector fields

    Hitoshi Ishii, Taiga Kumagai

    Discrete & Continuous Dynamical Systems - A   41 ( 4 ) 1519 - 1542  2021  [Refereed]

    DOI

  • Existence and Uniqueness of Viscosity Solutions of an Integro-differential Equation Arising in Option Pricing

    Hitoshi Ishii, Alexandre Roch

    SIAM Journal on Financial Mathematics   12 ( 2 ) 604 - 640  2021.01  [Refereed]

    DOI

  • Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle

    Isabeau Birindelli, Giulio Galise, Hitoshi Ishii

    Transactions of the American Mathematical Society   374 ( 1 ) 539 - 564  2020.10  [Refereed]

     View Summary

    <p>We investigate positivity sets of nonnegative supersolutions of the fully nonlinear elliptic equations <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma u comma upper D u comma upper D squared u right-parenthesis equals 0">
    <mml:semantics>
    <mml:mrow>
    <mml:mi>F</mml:mi>
    <mml:mo stretchy="false">(</mml:mo>
    <mml:mi>x</mml:mi>
    <mml:mo>,</mml:mo>
    <mml:mi>u</mml:mi>
    <mml:mo>,</mml:mo>
    <mml:mi>D</mml:mi>
    <mml:mi>u</mml:mi>
    <mml:mo>,</mml:mo>
    <mml:msup>
    <mml:mi>D</mml:mi>
    <mml:mn>2</mml:mn>
    </mml:msup>
    <mml:mi>u</mml:mi>
    <mml:mo stretchy="false">)</mml:mo>
    <mml:mo>=</mml:mo>
    <mml:mn>0</mml:mn>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">F(x,u,Du,D^2u)=0</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula> in <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega">
    <mml:semantics>
    <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
    <mml:annotation encoding="application/x-tex">\Omega</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula>, where <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega">
    <mml:semantics>
    <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
    <mml:annotation encoding="application/x-tex">\Omega</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula> is an open subset of <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript upper N">
    <mml:semantics>
    <mml:msup>
    <mml:mrow class="MJX-TeXAtom-ORD">
    <mml:mi mathvariant="double-struck">R</mml:mi>
    </mml:mrow>
    <mml:mi>N</mml:mi>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">\mathbb {R}^N</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula>, and the validity of the strong maximum principle for <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma u comma upper D u comma upper D squared u right-parenthesis equals f">
    <mml:semantics>
    <mml:mrow>
    <mml:mi>F</mml:mi>
    <mml:mo stretchy="false">(</mml:mo>
    <mml:mi>x</mml:mi>
    <mml:mo>,</mml:mo>
    <mml:mi>u</mml:mi>
    <mml:mo>,</mml:mo>
    <mml:mi>D</mml:mi>
    <mml:mi>u</mml:mi>
    <mml:mo>,</mml:mo>
    <mml:msup>
    <mml:mi>D</mml:mi>
    <mml:mn>2</mml:mn>
    </mml:msup>
    <mml:mi>u</mml:mi>
    <mml:mo stretchy="false">)</mml:mo>
    <mml:mo>=</mml:mo>
    <mml:mi>f</mml:mi>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">F(x,u,Du,D^2u)=f</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula> in <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega">
    <mml:semantics>
    <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
    <mml:annotation encoding="application/x-tex">\Omega</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula>, with <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of normal upper C left-parenthesis normal upper Omega right-parenthesis">
    <mml:semantics>
    <mml:mrow>
    <mml:mi>f</mml:mi>
    <mml:mo>∈<!-- ∈ --></mml:mo>
    <mml:mrow class="MJX-TeXAtom-ORD">
    <mml:mi mathvariant="normal">C</mml:mi>
    </mml:mrow>
    <mml:mo stretchy="false">(</mml:mo>
    <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
    <mml:mo stretchy="false">)</mml:mo>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">f\in \mathrm {C}(\Omega )</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula> being nonpositive. We obtain geometric characterizations of positivity sets <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet x element-of normal upper Omega colon u left-parenthesis x right-parenthesis greater-than 0 EndSet">
    <mml:semantics>
    <mml:mrow>
    <mml:mo fence="false" stretchy="false">{</mml:mo>
    <mml:mi>x</mml:mi>
    <mml:mo>∈<!-- ∈ --></mml:mo>
    <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
    <mml:mspace width="thinmathspace" />
    <mml:mo>:</mml:mo>
    <mml:mspace width="thinmathspace" />
    <mml:mi>u</mml:mi>
    <mml:mo stretchy="false">(</mml:mo>
    <mml:mi>x</mml:mi>
    <mml:mo stretchy="false">)</mml:mo>
    <mml:mo>&gt;</mml:mo>
    <mml:mn>0</mml:mn>
    <mml:mo fence="false" stretchy="false">}</mml:mo>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">\{x\in \Omega \,:\, u(x)&gt;0\}</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula> of nonnegative supersolutions <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u">
    <mml:semantics>
    <mml:mi>u</mml:mi>
    <mml:annotation encoding="application/x-tex">u</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula> and establish the strong maximum principle under some geometric assumption on the set <inline-formula content-type="math/mathml">
    <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet x element-of normal upper Omega colon f left-parenthesis x right-parenthesis equals 0 EndSet">
    <mml:semantics>
    <mml:mrow>
    <mml:mo fence="false" stretchy="false">{</mml:mo>
    <mml:mi>x</mml:mi>
    <mml:mo>∈<!-- ∈ --></mml:mo>
    <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
    <mml:mspace width="thinmathspace" />
    <mml:mo>:</mml:mo>
    <mml:mspace width="thinmathspace" />
    <mml:mi>f</mml:mi>
    <mml:mo stretchy="false">(</mml:mo>
    <mml:mi>x</mml:mi>
    <mml:mo stretchy="false">)</mml:mo>
    <mml:mo>=</mml:mo>
    <mml:mn>0</mml:mn>
    <mml:mo fence="false" stretchy="false">}</mml:mo>
    </mml:mrow>
    <mml:annotation encoding="application/x-tex">\{x\in \Omega \,:\, f(x)=0\}</mml:annotation>
    </mml:semantics>
    </mml:math>
    </inline-formula>.</p>

    DOI

  • The vanishing discount problem for monotone systems of Hamilton–Jacobi equations: part 2—nonlinear coupling

    Hitoshi Ishii, Liang Jin

    Calculus of Variations and Partial Differential Equations   59 ( 4 ) eng  2020.08  [Refereed]

    DOI

  • Towards a reversed Faber–Krahn inequality for the truncated Laplacian

    Isabeau Birindelli, Giulio Galise, Hitoshi Ishii

    Revista Matemática Iberoamericana   36 ( 3 ) 723 - 740  2019.09  [Refereed]

    DOI

  • Vanishing contact structure problem and convergence of the viscosity solutions

    Chen, Qinbo, Cheng, Wei, Ishii, Hitoshi, Zhao, Kai

    Comm. Partial Differential Equations   44 ( 9 ) 801 - 836  2019  [Refereed]

  • A family of degenerate elliptic operators: Maximum principle and its consequences

    Isabeau Birindelli, Giulio Galise, Hitoshi Ishii

    Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire   35 ( 2 ) 283 - 326  2018.03  [Refereed]

     View Summary

    In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of k eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions.

    DOI

  • On the Langevin equation with variable friction

    Hitoshi Ishii, Panagiotis E. Souganidis, Hung V. Tran

    CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   56 ( 6 )  2017.12  [Refereed]

     View Summary

    We study two asymptotic problems for the Langevin equation with variable friction coefficient. The first is the small mass asymptotic behavior, known as the Smoluchowski-Kramers approximation, of the Langevin equation with strictly positive variable friction. The second result is about the limiting behavior of the solution when the friction vanishes in regions of the domain. Previous works on this subject considered one dimensional settings with the conclusions based on explicit computations.

    DOI

  • The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems

    Hitoshi Ishii, Hiroyoshi Mitake, Hung V. Tran

    JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES   108 ( 3 ) 261 - 305  2017.09  [Refereed]

     View Summary

    In [17] (Part 1 of this series), we have introduced a variational approach to studying the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations in a torus. We develop this approach further here to handle boundary value problems. In particular, we establish new representation formulas for solutions of discount problems, critical values, and use them to prove convergence results for the vanishing discount problems. (C) 2016 Elsevier Masson SAS. All rights reserved.

    DOI

  • The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus

    Hitoshi Ishii, Hiroyoshi Mitake, Hung V. Tran

    JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES   108 ( 2 ) 125 - 149  2017.08  [Refereed]

     View Summary

    We develop a variational approach to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such partial differential equations, which are natural extensions of the Mather measures. Using the viscosity Mather measures, we prove that the whole family of solutions v(lambda) of the discount problem with the factor lambda &gt; 0 converges to a solution of the ergodic problem as lambda -&gt; 0. (C) 2016 Elsevier Masson SAS. All rights reserved.

    DOI

  • ON VISCOSITY SOLUTION OF HJB EQUATIONS WITH STATE CONSTRAINTS AND REFLECTION CONTROL

    Anup Biswas, Hitoshi Ishii, Subhamay Saha, Lin Wang

    SIAM JOURNAL ON CONTROL AND OPTIMIZATION   55 ( 1 ) 365 - 396  2017  [Refereed]

     View Summary

    Motivated by a control problem of a certain queueing network we consider a control problem where the dynamics is constrained in the nonnegative orthant R-+(d) of the d-dimensional Euclidean space and controlled by the reflections at the faces/boundaries. We define a discounted value function associated to this problem and show that the value function is a viscosity solution to a certain HJB equation in R-+(d) with nonlinear Neumann type boundary condition. Under certain conditions, we also characterize this value function as the unique solution to this HJB equation.

    DOI

  • Metastability for Parabolic Equations with Drift: Part II. The Quasilinear Case

    Hitoshi Ishii, Panagiotis E. Souganidis

    INDIANA UNIVERSITY MATHEMATICS JOURNAL   66 ( 1 ) 315 - 360  2017  [Refereed]

     View Summary

    This is the second part of our work on metastability results for parabolic equations with drift. The aim is to present a self-contained study, using partial differential equations methods, of the metastability properties of the solutions to quasi-linear parabolic equations with drift, and to obtain results similar to those in Freidlin and Koralov [6, 8].

    DOI

  • A convergence result for the ergodic problem for Hamilton-Jacobi equations with Neumann-type boundary conditions

    Eman S. Al-Aidarous, Ebraheem O. Alzahrani, Hitoshi Ishii, Arshad M. M. Younas

    PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS   146 ( 2 ) 225 - 242  2016.04  [Refereed]

     View Summary

    We consider the ergodic (or additive eigenvalue) problem for the Neumann- type boundary-value problem for Hamilton-Jacobi equations and the corresponding discounted problems. Denoting by u(lambda) the solution of the discounted problem with discount factor lambda &gt; 0, we establish the convergence of the whole family {u(lambda)} lambda &gt; 0 to a solution of the ergodic problem as.. 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton-Jacobi equations with the Neumann- type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

    DOI

  • Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

    Norihisa Ikoma, Hitoshi Ishii

    BULLETIN OF MATHEMATICAL SCIENCES   5 ( 3 ) 451 - 510  2015.10  [Refereed]  [Invited]

     View Summary

    This is a continuation of Ikoma and Ishii (Ann Inst H Poincar, Anal Non Lin,aire 29:783-812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous of degree one, on finite intervals or balls. In the multi-dimensional case, we consider only radial eigenpairs. Our eigenvalue problem has a general first-order boundary condition which includes, as a special case, the Dirichlet, Neumann and Robin boundary conditions. Given a nonnegative integer n, we prove the existence and uniqueness, modulo multiplication of the eigenfunction by a positive constant, of an eigenpair whose eigenfunction, as a radial function in the multi-dimensional case, has exactly n zeroes. When an eigenfunction has n zeroes, we call the corresponding eigenvalue of nth order. Furthermore, we establish results concerning comparison of two eigenvalues, characterizations of nth order eigenvalues via differential inequalities, the maximum principle for the boundary value problem in connection with the principal eigenvalue, and existence of a solution having n zeroes, as a radial function in the multi-dimensional case, of the boundary value problem with an inhomogeneous term.

    DOI

  • Metastability for Parabolic Equations with Drift: Part I

    Hitoshi Ishii, Panagiotis E. Souganidis

    INDIANA UNIVERSITY MATHEMATICS JOURNAL   64 ( 3 ) 875 - 913  2015  [Refereed]

     View Summary

    We study the exponentially long-time behavior of solutions to linear uniformly parabolic equations that are small perturbations of transport equations with vector fields having a globally stable (attractive) equilibrium in the domain. The result is that the solutions converge to a constant, which is either the initial value at the stable point or the boundary value at the minimum of the associated quasi-potential. Problems of this type were considered by Freidlin and Wentzell and Freidlin and Koralov, using probabilistic arguments related to the theory of large deviations. Our approach, which is self-contained, relies entirely on pde arguments, and is flexible to the extent that allows us to study a class of semilinear equations of similar structure. This note also prepares the ground for the forthcoming Part II of this work where we consider general quasilinear problems.

    DOI

  • Asymptotic analysis for the eikonal equation with the dynamical boundary conditions

    Eman S. Al-Aidarous, Ebraheem O. Alzahrani, Hitoshi Ishii, Arshad M. M. Younas

    MATHEMATISCHE NACHRICHTEN   287 ( 14-15 ) 1563 - 1588  2014.10  [Refereed]

     View Summary

    We study the dynamical boundary value problem for Hamilton-Jacobi equations of the eikonal type with a small parameter. We establish two results concerning the asymptotic behavior of solutions of the Hamilton-Jacobi equations: one concerns with the convergence of solutions as the parameter goes to zero and the other with the large-time asymptotics of solutions of the limit equation. (C) 2014 The Authors. Mathematische Nachrichten published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim.

    DOI

  • A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations

    Guy Barles, Hitoshi Ishii, Hiroyoshi Mitake

    BULLETIN OF MATHEMATICAL SCIENCES   3 ( 3 ) 363 - 388  2013.12  [Refereed]  [Invited]

     View Summary

    We introduce a new PDE approach to establishing the large time asymptotic behavior of solutions of Hamilton-Jacobi equations, which modifies and simplifies the previous ones (Barles et al. in Arch Ration Mech Anal 204(2):515-558, 2012; Barles and Souganidis in SIAM J Math Anal 31(4):925-939, 2000), under a refined "strict convexity" assumption on the Hamiltonians. Not only such "strict convexity" conditions generalize the corresponding requirements on the Hamiltonians in Barles and Souganidis (SLAM J Math Anal 31(4):925-939, 2000), but also one of the most refined our conditions covers the situation studied in Namah and Roquejoffre (Commun Partial Differ Equ 24(5-6):883-893, 1999).

    DOI

  • A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations

    Hitoshi Ishii

    HAMILTON-JACOBI EQUATIONS: APPROXIMATIONS, NUMERICAL ANALYSIS AND APPLICATIONS, CETRARO, ITALY 2011   2074   111 - 249  2013  [Refereed]  [Invited]

     View Summary

    We present an introduction to the theory of viscosity solutions of first-order partial differential equations and a review on the optimal control/dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, with the Neumann boundary condition. This article also includes some of basics of mathematical analysis related to the optimal control/dynamical approach for easy accessibility to the topics.

    DOI

  • Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls

    Norihisa Ikoma, Hitoshi Ishii

    ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   29 ( 5 ) 783 - 812  2012.09  [Refereed]

     View Summary

    We study the eigenvalue problem for positively homogeneous, of degree one, elliptic ODE on finite intervals and PDE on balls. We establish the existence and completeness results for principal and higher eigenpairs, i.e., pairs of an eigenvalue and its corresponding eigenfunction. (c) 2012 Elsevier Masson SAS. All rights reserved.

    DOI

  • UNIQUENESS SETS FOR MINIMIZATION FORMULAS

    Yasuhiro Fujita, Hitoshi Ishii

    DIFFERENTIAL AND INTEGRAL EQUATIONS   25 ( 5-6 ) 579 - 588  2012.05  [Refereed]

     View Summary

    In this paper, we consider minimization formulas which arise typically in optimal control and weak KAM theory for Hamilton-Jacobi equations. Given a minimization formula, we define a uniqueness set for the formula, which replaces the original region of minimization without changing its values. Our goal is to provide a necessary and sufficient condition that a given set be a uniqueness set. We also provide a characterization of the existence of a minimal uniqueness set with respect to set inclusion.

  • On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

    Guy Barles, Hitoshi Ishii, Hiroyoshi Mitake

    ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS   204 ( 2 ) 515 - 558  2012.05  [Refereed]

     View Summary

    In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the "weak KAM approach", which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets.

    DOI

  • A pde approach to small stochastic perturbations of Hamiltonian flows

    Hitoshi Ishii, Panagiotis E. Souganidis

    JOURNAL OF DIFFERENTIAL EQUATIONS   252 ( 2 ) 1748 - 1775  2012.01  [Refereed]

     View Summary

    In this note we present a unified approach, based on pde methods, for the study of averaging principles for (small) stochastic perturbations of Hamiltonian flows in two space dimensions. Such problems were introduced by Freidlin and Wentzell and have been the subject of extensive study in the last few years using probabilistic arguments. When the Hamiltonian flow has critical points, it exhibits complicated behavior near the critical points under a small stochastic perturbation. Asymptotically the slow (averaged) motion takes place on a graph. The issues are to identify both the equations on the sides and the boundary conditions at the vertices of the graph. Our approach is very general and applies also to degenerate anisotropic elliptic operators which could not be considered using the previous methodology. (C) 2011 Elsevier Inc. All rights reserved.

    DOI

  • Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions

    Hitoshi Ishii

    CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   42 ( 1-2 ) 189 - 209  2011.09  [Refereed]

     View Summary

    We study the long-time asymptotic behavior of solutions u of the Hamilton-Jacobi equation u(tau)(x, t) + H(x, Du(x, t)) = 0 in Omega x (0, infinity), where Omega is a bounded open subset of R(n), with Hamiltonian H = H(x, p) being convex and coercive in p, and establish the uniform convergence of u to an asymptotic solution as t -&gt; infinity.

    DOI

  • Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions

    Hitoshi Ishii

    JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES   95 ( 1 ) 99 - 135  2011.01  [Refereed]

     View Summary

    We study convex Hamilton-Jacobi equations H(x, Du) = 0 and u(t) + H(x, Du) = 0 in a bounded domain Omega of R-n with the Neumann type boundary condition D(gamma)u = g in the viewpoint of weak KAM theory, where gamma is a vector field on the boundary partial derivative Omega pointing a direction oblique to partial derivative Omega. We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex Hamilton-Jacobi equations, some comparison and existence results for convex and coercive Hamilton-Jacobi equations with the Neumann type boundary condition as well as existence results for the Skorokhod problem. We define the Aubry set associated with the Neumann type boundary problem and establish some properties of the Aubry set including the existence results for the "calibrated" extremals for the corresponding action functional (or variational problem). (C) 2010 Elsevier Masson SAS. All rights reserved.

    DOI

  • A class of integral equations and approximation of p-Laplace equations

    Hitoshi Ishii, Gou Nakamura

    CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   37 ( 3-4 ) 485 - 522  2010.03  [Refereed]

     View Summary

    Let Omega subset of R(n) be a bounded domain, and let 1 &lt; p &lt; 8 and sigma &lt; p. We study the nonlinear singular integral equation
    M[u](x) = f(0)(x) in Omega
    with the boundary condition u = g(0) on partial derivative Omega, where f(0) is an element of C(&lt;(Omega)over bar&gt;) and g(0) is an element of C(partial derivative Omega) are given functions and M is the singular integral operator given by
    M[u](x) = p. v. integral(B(0, rho(x))) p - sigma/vertical bar z vertical bar(n+sigma) vertical bar u(x + z) - u(x)vertical bar(p) (2)(u(x + z) - u(x)) dz,
    with some choice of rho is an element of C(Omega) having the property, 0 &lt; rho(x) &lt;= dist (x, partial derivative Omega). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on &lt;(Omega)over bar&gt; as sigma -&gt; p, of the solution us of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation nu Delta(p)u = f(0) in Omega with the Dirichlet condition u = g(0) on partial derivative Omega, where the factor nu is a positive constant (see (7.2)).

    DOI

  • Long-time Behavior of Solutions of Hamilton-Jacobi Equations with Convex and Coercive Hamiltonians

    Naoyuki Ichihara, Hitoshi Ishii

    ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS   194 ( 2 ) 383 - 419  2009.11  [Refereed]

     View Summary

    We investigate the long-time behavior of viscosity solutions of Hamilton-Jacobi equations in R(n) with convex and coercive Hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity. We apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples. We take a dynamical approach, based on tools from weak KAM theory such as extremal curves, Aubry sets and representation formulas for solutions, for these investigations.

    DOI

  • Two remarks on periodic solutions of Hamilton-Jacobi equations

    Hitoshi Ishii, Hiroyoshi Mitake

    Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions     97 - 119  2009.01  [Refereed]

     View Summary

    We show firstly the equivalence between existence of a periodic solution of the Hamilton-Jacobi equation 'formula presented' is a bounded domain of 'formula presented', with the Dirichlet boundary condition 'formula presented' and that of a subsolution of the stationary problem 'formula presented' under the assumptions that the function 'formula presented' is periodic in t and H is coercive. Here 'formula presented' denotes the average of f over the period. This proposition is a variant of a recent result for 'formula presented' due to Bostan-Namah, and we give a different and simpler approach to such an equivalence. Secondly, we establish that any periodic solution u(x, t) of the problem, ut + H(x, Du) = 0 in 'formula presented' and 'formula presented', is constant in t on the Aubry set for H. Here H is assumed to be convex, coercive and strictly convex in a sense.

    DOI

  • Asymptotic solutions of Hamilton-Jacobi equations for large time and related topics

    Hitoshi Ishii

    ICIAM 07: 6TH INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS     193 - 217  2009  [Refereed]

     View Summary

    We discuss the recent developments related to the large-time asymptotic behavior of solutions of Hamilton-Jacobi equations.

  • Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space

    Hitoshi Ishii

    ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   25 ( 2 ) 231 - 266  2008

     View Summary

    We study the large time behavior of solutions of the Cauchy problem for the Hamilton-Jacobi equation u(t) + H(x, Du) = 0 in R-n x (0, infinity), where H(x, p) is continuous on R-n x R-n and convex in p. We establish a general convergence result for viscosity solutions u(x, t) of the Cauchy problem as t -&gt; infinity . (C) 2007 Elsevier Masson SAS. All rights reserved.

    DOI

  • Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians

    Naoyuki Ichihara, Hitoshi Ishii

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   33 ( 5 ) 784 - 807  2008

     View Summary

    We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations in n. We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem "converges" to an asymptotic solution for any lower semi-almost periodic initial function.

    DOI

  • The large-time behavior of solutions of Hamilton-Jacobi equations on the real line

    N. Ichihara, H. Ishii

    Methods Appl. Anal.   15 ( 2 ) 223 - 242  2008

  • Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians

    Hitoshi Ishii, Hiroyoshi Mitake

    INDIANA UNIVERSITY MATHEMATICS JOURNAL   56 ( 5 ) 2159 - 2183  2007

     View Summary

    We establish general representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians. In order to treat representation formulas on general domains, we introduce a notion of ideal boundary similar to the Martin boundary [21] in potential theory. We apply such representation formulas to investigate maximal solutions, in certain classes of functions, of Hamilton-Jacobi equations. Part of the results in this paper has been announced in [22].

    DOI

  • Asymptotic solutions of viscous Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator

    Y Fujita, H Ishii, P Loreti

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   31 ( 6 ) 827 - 848  2006.06

     View Summary

    We study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein-Uhlenbeck operator in R-N . The long time behavior in the main results is stated with help of the corresponding to ergodic problem, which complements, in the case of unbounded domains, the recent developments on long time behaviors of solutions of (viscous) Hamilton-Jacobi equations due to Namah (1996), Namah and Roquejoffre (1999), Roquejoffre (1998), Fathi (1998), Barles and Souganidis (2000, 2001). We also establish existence and uniqueness results for solutions of the Cauchy problem and ergodic problem for semilinear parabolic equations with the Ornstein-Uhlenbeck operator.

    DOI

  • Asymptotic solutions of Hamilton-Jacobi equations in Euclidean n space

    Yasuhiro Fujita, Hitoshi Ishii, Paola Loreti

    INDIANA UNIVERSITY MATHEMATICS JOURNAL   55 ( 5 ) 1671 - 1700  2006

     View Summary

    We study the asymptotic behavior of the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation ut + alpha x (.) Du + H(Du) = f(x) in R-n X (0, infinity), where alpha is a positive constant and H is a convex function on Rn, and establish a convergence result for the viscosity solution u(x, t) as t -&gt; infinity.

    DOI

  • Convexified Gauss curvature flow of sets: A stochastic approximation

    Hitoshi Ishii, Toshio Mikami

    SIAM Journal on Mathematical Analysis   36 ( 2 ) 552 - 579  2005

     View Summary

    We construct a discrete stochastic approximation of a convexified Gauss curvature flow of boundaries of bounded open sets in an anisotropic external field. We also show that a weak solution to the PDE which describes the motion of a bounded open set is unique and is a viscosity solution of it. © 2004 Society for Industrial and Applied Mathematics.

    DOI

  • Limits of solutions of p-laplace equations as p goes to infinity and related variational problems

    H Ishii, P Loreti

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   37 ( 2 ) 411 - 437  2005

     View Summary

    We show that the convergence, as p --&gt; infinity, of the solution u(p) of the Dirichlet problem for -Delta p(u)(x) = f(x) in a bounded domain Omega subset of R-n with zero-Dirichlet boundary condition and with continuous f in the following cases: (i) one-dimensional case, radial cases; (ii) the case of no balanced family; and (iii) two cases with vanishing integral. We also give some properties of the maximizers for the functional integral(Omega) f(x)v(x) dx in the space of functions v is an element of C((Omega) over bar) boolean AND W-1,W-infinity (Omega) satisfying v\(theta Omega) = 0 and parallel to Dv parallel to(L infinity(Omega)) &lt;= 1.

    DOI

  • Nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain

    H Ishii, MH Sato

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   57 ( 7-8 ) 1077 - 1098  2004.06

     View Summary

    We establish comparison and existence theorems of viscosity solutions of the initial-boundary value problem for some singular degenerate parabolic partial differential equations with nonlinear oblique derivative boundary conditions. The theorems cover the capillary problem for the mean curvature flow equation and apply to more general Neumann-type boundary problems for parabolic equations in the level set approach to motion of hypersurfaces with velocity depending on the normal direction and curvature. (C) 2004 Elsevier Ltd. All rights reserved.

    DOI

  • Nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain

    H Ishii, MH Sato

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   57 ( 7-8 ) 1077 - 1098  2004.06

     View Summary

    We establish comparison and existence theorems of viscosity solutions of the initial-boundary value problem for some singular degenerate parabolic partial differential equations with nonlinear oblique derivative boundary conditions. The theorems cover the capillary problem for the mean curvature flow equation and apply to more general Neumann-type boundary problems for parabolic equations in the level set approach to motion of hypersurfaces with velocity depending on the normal direction and curvature. (C) 2004 Elsevier Ltd. All rights reserved.

    DOI

  • Motion of a graph by R-curvature

    H Ishii, T Mikami

    ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS   171 ( 1 ) 1 - 23  2004.01

     View Summary

    We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R-d, by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions.

    DOI

  • Convexified gauss curvature flow of sets: A stochastic approximation

    H Ishii, T Mikami

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   36 ( 2 ) 552 - 579  2004

     View Summary

    We construct a discrete stochastic approximation of a convexified Gauss curvature flow of boundaries of bounded open sets in an anisotropic external field. We also show that a weak solution to the PDE which describes the motion of a bounded open set is unique and is a viscosity solution of it.

    DOI

  • A level set approach to the wearing process of a nonconvex stone

    H Ishii, T Mikami

    CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   19 ( 1 ) 53 - 93  2004

     View Summary

    We study the geometric evolution of a nonconvex stone by the wearing process via the partial differential equation methods. We use the so-called level set approach to this geometric evolution of a set. We establish a comparison theorem, an existence theorem, and some stability properties of solutions of the partial differential equation arising in this level set approach, and define the flow of a set by the wearing process via the level set approach.

    DOI

  • Motion of a graph by R-curvature

    H Ishii, T Mikami

    ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS   171 ( 1 ) 1 - 23  2004.01

     View Summary

    We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R-d, by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions.

    DOI

  • A level set approach to the wearing process of a nonconvex stone

    H Ishii, T Mikami

    CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   19 ( 1 ) 53 - 93  2004

     View Summary

    We study the geometric evolution of a nonconvex stone by the wearing process via the partial differential equation methods. We use the so-called level set approach to this geometric evolution of a set. We establish a comparison theorem, an existence theorem, and some stability properties of solutions of the partial differential equation arising in this level set approach, and define the flow of a set by the wearing process via the level set approach.

    DOI

  • Relaxation of Hamilton-Jacobi equations

    H Ishii, P Loreti

    ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS   169 ( 4 ) 265 - 304  2003.09

     View Summary

    We study the relaxation of Hamilton-Jacobi equations. The relaxation in our terminology is the following phenomenon: the pointwise supremum over a certain collection of subsolutions, in the almost everywhere sense, of a Hamilton-Jacobi equation yields a viscosity solution of the ``convexified'' Hamilton-Jacobi equation. This phenomenon has recently been observed in [13] in eikonal equations. We show in this paper that this relaxation is a common phenomenon for a wide range of Hamilton-Jacobi equations.

    DOI

  • Relaxation of Hamilton-Jacobi equations

    H Ishii, P Loreti

    ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS   169 ( 4 ) 265 - 304  2003.09

     View Summary

    We study the relaxation of Hamilton-Jacobi equations. The relaxation in our terminology is the following phenomenon: the pointwise supremum over a certain collection of subsolutions, in the almost everywhere sense, of a Hamilton-Jacobi equation yields a viscosity solution of the ``convexified'' Hamilton-Jacobi equation. This phenomenon has recently been observed in [13] in eikonal equations. We show in this paper that this relaxation is a common phenomenon for a wide range of Hamilton-Jacobi equations.

    DOI

  • Simultaneous Effects of Homogenization and Vanishing Viscosity in Fully Nonlinear Elliptic Equations

    Hitoshi Ishii

    Funkcialaj Ekvacioj   46 ( 1 ) 63 - 88  2003

    DOI

  • Relaxation in an L-infinity-optimization problem

    H Ishii, P Loreti

    PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS   133 ( 3 ) 599 - 615  2003

     View Summary

    Let Omega be an open bounded subset of Rn and f a continuous function on Omega satisfying f(x) &gt; 0 for all x is an element of Omega. We consider the maximization problem for the integral fOmega f(x)u(x) dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on partial derivativeOmega and to the gradient constraint of the form H(Du(x)) less than or equal to 1, and prove that the supremum is 'achieved' by the viscosity solution of H(Du(x)) = 1 in Omega and u = 0 on partial derivativeOmega, where H denotes the convex envelope of H. This result is applied to an asymptotic problem, as p --&gt; infinity, for quasi-minimizers of the integral
    integral(Omega)[1/p H(Du(x))(p) - f(x)u(x)] dx.
    An asymptotic problem as k --&gt; infinity for
    inf integral(Omega)[k dist(Du(x),K) - f(x)u(x)] dx
    is also considered, where the infimum is taken all over u is an element of W-0(1,1)(Omega) and the set K is given by {xi \ H (xi) less than or equal to 1}.

  • Asymptotic analysis for a class of infinite systems of first-order PDE: nonlinear parabolic PDE in the singular limit

    H. Ishii, K. Shimano

    Comm. Partial Differential Equations   28 ( 1-2 ) 409 - 438  2003

  • Simultaneous effects of homogenization and vanishing viscosity in fully nonlinear elliptic equations

    K. Horie, H. Ishii

    Funkcial. Ekvac.   46 ( 1 ) 63 - 88  2003

  • Relaxation in an L-infinity-optimization problem

    H Ishii, P Loreti

    PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS   133 ( 3 ) 599 - 615  2003

     View Summary

    Let Omega be an open bounded subset of Rn and f a continuous function on Omega satisfying f(x) &gt; 0 for all x is an element of Omega. We consider the maximization problem for the integral fOmega f(x)u(x) dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on partial derivativeOmega and to the gradient constraint of the form H(Du(x)) less than or equal to 1, and prove that the supremum is 'achieved' by the viscosity solution of H(Du(x)) = 1 in Omega and u = 0 on partial derivativeOmega, where H denotes the convex envelope of H. This result is applied to an asymptotic problem, as p --&gt; infinity, for quasi-minimizers of the integral
    integral(Omega)[1/p H(Du(x))(p) - f(x)u(x)] dx.
    An asymptotic problem as k --&gt; infinity for
    inf integral(Omega)[k dist(Du(x),K) - f(x)u(x)] dx
    is also considered, where the infimum is taken all over u is an element of W-0(1,1)(Omega) and the set K is given by {xi \ H (xi) less than or equal to 1}.

  • Asymptotic analysis for a class of infinite systems of first-order PDE: Nonlinear parabolic PDE in the singular limit

    H Ishii, K Shimano

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   28 ( 1-2 ) 409 - 438  2003

     View Summary

    We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem.

    DOI

  • A two-dimensional random crystalline algorithm for Gauss curvature flow

    H. Ishii, T. Mikami

    Advances in Applied Probability   34 ( 3 ) 491 - 504  2002.09

     View Summary

    A two-dimensional random crystalline algorithm was proposed for Gauss curvature flow. Gauss curvature flow of smooth closed convex hypersurfaces in Rd+1 was defined. A discrete-time approximation scheme for Gauss curvature flow was briefly described.

    DOI

  • A two-dimensional random crystalline algorithm for Gauss curvature flow

    H Ishii, T Mikami

    ADVANCES IN APPLIED PROBABILITY   34 ( 3 ) 491 - 504  2002.09

     View Summary

    We propose and study a random crystalline algorithm (a discrete approximation) of the Gauss curvature flow of smooth simple closed convex curves in R-2 as a stepping stone to the full understanding of such phenomena as the wearing process of stones on a beach.

    DOI

  • A class of Stochastic optimal control problems with state constraint

    H Ishii, P Loreti

    INDIANA UNIVERSITY MATHEMATICS JOURNAL   51 ( 5 ) 1167 - 1196  2002

     View Summary

    We investigate, via the dynamic programming approach, optimal control problems of infinite horizon with state constraint, where the state X-t is given as a solution of a controlled stochastic differential equation and the state constraint is described either by the condition that X-t is an element of (G) over bar for all t &gt; 0 or by the condition that X-t is an element of G for all t &gt; 0, where G be a given open subset of R-N. Under the assumption that for each z is an element of partial derivativeG there exists a, G A, where A denotes the control set, such that the diffusion matrix sigma (x, a) vanishes for a = a(z) and for x is an element of partial derivativeG in a neighborhood of z and the drift vector b(x, a) directs inside of G at z for a = az and x = z as well as some other mild assumptions, we establish the unique existence of a continuous viscosity solution of the state constraint problem for the associated Hamilton-jacobi-Bellman equation, prove that the value functions V associated with the constraint (G) over bar, V-r of the relaxed problem associated with the constraint (G) over bar, and V-o associated with the constraint G, satisfy in the viscosity sense the state constraint problem, and establish Holder regularity results for the viscosity solution of the state constraint problem.

  • Fully nonlinear oblique derivative problems for singular degenerate parabolic equations

    H. Ishii

    数理解析研究所講究録   1287   164 - 170  2002

  • A class of Stochastic optimal control problems with state constraint

    H Ishii, P Loreti

    INDIANA UNIVERSITY MATHEMATICS JOURNAL   51 ( 5 ) 1167 - 1196  2002

     View Summary

    We investigate, via the dynamic programming approach, optimal control problems of infinite horizon with state constraint, where the state X-t is given as a solution of a controlled stochastic differential equation and the state constraint is described either by the condition that X-t is an element of (G) over bar for all t &gt; 0 or by the condition that X-t is an element of G for all t &gt; 0, where G be a given open subset of R-N. Under the assumption that for each z is an element of partial derivativeG there exists a, G A, where A denotes the control set, such that the diffusion matrix sigma (x, a) vanishes for a = a(z) and for x is an element of partial derivativeG in a neighborhood of z and the drift vector b(x, a) directs inside of G at z for a = az and x = z as well as some other mild assumptions, we establish the unique existence of a continuous viscosity solution of the state constraint problem for the associated Hamilton-jacobi-Bellman equation, prove that the value functions V associated with the constraint (G) over bar, V-r of the relaxed problem associated with the constraint (G) over bar, and V-o associated with the constraint G, satisfy in the viscosity sense the state constraint problem, and establish Holder regularity results for the viscosity solution of the state constraint problem.

  • Fully nonlinear oblique derivative problems for singular degenerate parabolic equations

    H. Ishii

    数理解析研究所講究録   1287   164 - 170  2002

  • A mathematical model of the wearing process of a nonconvex stone

    H Ishii, T Mikami

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   33 ( 4 ) 860 - 876  2001.12

     View Summary

    We formulate the wearing process of a nonconvex stone in terms of partial differential equations (PDEs). We establish a comparison theorem, an existence theorem, and some stability properties of solutions of this PDE.

    DOI

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

    H Ishii, K Ishii

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   33 ( 2 ) 369 - 389  2001.09

     View Summary

    We show that the algorithm considered by Ishii [GAKUTO Internat. Ser. Math. Sci. Appl. 5, Gakkotosho, Tokyo, 1995, pp. 111-127] and Ishii, Pires, and Souganidis [J. Math. Soc. Japan, 50 (1999), pp. 267-308] can be applied to motion by mean curvature with right-angle boundary condition.

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

    H Ishii, K Ishii

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   33 ( 2 ) 369 - 389  2001.09

     View Summary

    We show that the algorithm considered by Ishii [GAKUTO Internat. Ser. Math. Sci. Appl. 5, Gakkotosho, Tokyo, 1995, pp. 111-127] and Ishii, Pires, and Souganidis [J. Math. Soc. Japan, 50 (1999), pp. 267-308] can be applied to motion by mean curvature with right-angle boundary condition.

  • Hamilton-Jacobi equations with partial gradient and application to homogenization

    O Alvarez, H Ishii

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   26 ( 5-6 ) 983 - 1002  2001

     View Summary

    The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of R-n
    H (x, u, D(x ')u) = 0 in Omega, u = g on partial derivative Omega,
    where D(x ')u is the partial gradient of the scalar function u with respect to the first n ' variables (n ' less than or equal to n), has a viscosity solution which is unique a.e. When applied to the periodic homogenization of Hamilton-Jacobi equations in a perforated set, the result yields the a.e. convergence of the solutions of the problem at scale epsilon as epsilon --&gt; 0 to the solution of the homogenized Hamilton-Jacobi equation.

  • On the rate of convergence in homogenization of Hamilton-Jacobi equations

    I. Capuzzo Dolcetta, H. Ishii

    Indiana Univ. Math. J.   50 ( 3 ) 1113 - 1129  2001

  • A generalization of a theorem of Barron and Jensen and a comparison theorem for lower semicontinuous viscosity solutions

    H Ishii

    PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS   131 ( 1 ) 137 - 154  2001

     View Summary

    We extend and clarify some of observations due to Barren and Jensen concerning the relation between subdifferentials and superdifferentials of a function and extend the comparison principle far semicontinuous solutions of Hamilton-Jacobi equations with convex Hamiltonians to that in infinite-dimensional Hilbert spaces.

  • Hamilton-Jacobi equations with partial gradient and application to homogenization

    O Alvarez, H Ishii

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   26 ( 5-6 ) 983 - 1002  2001

     View Summary

    The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of R-n
    H (x, u, D(x ')u) = 0 in Omega, u = g on partial derivative Omega,
    where D(x ')u is the partial gradient of the scalar function u with respect to the first n ' variables (n ' less than or equal to n), has a viscosity solution which is unique a.e. When applied to the periodic homogenization of Hamilton-Jacobi equations in a perforated set, the result yields the a.e. convergence of the solutions of the problem at scale epsilon as epsilon --&gt; 0 to the solution of the homogenized Hamilton-Jacobi equation.

  • On the rate of convergence in homogenization of Hamilton-Jacobi equations

    I. Capuzzo Dolcetta, H. Ishii

    Indiana Univ. Math. J.   50 ( 3 ) 1113 - 1129  2001

  • A mathematical model of the wearing process of a nonconvex stone

    Hitoshi Ishii, Toshio Mikami

    SIAM Journal on Mathematical Analysis   33 ( 4 ) 860 - 876  2001

     View Summary

    We formulate the wearing process of a nonconvex stone in terms of partial differential equations (PDEs). We establish a comparison theorem, an existence theorem, and some stability properties of solutions of this PDE.

    DOI

  • A generalization of a theorem of Barron and Jensen and a comparison theorem for lower semicontinuous viscosity

    H. Ishii

    Proc. Roy. Soc. Edinburgh Sect. A   131 ( 1 ) 137 - 154  2001

  • A characterization of the existence of solutions for Hamilton-Jacobi equations in ergodic control problems with applications

    M Arisawa, H Ishii, PL Lions

    APPLIED MATHEMATICS AND OPTIMIZATION   42 ( 1 ) 35 - 50  2000.07

     View Summary

    We give a characterization of the existence of bounded solutions for Hamilton-Jacobi equations in ergodic control problems with state-constraint. This result is applied to the reexamination of the counterexample given in [5] concerning the existence of solutions for ergodic control problems in infinite-dimensional Hilbert spaces and also establishing results on effective Hamiltonians in periodic homogenization of Hamilton-Jacobi equations.

    DOI

  • On epsilon-optimal controls for state constraint problems

    H Ishii, S Koike

    ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   17 ( 4 ) 473 - 502  2000.07

     View Summary

    We present a method of constructing E-optimal controls in the feedback form for state constraint problems.
    Our method is as follows: We first find feedback laws directly from the associated Hamilton-Jacobi-Bellman equation and an approximation of the value function by the inf-convolution. We then construct piece-wise constant controls so that corresponding cost functionals approximate the value function of state constraint problems. (C) 2000 Editions scientifiques et medicales Elsevier SAS.

  • A PDE approach to stochastic invariance

    H Ishii, P Loreti, ME Tessitore

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   6 ( 3 ) 651 - 664  2000.07

     View Summary

    We study an invariance property for a controlled stochastic differential equation and give a few of its characterizations in connection with the corresponding Hamilton-Jacobi-Bellman equation.

  • A characterization of the existence of solutions for Hamilton-Jacobi equations in ergodic control problems with applications

    M Arisawa, H Ishii, PL Lions

    APPLIED MATHEMATICS AND OPTIMIZATION   42 ( 1 ) 35 - 50  2000.07

     View Summary

    We give a characterization of the existence of bounded solutions for Hamilton-Jacobi equations in ergodic control problems with state-constraint. This result is applied to the reexamination of the counterexample given in [5] concerning the existence of solutions for ergodic control problems in infinite-dimensional Hilbert spaces and also establishing results on effective Hamiltonians in periodic homogenization of Hamilton-Jacobi equations.

    DOI

  • On epsilon-optimal controls for state constraint problems

    H Ishii, S Koike

    ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   17 ( 4 ) 473 - 502  2000.07

     View Summary

    We present a method of constructing E-optimal controls in the feedback form for state constraint problems.
    Our method is as follows: We first find feedback laws directly from the associated Hamilton-Jacobi-Bellman equation and an approximation of the value function by the inf-convolution. We then construct piece-wise constant controls so that corresponding cost functionals approximate the value function of state constraint problems. (C) 2000 Editions scientifiques et medicales Elsevier SAS.

  • A PDE approach to stochastic invariance

    H Ishii, P Loreti, ME Tessitore

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   6 ( 3 ) 651 - 664  2000.07

     View Summary

    We study an invariance property for a controlled stochastic differential equation and give a few of its characterizations in connection with the corresponding Hamilton-Jacobi-Bellman equation.

  • A Dirichlet type problem for nonlinear degenerate elliptic equations arising in

    M. Bardi, P. Goatin, H. Ishii

    Adv. Math. Sci. Appl.   10 ( 1 ) 329 - 352  2000

  • A Dirichlet type problem for nonlinear degenerate elliptic equations arising in

    M. Bardi, P. Goatin, H. Ishii

    Adv. Math. Sci. Appl.   10 ( 1 ) 329 - 352  2000

  • On the rate of convergence in homogenization of Hamilton-Jacobi equations

    I. Capuzzo Dolcetta, H. Ishii

    International Conference on Differential Equations/World Sci. Publishing    2000

  • Gauss curvature flow and its approximation

    H. Ishii

    Free boundary problems : theory and applications, GAKUTO Internat. Ser. Math. Sci. Appl.    2000

  • Threshold dynamics type approximation schemes for propagating fronts

    H Ishii, GE Pires, PE Souganidis

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   51 ( 2 ) 267 - 308  1999.04

     View Summary

    We study the convergence of general threshold dynamics type approximation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. We also present results about the asymptotic shape of fronts propagating by threshold dynamics. Our results generalize and extend models introduced in the theories of cellular automaton and motion by mean curvature.

    DOI

  • Threshold dynamics type approximation schemes for propagating fronts

    H Ishii, GE Pires, PE Souganidis

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   51 ( 2 ) 267 - 308  1999.04

     View Summary

    We study the convergence of general threshold dynamics type approximation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. We also present results about the asymptotic shape of fronts propagating by threshold dynamics. Our results generalize and extend models introduced in the theories of cellular automaton and motion by mean curvature.

    DOI

  • Hopf-Lax formulas for semicontinuous data

    O. Alvarez, E. N. Barron, H. Ishii

    Indiana Univ. Math. J.   48 ( 3 ) 993 - 1035  1999

  • Hopf-Lax formulas for Hamilton-Jacobi equations with semicontinuous initial data

    H. Ishii

    Singularity theory and differential equations   1111   144 - 156  1999

  • Homogenization of the Cauchy problem for Hamilton-Jacobi equations

    H. Ishii

    Stochastic analysis, control, optimization and applications     305 - 324  1999

  • Waiting time effects for Gauss curvature flows

    D. Chopp, L. C. Evans, H. Ishii

    Indiana Univ. Math. J.   48 ( 1 ) 311 - 334  1999

  • Hopf-Lax formulas for semicontinuous data

    O. Alvarez, E. N. Barron, H. Ishii

    Indiana Univ. Math. J.   48 ( 3 ) 993 - 1035  1999

  • Hopf-Lax formulas for Hamilton-Jacobi equations with semicontinuous initial data

    H. Ishii

    Singularity theory and differential equations   1111   144 - 156  1999

  • Waiting time effects for Gauss curvature flows

    D. Chopp, L. C. Evans, H. Ishii

    Indiana Univ. Math. J.   48 ( 1 ) 311 - 334  1999

  • Homogenization of the Cauchy problem for Hamilton-Jacobi equations

    H. Ishii

    Systems Control Found. Appl./Birkhauser    1999

  • Some properties of ergodic attractors for controlled dynamical systems

    M. Arisawa, H. Ishii

    Discrete Contin. Dynam. Systems   4 ( 1 ) 43 - 54  1998

  • Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure

    K. Horie, H. Ishii

    Indiana Univ. Math. J.   47 ( 3 ) 1011 - 1058  1998

  • An approximation scheme for Gauss curvature flow

    H. Ishii

    数理解析研究所講究録   No. 1061   108 - 123  1998

  • Some properties of ergodic attractors for controlled dynamical systems

    M. Arisawa, H. Ishii

    Discrete Contin. Dynam. Systems   4 ( 1 ) 43 - 54  1998

  • Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure

    K. Horie, H. Ishii

    Indiana Univ. Math. J.   47 ( 3 ) 1011 - 1058  1998

  • An approximation scheme for Gauss curvature flow

    H. Ishii

    数理解析研究所講究録   No. 1061   108 - 123  1998

  • The level set method for etching and deposition

    D Adalsteinsson, LC Evans, H Ishii

    MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES   7 ( 8 ) 1153 - 1186  1997.12

     View Summary

    We provide a rigorous interpretation of the level set approach to certain nonlocal geometric motions modelling etching effects in manufacture. The shadowing of certain parts of a surface by other parts gives rise to a nonlocal Hamilton-Jacobi type PDE, with a multivalued Hamiltonian. We also show that deposition effects do not fall within the conventional level set framework, and accordingly must be reinterpreted for numerical implementation.

  • The level set method for etching and deposition

    D Adalsteinsson, LC Evans, H Ishii

    MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES   7 ( 8 ) 1153 - 1186  1997.12

     View Summary

    We provide a rigorous interpretation of the level set approach to certain nonlocal geometric motions modelling etching effects in manufacture. The shadowing of certain parts of a surface by other parts gives rise to a nonlocal Hamilton-Jacobi type PDE, with a multivalued Hamiltonian. We also show that deposition effects do not fall within the conventional level set framework, and accordingly must be reinterpreted for numerical implementation.

  • Un contre-exemple pour la theorie du controle ergodique en dimension infinie

    M. Arisawa, H. Ishii, P.-L. Lions

    C. R. Acad. Sci. Paris Ser. I Math.   325 ( 1 ) 37 - 41  1997

  • Viscosity solutions and their applications

    H. Ishii

    Sugaku Expositions   10 ( 2 ) 123 - 141  1997

  • Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above

    H. Ishii

    Appl. Anal.   67; 3-4, pp. 357--372  1997

  • Un contre-exemple pour la theorie du controle ergodique en dimension infinie

    M. Arisawa, H. Ishii, P.-L. Lions

    C. R. Acad. Sci. Paris Ser. I Math.   325 ( 1 ) 37 - 41  1997

  • Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above

    H. Ishii

    Appl. Anal.   67 ( 3-4 ) 357 - 372  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

     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

  • The Tate Institute of Fundamental Research, Bangarole(Overseas Information)

    Ishii Hitoshi

    Bulletin of the Japan Society for Industrial and Applied Mathematics   6 ( 1 ) 76 - 80  1996

    DOI CiNii

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

    H. Ishii, S. Koike

    SIAM J. Control Optim.   34 ( 2 ) 554 - 571  1996

    DOI

  • Viscosity solutions of nonlinear partial differential equations

    H. Ishii

    Sugaku Expositions   9 ( 2 ) 135 - 152  1996

  • Degenerate parabolic PDEs with discontinuities and generalized evolutions of surfaces

    H. Ishii

    Adv. Differential Equations   1 ( 1 ) 51 - 72  1996

  • Degenerate parabolic PDEs with discontinuities and generalized evolutions of surfaces

    H. Ishii

    Adv. Differential Equations   1 ( 1 ) 51 - 72  1996

  • A singular limit on risk sensitive control and semi-classical analysis

    H. Ishii, H. Nagai, F. Teramoto

       1996

  • GENERALIZED MOTION OF NONCOMPACT HYPERSURFACES WITH VELOCITY HAVING ARBITRARY GROWTH ON THE CURVATURE TENSOR

    H ISHII, P SOUGANIDIS

    TOHOKU MATHEMATICAL JOURNAL   47 ( 2 ) 227 - 250  1995.06

     View Summary

    In this note we study the generalized motion of noncompact hyper-surfaces with normal velocity depending on the normal direction and the curvature tensor. This work extends the by-now-classical works of Evans and Spruck (for mean curvature) and Chen, Giga and Goto (for general motions with sublinear curvature dependence), because it allows general dependence on the curvature tensor. It also allows a general treatment of the generalized evolution including noncompact hypersurfaces. A number of results regarding no interior, convexity, etc. are also presented.

  • GENERALIZED MOTION OF NONCOMPACT HYPERSURFACES WITH VELOCITY HAVING ARBITRARY GROWTH ON THE CURVATURE TENSOR

    H ISHII, P SOUGANIDIS

    TOHOKU MATHEMATICAL JOURNAL   47 ( 2 ) 227 - 250  1995.06

     View Summary

    In this note we study the generalized motion of noncompact hyper-surfaces with normal velocity depending on the normal direction and the curvature tensor. This work extends the by-now-classical works of Evans and Spruck (for mean curvature) and Chen, Giga and Goto (for general motions with sublinear curvature dependence), because it allows general dependence on the curvature tensor. It also allows a general treatment of the generalized evolution including noncompact hypersurfaces. A number of results regarding no interior, convexity, etc. are also presented.

  • UNIQUENESS RESULTS FOR A CLASS OF HAMILTON-JACOBI EQUATIONS WITH SINGULAR COEFFICIENTS

    H ISHII, M RAMASWAMY

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   20 ( 11-12 ) 2187 - 2213  1995

     View Summary

    We establish uniqueness or comparison results for a class of Hamilton-Jacobi equations and give characterizations of maximal solutions of Hamilton-Jacobi equations. The results are applied to characterizing value functions of exit time problems in optimal control.

  • On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions

    H. Ishii

    Funkcial. Ekvac.   38 ( 1 ) 101 - 120  1995

  • UNIQUENESS RESULTS FOR A CLASS OF HAMILTON-JACOBI EQUATIONS WITH SINGULAR COEFFICIENTS

    H ISHII, M RAMASWAMY

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   20 ( 11-12 ) 2187 - 2213  1995

     View Summary

    We establish uniqueness or comparison results for a class of Hamilton-Jacobi equations and give characterizations of maximal solutions of Hamilton-Jacobi equations. The results are applied to characterizing value functions of exit time problems in optimal control.

  • Viscosity solutions and their applications

    H. Ishii

    Sugaku   47 ( 2 ) 97 - 110  1995

  • On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions

    H. Ishii

    Funkcial. Ekvac.   38 ( 1 ) 101 - 120  1995

  • A generalization of the Bence, Merriman and Osher algorithm for motion by mean curvature

    H. Ishii

    Curvature flows and related topics    1995

  • 非線形偏微分方程式の粘性解について

    石井 仁司

    数学   46 ( 2 ) 144 - 157  1994.05

    DOI CiNii

  • On the uniqueness and existence of solutions of fully nonlinear parabolic PDEs under the Osgood type condition

    H. Ishii, K. Kobayasi

    Differential Integral Equations   7 ( 3-4 ) 909 - 920  1994

  • On the uniqueness and existence of solutions of fully nonlinear parabolic PDEs under the Osgood type condition

    H. Ishii, K. Kobayasi

    Differential Integral Equations   7 ( 3-4 ) 909 - 920  1994

  • Viscosity solutions of nonlinear partial differential equations

    H. Ishii

    数学 (Sugaku Expositions; 1996)   46 ( 2 ) 144 - 151  1994

  • The maximum principle for degenerate parabolic PDEs with singularities

    H. Ishii

    Miniconference on Analysis and Applications    1994

  • SDES WITH OBLIQUE REFLECTION ON NONSMOOTH DOMAINS

    P DUPUIS, H ISHII

    ANNALS OF PROBABILITY   21 ( 1 ) 554 - 580  1993.01

     View Summary

    In this paper we consider stochastic differential equations with reflecting boundary conditions for domains that might have corners and for which the allowed directions of reflection at a point on the boundary of the domain are possibly oblique. The main results are strong existence and uniqueness for solutions of such equations. A key ingredient is a family of relatively regular functions appropriate to the given domain and directions of reflection. Two cases are treated in the paper. In the first case the direction of reflection is single valued and varies smoothly, and the main new feature is that the boundary of the domain may be nonsmooth. In the second case the domain is taken to be the intersection of a finite number of domains with relatively smooth boundary, and at the resulting corner points more than one oblique direction is allowed.

  • Uniqueness of solutions to the Cauchy problem for υt-υΔυ+γ|∇υ|2=0

    I. Fukuda, H. Ishii, M. Tsutsumi

    Differential Integral Equations   6 ( 6 ) 1231 - 1252  1993

  • Viscosity solutions of functional -differential equations

    H. Ishii, S. Koike

    Adv. Math. Sci. Appl.   3   191 - 218  1993

  • Viscosity solutions of nonlinear second-order partial differential equations in hilbert spaces

    Ishii Hitoshi

    Communications in Partial Differential Equations   18 ( 3-4 ) 601 - 650  1993.01

    DOI

  • SDES WITH OBLIQUE REFLECTION ON NONSMOOTH DOMAINS

    P DUPUIS, H ISHII

    ANNALS OF PROBABILITY   21 ( 1 ) 554 - 580  1993.01

     View Summary

    In this paper we consider stochastic differential equations with reflecting boundary conditions for domains that might have corners and for which the allowed directions of reflection at a point on the boundary of the domain are possibly oblique. The main results are strong existence and uniqueness for solutions of such equations. A key ingredient is a family of relatively regular functions appropriate to the given domain and directions of reflection. Two cases are treated in the paper. In the first case the direction of reflection is single valued and varies smoothly, and the main new feature is that the boundary of the domain may be nonsmooth. In the second case the domain is taken to be the intersection of a finite number of domains with relatively smooth boundary, and at the resulting corner points more than one oblique direction is allowed.

  • Uniqueness of solutions to the Cauchy problem for υt-υΔυ+γ|∇υ|2=0

    I. Fukuda, H. Ishii, M. Tsutsumi

    Differential Integral Equations   6 ( 6 ) 1231 - 1252  1993

  • Viscosity solutions of functional -differential equations

    H. Ishii, S. Koike

    Adv. Math. Sci. Appl.   3   191 - 218  1993

  • VISCOSITY SOLUTIONS OF NONLINEAR 2ND-ORDER PARTIAL-DIFFERENTIAL EQUATIONS IN HILBERT-SPACES

    H ISHII

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   18 ( 3-4 ) 601 - 650  1993

  • GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR INTERFACE EQUATIONS COUPLED WITH DIFFUSION-EQUATIONS

    Y GIGA, S GOTO, H ISHII

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   23 ( 4 ) 821 - 835  1992.07

     View Summary

    A weak formulation for an interface dynamics coupled with a diffusion equation is introduced. A global-in-time weak solution is constructed for an arbitrary initial data under a periodic boundary condition. The result applies to the interface equation obtained as a certain singular limit of some reaction-diffusion systems including the activator-inhibitor model.

  • USERS GUIDE TO VISCOSITY SOLUTIONS OF 2ND-ORDER PARTIAL-DIFFERENTIAL EQUATIONS

    MG CRANDALL, H ISHII, PL LIONS

    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY   27 ( 1 ) 1 - 67  1992.07

     View Summary

    The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.

  • GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR INTERFACE EQUATIONS COUPLED WITH DIFFUSION-EQUATIONS

    Y GIGA, S GOTO, H ISHII

    SIAM JOURNAL ON MATHEMATICAL ANALYSIS   23 ( 4 ) 821 - 835  1992.07

     View Summary

    A weak formulation for an interface dynamics coupled with a diffusion equation is introduced. A global-in-time weak solution is constructed for an arbitrary initial data under a periodic boundary condition. The result applies to the interface equation obtained as a certain singular limit of some reaction-diffusion systems including the activator-inhibitor model.

  • USERS GUIDE TO VISCOSITY SOLUTIONS OF 2ND-ORDER PARTIAL-DIFFERENTIAL EQUATIONS

    MG CRANDALL, H ISHII, PL LIONS

    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY   27 ( 1 ) 1 - 67  1992.07

     View Summary

    The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.

  • VISCOSITY SOLUTIONS FOR A CLASS OF HAMILTON-JACOBI EQUATIONS IN HILBERT-SPACES

    H ISHII

    JOURNAL OF FUNCTIONAL ANALYSIS   105 ( 2 ) 301 - 341  1992.05

  • Perron's method for monotone systems of second-order elliptic partial differential equations

    H. Ishii

    Differential Integral Equations   5 ( 1 ) 1 - 24  1992

  • Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces

    Hitoshi Ishii

    J. Funct. Anal.   105 ( 2 ) 301 - 341  1992

  • Perron's method for monotone systems of second-order elliptic partial differential equations

    H. Ishii

    Differential Integral Equations   5 ( 1 ) 1 - 24  1992

  • FULLY NONLINEAR OBLIQUE DERIVATIVE PROBLEMS FOR NONLINEAR 2ND-ORDER ELLIPTIC PDES

    H ISHII

    DUKE MATHEMATICAL JOURNAL   62 ( 3 ) 633 - 661  1991.04

  • Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains

    Y. Giga, S. Goto, H. Ishii, M.-H. Sato

    Indiana Univ. Math. J.   40 ( 2 ) 443 - 470  1991

  • On oblique derivative problems for fully nonlinear second-order elliptic PDE’s on domains with corners

    Paul Dupuis, Hitoshi Ishii

    Hokkaido Mathematical Journal   20 ( 1 ) 135 - 164  1991

    DOI

  • On Lipschitz continuity of thesolution mapping to the Skorokhod problem, with applications

    P. Dupuis, H. Ishii

    Stochastics Stochastics Rep.   35 ( 1 ) 31 - 62  1991

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

    Hitoshi Ishii, Shigeaki Koike

    Funkcial. Ekvac.   34 ( 1 ) 143 - 155  1991

  • VISCOSITY SOLUTIONS FOR MONOTONE SYSTEMS OF 2ND-ORDER ELLIPTIC PDES

    H ISHII, S KOIKE

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   16 ( 6-7 ) 1095 - 1128  1991

  • REMARKS ON ELLIPTIC SINGULAR PERTURBATION PROBLEMS

    H ISHII, S KOIKE

    APPLIED MATHEMATICS AND OPTIMIZATION   23 ( 1 ) 1 - 15  1991.01

     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.

    DOI

  • VISCOSITY SOLUTIONS OF THE BELLMAN EQUATION ON AN ATTAINABLE SET

    H ISHII, JL MENALDI, L ZAREMBA

    PROBLEMS OF CONTROL AND INFORMATION THEORY-PROBLEMY UPRAVLENIYA I TEORII INFORMATSII   20 ( 5 ) 317 - 328  1991

     View Summary

    By an appropriate modification of the viscosity solution concept, we introduce a notion solution of a PDE that is applicable, among others, to the Bellman equation and first general classes of optimal control problems with the only restriction on a payoff functional that the stopping time is bounded by a fixed number T. We consider this PDE on the attainable set from OMEGA-0, a set Of given initial conditions. We prove both existence and uniqueness results for optimal control problems. The approach is illustrated with several examples and comments.

  • Fully nonlinear oblique derivative problems for nonlinear second-order elliptic pde’s

    Hitoshi Ishii

    Duke Mathematical Journal   62 ( 3 ) 633 - 661  1991

    DOI

  • Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains

    Y. Giga, S. Goto, H. Ishii, M.-H. Sato

    Indiana Univ. Math. J.   40 ( 2 ) 443 - 470  1991

  • On oblique derivative problems for fully nonlinear second-order elliptic PDE’s on domains with corners

    Paul Dupuis, Hitoshi Ishii

    Hokkaido Mathematical Journal   20 ( 1 ) 135 - 164  1991

    DOI

  • On Lipschitz continuity of thesolution mapping to the Skorokhod problem, with applications

    P. Dupuis, H. Ishii

    Stochastics Stochastics Rep.   35 ( 1 ) 31 - 62  1991

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

    Hitoshi Ishii, Shigeaki Koike

    Funkcial. Ekvac.   34 ( 1 ) 143 - 155  1991

  • VISCOSITY SOLUTIONS FOR MONOTONE SYSTEMS OF 2ND-ORDER ELLIPTIC PDES

    H ISHII, S KOIKE

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   16 ( 6-7 ) 1095 - 1128  1991

  • REMARKS ON ELLIPTIC SINGULAR PERTURBATION PROBLEMS

    H ISHII, S KOIKE

    APPLIED MATHEMATICS AND OPTIMIZATION   23 ( 1 ) 1 - 15  1991.01

     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.

    DOI

  • VISCOSITY SOLUTIONS OF THE BELLMAN EQUATION ON AN ATTAINABLE SET

    H ISHII, JL MENALDI, L ZAREMBA

    PROBLEMS OF CONTROL AND INFORMATION THEORY-PROBLEMY UPRAVLENIYA I TEORII INFORMATSII   20 ( 5 ) 317 - 328  1991

     View Summary

    By an appropriate modification of the viscosity solution concept, we introduce a notion solution of a PDE that is applicable, among others, to the Bellman equation and first general classes of optimal control problems with the only restriction on a payoff functional that the stopping time is bounded by a fixed number T. We consider this PDE on the attainable set from OMEGA-0, a set Of given initial conditions. We prove both existence and uniqueness results for optimal control problems. The approach is illustrated with several examples and comments.

  • ON OBLIQUE DERIVATIVE PROBLEMS FOR FULLY NONLINEAR 2ND-ORDER ELLIPTIC PARTIAL-DIFFERENTIAL EQUATIONS ON NONSMOOTH DOMAINS

    P DUPUIS, H ISHII

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   15 ( 12 ) 1123 - 1138  1990.12

  • ON OBLIQUE DERIVATIVE PROBLEMS FOR FULLY NONLINEAR 2ND-ORDER ELLIPTIC PARTIAL-DIFFERENTIAL EQUATIONS ON NONSMOOTH DOMAINS

    P DUPUIS, H ISHII

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   15 ( 12 ) 1123 - 1138  1990.12

  • A VISCOSITY SOLUTION APPROACH TO THE ASYMPTOTIC ANALYSIS OF QUEUING-SYSTEMS

    P DUPUIS, H ISHII, HM SONER

    ANNALS OF PROBABILITY   18 ( 1 ) 226 - 255  1990.01

  • The maximum principle for semicontinuous functions

    M. G. Crandall, H. Ishii

    Differential Integral Equations   3 ( 6 ) 1001 - 1014  1990

  • VISCOSITY SOLUTIONS OF FULLY NONLINEAR 2ND-ORDER ELLIPTIC PARTIAL-DIFFERENTIAL EQUATIONS

    H ISHII, PL LIONS

    JOURNAL OF DIFFERENTIAL EQUATIONS   83 ( 1 ) 26 - 78  1990.01

  • A VISCOSITY SOLUTION APPROACH TO THE ASYMPTOTIC ANALYSIS OF QUEUING-SYSTEMS

    P DUPUIS, H ISHII, HM SONER

    ANNALS OF PROBABILITY   18 ( 1 ) 226 - 255  1990.01

  • The maximum principle for semicontinuous functions

    M. G. Crandall, H. Ishii

    Differential Integral Equations   3 ( 6 ) 1001 - 1014  1990

  • VISCOSITY SOLUTIONS OF FULLY NONLINEAR 2ND-ORDER ELLIPTIC PARTIAL-DIFFERENTIAL EQUATIONS

    H ISHII, PL LIONS

    JOURNAL OF DIFFERENTIAL EQUATIONS   83 ( 1 ) 26 - 78  1990.01

  • A REMARK ON A SYSTEM OF INEQUALITIES WITH BILATERAL OBSTACLES

    H ISHII, N YAMADA

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   13 ( 11 ) 1295 - 1301  1989.11

  • A REMARK ON A SYSTEM OF INEQUALITIES WITH BILATERAL OBSTACLES

    H ISHII, N YAMADA

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   13 ( 11 ) 1295 - 1301  1989.11

  • THE BELLMAN EQUATION FOR MINIMIZING THE MAXIMUM COST

    EN BARRON, H ISHII

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   13 ( 9 ) 1067 - 1090  1989.09

  • THE BELLMAN EQUATION FOR MINIMIZING THE MAXIMUM COST

    EN BARRON, H ISHII

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   13 ( 9 ) 1067 - 1090  1989.09

  • On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's

    Hitoshi Ishii

    Communications on Pure and Applied Mathematics   42 ( 1 ) 15 - 45  1989

     View Summary

    We prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations. One of them extends some recent uniqueness results by Jensen. Some establish the uniqueness of solutions for second‐order Isaacs' equations and hence include the uniqueness results for Bellman equations by P.‐L. Lions. Our comparison results apply even for discontinuous solutions and so Perron's method readily yields the existence of continuous solutions. Copyright © 1989 Wiley Periodicals, Inc., A Wiley Company

    DOI

  • A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations

    Hitoshi Ishii

    Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)   16 ( 1 ) 105 - 135  1989

  • ON UNIQUENESS AND EXISTENCE OF VISCOSITY SOLUTIONS OF FULLY NONLINEAR 2ND-ORDER ELLIPTIC PDES

    H ISHII

    COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS   42 ( 1 ) 15 - 45  1989.01

  • A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations

    Hitoshi Ishii

    Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)   16 ( 1 ) 105 - 135  1989

  • REPRESENTATION OF SOLUTIONS OF HAMILTON-JACOBI EQUATIONS

    H ISHII

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   12 ( 2 ) 121 - 146  1988.02

  • LARGE DEVIATION BEHAVIOR OF 2 COMPETING QUEUES

    P DUPUIS, H ISHII

    PROCEEDINGS OF THE 22ND CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS, VOLS 1 & 2     1009 - 1013  1988  [Refereed]

  • Representation of solutions of Hamilton-Jacobi equations

    Hitoshi Ishii

    Nonlinear Anal.   12 ( 2 ) 121 - 146  1988

  • UNIQUENESS OF VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS REVISITED

    MG CRANDALL, H ISHII, PL LIONS

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   39 ( 4 ) 581 - 607  1987.10

  • UNIQUENESS OF VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS REVISITED

    MG CRANDALL, H ISHII, PL LIONS

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   39 ( 4 ) 581 - 607  1987.10

  • PERRON METHOD FOR HAMILTON-JACOBI EQUATIONS

    H ISHII

    DUKE MATHEMATICAL JOURNAL   55 ( 2 ) 369 - 384  1987.06

  • A SIMPLE, DIRECT PROOF OF UNIQUENESS FOR SOLUTIONS OF THE HAMILTON-JACOBI EQUATIONS OF EIKONAL TYPE

    H ISHII

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   100 ( 2 ) 247 - 251  1987.06

  • A simple, direct proof of uniqueness for solutions of the hamilton-jacobi equations of eikonal type

    Hitoshi Ishii

    Proceedings of the American Mathematical Society   100 ( 2 ) 247 - 251  1987

     View Summary

    We present a new, direct proof of the uniqueness theorem for a class of Hamilton-Jacobi equations including the eikonal equation in geometric optics. © 1987 American Mathematical Society.

    DOI

  • Perron’s method for Hamilton-Jacobi equations

    Hitoshi Ishii

    Duke Mathematical Journal   55 ( 2 ) 369 - 384  1987

    DOI

  • Existence and uniqueness of solutions of Hamilton-Jacobi equations

    Hitoshi Ishii

    Funkcial. Ekvac.   29 ( 2 ) 167 - 188  1986

  • Existence and uniqueness of solutions of Hamilton-Jacobi equations

    Hitoshi Ishii

    Funkcial. Ekvac.   29 ( 2 ) 167 - 188  1986

  • A PDE APPROACH TO SOME ASYMPTOTIC PROBLEMS CONCERNING RANDOM DIFFERENTIAL-EQUATIONS WITH SMALL NOISE INTENSITIES

    LC EVANS, H ISHII

    ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   2 ( 1 ) 1 - 20  1985

  • Boundary estimates for bounded solutions of a classical Yang-Mills equation

    H.Ishii, K.Nakamitsu

    Math. Japon.   30 ( 2 ) 199 - 215  1985

  • A nonlinear diffusion equation in phytoplankton dynamics with self-shading effect

    Hitoshi Ishii, Izumi Takagi

    Mathematics in biology and medicine, Lecture Notes in Biomath.   57   66 - 71  1985

  • Theory of the existence and uniqueness of viscosity solutions of Hamilton-Jacobi equations and its applications

    Hitoshi Ishii

    数理解析研究所講究録   559   162 - 181  1985

  • Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets.

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   28   33 - 77  1985

  • A PDE APPROACH TO SOME ASYMPTOTIC PROBLEMS CONCERNING RANDOM DIFFERENTIAL-EQUATIONS WITH SMALL NOISE INTENSITIES

    LC EVANS, H ISHII

    ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   2 ( 1 ) 1 - 20  1985

  • Boundary estimates for bounded solutions of a classical Yang-Mills equation

    H.Ishii, K.Nakamitsu

    Math. Japon.   30 ( 2 ) 199 - 215  1985

  • A nonlinear diffusion equation in phytoplankton dynamics with self-shading effect

    Hitoshi Ishii, Izumi Takagi

    Mathematics in biology and medicine, Lecture Notes in Biomath.   57   66 - 71  1985

  • Theory of the existence and uniqueness of viscosity solutions of Hamilton-Jacobi equations and its applications

    Hitoshi Ishii

    数理解析研究所講究録   559   162 - 181  1985

  • Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   28   33 - 77  1985

  • Recent developments in the theory of Hamilton-Jacobi equations

    H. Ishii

    Essays in celebration of the 100th anniversary of Chuo University    1985

  • On Representation of Solutions of Hamilton-Jacobi Equations with Convex Hamiltonians

    Hitoshi Ishii

    North-Holland Mathematics Studies   128 ( C ) 15 - 52  1985.01

     View Summary

    Recently, Crandall and Lions introduced the notion of viscosity solution for Hamilton–Jacobi equations to settle the uniqueness problem of generalized solutions of Hamilton–Jacobi equations. The existence of viscosity solutions of Hamilton–Jacobi equations was established under the same hypotheses on the Hamiltonians as those for the uniqueness of viscosity solutions. Thus, the chapter presents the Hamiltonian as a max or max–min of linear functions of p, and proves the uniform continuity of the value function of the associated optimal control or differential game problem. © 1985, Elsevier Inc. All rights reserved.

    DOI

  • DIFFERENTIAL-GAMES AND NONLINEAR 1ST ORDER PDE ON BOUNDED DOMAINS

    LC EVANS, H ISHII

    MANUSCRIPTA MATHEMATICA   49 ( 2 ) 109 - 139  1984

  • APPROXIMATE SOLUTIONS OF THE BELLMAN EQUATION OF DETERMINISTIC CONTROL-THEORY

    IC DOLCETTA, H ISHII

    APPLIED MATHEMATICS AND OPTIMIZATION   11 ( 2 ) 161 - 181  1984

  • Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations

    Hitoshi Ishii

    Indiana Univ. Math. J.   33 ( 5 ) 721 - 748  1984

  • DIFFERENTIAL-GAMES AND NONLINEAR 1ST ORDER PDE ON BOUNDED DOMAINS

    LC EVANS, H ISHII

    MANUSCRIPTA MATHEMATICA   49 ( 2 ) 109 - 139  1984

  • APPROXIMATE SOLUTIONS OF THE BELLMAN EQUATION OF DETERMINISTIC CONTROL-THEORY

    IC DOLCETTA, H ISHII

    APPLIED MATHEMATICS AND OPTIMIZATION   11 ( 2 ) 161 - 181  1984

  • UNIQUENESS OF UNBOUNDED VISCOSITY SOLUTION OF HAMILTON-JACOBI EQUATIONS

    H ISHII

    INDIANA UNIVERSITY MATHEMATICS JOURNAL   33 ( 5 ) 721 - 748  1984

  • BOUNDARY-REGULARITY AND UNIQUENESS FOR AN ELLIPTIC EQUATION WITH GRADIENT CONSTRAINT

    H ISHII, S KOIKE

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   8 ( 4 ) 317 - 346  1983

  • Remarks on existence of viscosity solutions of Hamilton-Jacobi equations

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   26   5 - 24  1983

  • BOUNDARY-REGULARITY AND UNIQUENESS FOR AN ELLIPTIC EQUATION WITH GRADIENT CONSTRAINT

    H ISHII, S KOIKE

    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   8 ( 4 ) 317 - 346  1983

  • Remarks on existence of viscosity solutions of Hamilton-Jacobi equations

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   26   5 - 24  1983

  • GLOBAL STABILITY OF STATIONARY SOLUTIONS TO A NON-LINEAR DIFFUSION EQUATION IN PHYTOPLANKTON DYNAMICS

    H ISHII, TAKAGI, I

    JOURNAL OF MATHEMATICAL BIOLOGY   16 ( 1 ) 1 - 24  1982

  • On the existence of almost periodic complete trajectories for contractive almost periodic processes

    Hitoshi Ishii

    J. Differential Equations   43 ( 1 ) 66 - 72  1982

  • GLOBAL STABILITY OF STATIONARY SOLUTIONS TO A NON-LINEAR DIFFUSION EQUATION IN PHYTOPLANKTON DYNAMICS

    H ISHII, TAKAGI, I

    JOURNAL OF MATHEMATICAL BIOLOGY   16 ( 1 ) 1 - 24  1982

  • ON THE EXISTENCE OF ALMOST PERIODIC COMPLETE TRAJECTORIES FOR CONTRACTIVE ALMOST PERIODIC PROCESSES

    H ISHII

    JOURNAL OF DIFFERENTIAL EQUATIONS   43 ( 1 ) 66 - 72  1982

  • On a certain estimate of the free boundary in the Stefan problem

    Hitoshi Ishii

    Journal of Differential Equations   42 ( 1 ) 106 - 115  1981

    DOI

  • ON A CERTAIN ESTIMATE OF THE FREE-BOUNDARY IN THE STEFAN PROBLEM

    H ISHII

    JOURNAL OF DIFFERENTIAL EQUATIONS   42 ( 1 ) 106 - 115  1981

  • Remarks on evolution equations with almost periodic forcing terms

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   23   55 - 71  1980

  • Erratum : "Asymptotic stability of almost periodic solutions of a free boundary problem arising in hydraulics"

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   23   83 - 83  1980

  • Asymptotic stability and existence of almost periodic solutions for the one-dimensional two-phase Stefan problem

    Hitoshi Ishii

    Math. Japon.   25 ( 4 ) 379 - 393  1980

  • Remarks on evolution equations with almost periodic forcing terms

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   23   55 - 71  1980

  • Erratum : "Asymptotic stability of almost periodic solutions of a free boundary problem arising in hydraulics"

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   23   83 - 83  1980

  • Asymptotic stability and existence of almost periodic solutions for the one-dimensional two-phase Stefan problem

    Hitoshi Ishii

    Math. Japon.   25 ( 4 ) 379 - 393  1980

  • Asymptotic stability of almost periodic solutions of a free boundary problem arising in hydraulics

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   22   73 - 95  1979

  • Asymptotic stability of almost periodic solutions of a free boundary problem arising in hydraulics

    Hitoshi Ishii

    Bull. Fac. Sci. Engrg. Chuo Univ.   22   73 - 95  1979

  • Asymptotic stability and blowing up of solutions of some nonlinear equations

    Hitoshi Ishii

    J. Differential Equations   26 ( 2 ) 291 - 319  1977

  • Asymptotic stability and blowing up of solutions of some nonlinear equations

    Hitoshi Ishii

    J. Differential Equations   26 ( 2 ) 291 - 319  1977

  • On the solutions of the Navier-Stokes equations of slightly compressible fluids

    R. Iino, H. Ishii, Y. Machino

    Bull. Sci. Engrg. Res. Lab. Waseda Univ.   69   74 - 79  1975

  • Some uniqueness theorems for first order hyperbolic systems

    Hitoshi Ishii, Yoshinori Sagisaka, Masayoshi Tsutsumi

    Publ. Res. Inst. Math. Sci.   11 ( 2 ) 403 - 415  1975

  • On some perturbation of the Navier-Stokes equations in Lp spaces

    Hitoshi Ishii

    Funkcial. Ekvac.   18 ( 1 ) 73 - 83  1975

  • On the solutions of the Navier-Stokes equations of slightly compressible fluids

    R. Iino, H. Ishii, Y. Machino

    Bull. Sci. Engrg. Res. Lab. Waseda Univ.   69   74 - 79  1975

  • Some uniqueness theorems for first order hyperbolic systems

    Hitoshi Ishii, Yoshinori Sagisaka, Masayoshi Tsutsumi

    Publ. Res. Inst. Math. Sci.   11 ( 2 ) 403 - 415  1975

  • On some perturbation of the Navier-Stokes equations in Lp spaces

    Hitoshi Ishii

    Funkcial. Ekvac.   18 ( 1 ) 73 - 83  1975

  • On some Fourier multipliers and partial differential equations.

    Hitoshi Ishii

    Math. Japon.   19 ( 3 ) 139 - 163  1974

  • On some Fourier multipliers and partial differential equations

    Hitoshi Ishii

    Math. Japon.   19 ( 3 ) 139 - 163  1974

  • Estimates from Wp, α to Wq, β for the solutions of the Petrovskii well posed Cauchy problems.

    Hitoshi Ishii

    Proc. Japan Acad.   49   705 - 710  1973

  • Estimates from Wp, α to Wq, β for the solutions of the Petrovskii well posed Cauchy problems

    Hitoshi Ishii

    Proc. Japan Acad.   49   705 - 710  1973

▼display all

Books and Other Publications

  • プリンストン数学大全

    HITOSHI ISHII( Part: Joint translator)

    朝倉書店  2015.11

  • Hamilton-Jacobi equations: approximations, numerical analysis and applications

    HITOSHI ISHII( Part: Joint author, pp. 111–249)

    Springer, Heidelberg; Fondazione C.I.M.E.  2013

  • 応用解析ハンドブック

    HITOSHI ISHII( Part: Joint author, pp. 311-374)

    シュプリンガー・ジャパン  2010.02

  • Recent progress on reaction-diffusion systems and viscosity solutions

    HITOSHI ISHII( Part: Joint editor)

    World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ  2009

Misc

Awards

  • Kodaira Kunihiko Prize

    2019.09   Mathematical Society of Japan   Viscosity solution theory for fully nonlinear partial differential equations

    Winner: HITOSHI ISHII

  • Okuma Memorial Academic Commemorative Prize

    2017.11   Waseda University   Founding of the viscous solution theory of nonlinear partial differential equations and its application

    Winner: HITOSHI ISHII

  • Fellow

    2012.01   American Mathematical Society  

    Winner: HITOSHI ISHII

  • Highly cited researcher

    2002   Thompson ISI  

    Winner: HITOSHI ISHII

  • Autumn Prize

    1994.09   Mathematical Society of Japan  

    Winner: HITOSHI ISHII

Research Projects

  • Dynamic game analysis of international environmental agreements

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

    Project Year :

    2021.04
    -
    2025.03
     

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

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

    Project Year :

    2020.04
    -
    2025.03
     

  • Advancement in viscosity solution theory: asymptotic and boundary value problems

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

    Project Year :

    2020.04
    -
    2023.03
     

  • Discount rate and international environmental agreement in climate changeOngoing

    日本学術振興会  科学研究費助成事業(基盤研究(B))

    Project Year :

    2018.04
    -
    2021.03
     

    KEN-ICHI AKAKO

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

    日本学術振興会  科学研究費助成事業(基盤研究(B))

    Project Year :

    2016.04
    -
    2020.03
     

    HITOSHI ISHII

  • Deepening of the theory of viscosity solutions and its applications

    日本学術振興会  科学研究費助成事業(基盤研究(A))

    Project Year :

    2011.04
    -
    2016.03
     

    HITOSHI ISHII

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

    日本学術振興会  科学研究費助成事業(基盤研究(A))

    Project Year :

    2006.04
    -
    2010.03
     

    HITOSHI ISHII

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

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

    Project Year :

    2006
    -
    2009
     

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

     View Summary

    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.

  • 粘性解の理論と応用の研究

    日本学術振興会  科学研究費助成事業(基盤研究(B))

    Project Year :

    2003.04
    -
    2006.03
     

    石井 仁司

  • 粘性解と変分問題

    日本学術振興会  科学研究費助成事業(萌芽研究)

    Project Year :

    2002.04
    -
    2005.03
     

    石井仁司

  • 粘性解と変分問題

    日本学術振興会  科学研究費助成事業 萌芽研究

    Project Year :

    2002
    -
    2004
     

    石井 仁司

     View Summary

    各地で開かれた研究集会に参加し,国内研究協力者と研究打ち合わせ・共同研究を行い,海外からの研究協力者を招へいしながら,次のような成果を得た.Ornstein-Uhlenbeck作用素の項を持つ粘性ハミルトン・ヤコビ方程式u_t-Δu+αx・Du+H(Du)=f(x)の解について研究し,初期値問題の可解性,時間無限大における解の漸近挙動に関する詳しい結果を得た.この研究はNamah, Fathi, Roquejoffre, Barles・Souganidisの最近の時間無限大における同様な研究を推し進めるもので,非有界領域の場合を扱った点に重要さがある.この研究では,さらに解を構成する際に,まず粘性解の存在を示し,この粘性解が古典解であることを示すという手順が取られている。そのために粘性解が古典解であることを示すことが重要であるが,このための一つの自然な方法を提示している。Hamilton-Jacobi方程式u_t+αx・Du+H(Du)fx)についても,Hが凸関数の場合に全空間上での解の時間無限大での漸近挙動について一般的な仮定のもとで収束定理を得ることが出来た.この漸近挙動の考察において弱KAM定理で導入されたHamilton-Jacobi方程式のAubry集合が重要な役割を果たす。この集合を特定し,最適制御の値関数として解を捉え,時間無限大での解の挙動を解析した.これまでの研究でHamilton-Jacobi方程式に対する緩和法を導入し(より正確には,緩和現象の発現数学的に捉え),比較的一般の非凸なHamiltonianを持つHamilton-Jacobi方程式に対して緩和現象の発現を示した.特に,今年度は,初期値問題を考察し,緩和現象の発現のための,確認し易い十分条件を確立した.

  • 微分方程式の粘性解とその応用の研究

    日本学術振興会  科学研究費助成事業(基盤研究(B))

    Project Year :

    2000.04
    -
    2003.03
     

    石井 仁司

  • 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(基盤研究(B))

    Project Year :

    1997.04
    -
    2000.03
     

    石井 仁司

  • 超曲面の曲率流における待ち時間の研究

    日本学術振興会  科学研究費助成事業(萌芽的研究)

    Project Year :

    1997.04
    -
    1999.04
     

    石井 仁司

  • 超曲面の曲率流における待ち時間の研究

    日本学術振興会  科学研究費助成事業 萌芽的研究

    Project Year :

    1997
    -
    1998
     

    石井 仁司, 倉田 和浩

     View Summary

    研究課題について、国内の研究集会において、得られた結果の公表と研究討議を行い、また、オーストラリアのミッション・ビーチで行われた国際研究集会において研究成果の公表を行いながら、研究を進め、以下のような成果をあげた。1.超曲面のガウス曲率流について、初期曲面の平坦部分が動き出すまでの待ち時間に関する基本的研究について、昨年度に得られた結果を中心にまとめ上げた。おもな結果は以下のものである。(1)超曲面が平坦部分を持つならばガウス曲率流は必ずC^2級ではなくなる。(2)初期曲面の一点において、二つ以上の主曲率が0であれば、その点の待ち時間は正である。(3)初期曲面のある点について、その点で高々一つの主曲率が0であれば、その点の待ち時間は0である。2.ガウス曲率流は海岸での岩石の磨耗過程を記述する数学モデルである。ガウス曲率流は凸曲面に対してのみ、この磨耗のモデルとして意味を持ち、曲面が凸でない場合には磨耗のモデルとしては不適切である。さらに、数学的にも凸でない場合には、一般的に言えば、与えられた初期曲面に対してガウス曲率流は存在しない。曲面が凸である場合の磨耗のモデルであるガウス曲率流に対応する、曲面が凸でない場合の磨耗の数学的モデルを提案し、その存在と一意性と安定性を証明した。この結果については、現在論文として纏めている。3.ガウス曲率流に対する一つの幾何学的近似アルゴリズムを発見し、その近似アルゴリズムのガウス曲率流への収束を証明した。4.ガウス曲率流あるいは岩石の磨耗の確率近似モデルを提案し、凸曲線の場合にそのガウス曲率流への収束を証明した。

  • Hamilton-Jacobi 方程式に対する特異摂動問題の研究

    日本学術振興会  科学研究費助成事業(基盤研究(C))

    Project Year :

    1996.04
    -
    1997.03
     

    石井 仁司

  • 非線形偏微分方程式の粘性解とその応用の研究

    日本学術振興会  科学研究費助成事業(基盤研究(C))

    Project Year :

    1995.04
    -
    1996.03
     

    石井 仁司

  • 粘性解とその応用に関する共同研究

    日本学術振興会  科学研究費助成事業(基盤研究(C))

    Project Year :

    1995.04
    -
    1996.03
     

    石井 仁司

  • Hamilton-Jacobi 方程式に対する特異摂動問題の研究

    日本学術振興会  科学研究費助成事業 基盤研究(C)

    Project Year :

    1996
     
     
     

    石井 仁司, 岩野 正宏, 西岡 國雄, 倉田 和浩, 酒井 良, 望月 清

     View Summary

    Hamilton-Jacobi方程式について,特異摂動問題の研究を行った.各地の専門研究者との研究打ち合わせを行い,また関連ある研究会等に参加しながら研究を進めた.また,摂動試験関数法の考案者であるL.C.Evans教授を8月に招聘し,本研究による成果の評価,研究方針の検討,最新結果の供与等の寄与を得た.さらに,石井が12月に米国,Berkeleyを訪問し,本研究による成果を公表し,Evans教授,M.G.Crandall教授と本研究について,評価,検討する機会を得た.
    Hamilton-Jacobi方程式に対する特異摂動問題を均質化理論の観点から研究した.特に,Hamilton-Jacobi方程式を考える領域がパラメータに依存する場合の研究に重点を置いて行った.本研究においてはこの問題に対して,粘性解の方法,特にEvansによる摂動試験関数法を改良し適用し,研究を進めた.周期的均質化を考察したが,本研究による一つの成果は微分方程式を考える領域に対する条件の確立である.それは,この周期的領域をトーラスに上に標準射影で射影した時に連結になるというものである.これまでの多くの研究では領域の連結性を仮定していたが,これはより広いクラスの領域を均質化理論で扱えるというものである.どのような境界条件が取り扱えるかという問題は基本的であるが,一つには非線形Neumann型の境界条件,もう一つとしてDirichiet型の境界条件の場合にeffective Hamiltonianを決定し,さらに均質化における(粘性)解の収束を一様収束の位相で証明した.これが主なる成果である.この他,関連してHamilton-Jacobi方程式に対する初期値問題の解に対する新しい比較定理の証明に成功した.また,曲面の時間発展の数学的定式化の一つである等高面法の可能性を知る上で重要な知見として,等高面法で記述され得る曲面の時間発展の特徴付けの研究を行い,この特徴付けに成功した.

  • 非線形楕円型及び放物型偏微分方程式の研究 研究課題

    日本学術振興会  科学研究費助成事業(基盤研究(C))

    Project Year :

    1992.04
    -
    1993.03
     

    石井 仁司

  • 非線形退化楕円型偏微分方程式の研究

    日本学術振興会  科学研究費助成事業(基盤研究(C))

    Project Year :

    1990.04
    -
    1992.03
     

    石井 仁司

  • ハミルトン・ヤコビ方程式の研究

    日本学術振興会  科学研究費助成事業(奨励研究(A))

    Project Year :

    1984.04
    -
    1985.03
     

    石井 仁司

  • 非線形偏微分方程式の解の周期性・概周期性に関する研究

    日本学術振興会  科学研究費助成事業(奨励研究(A))

    Project Year :

    1980.04
    -
    1981.03
     

    石井 仁司

  • Study on periodicity and almost periodicity of solutions of nonlinear partial differential equations

  • A study of Hamilton-Jacobi equations

  • Some investigations on Riemann surfaces

  • On spectrum of Riemannian manifold

  • A study on nonlinear degenerate elliptic partial differential equations

  • On the deformations of cyclic Galois coverings of algebraic curves

  • Initial value problem for partial differential equations

  • Comprehensive Researches of Differential Equations

  • Study of irregular singularities of differential equations

  • Deformation theory of group schemes and Construction of extensions

  • Synthetic study on differential equations

  • Joint Study on Viscosity Solutions and Their Applications

  • Study on singular perturbation problem for Hamilton-Jacobi equation

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

  • Nonlinear Evolution Equations and Elliptic Equations

  • Theory and applications of viscosity solutions

  • Free boundary problems in potential theory

  • Phase transition and free boundary problem

  • Study of Solutions to Partial Differential Equations, Variational problems and Inverse. Problems

  • Changes of configuration in free boundary problems

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

  • Study on Nonlinear Evolution Equations and Nonlinear Elliptic Equations

  • Research on viscosity solutions of differential equations and their applications

  • Bellman equations of risk-sensitive stochastic and their applications

  • Nonlinear elliptic and parabolic PDEs, theories and applications

  • Viscosity solutions and variational problems

  • Research on the theory of viscosity solutions and its applications

  • Synthetic study for nonlinear evolution equations and eonlinear elliptic equations

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

  • Expected utility maximiaation problems and stochastic control

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

  • Structures created and preserved in nonlinear diffusion field

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

  • Viscosity solution theory for fully nonlinear equations and its applications

  • Development of Analysis on Evolving Pattern for Complicated Phenomena

  • Synthetic study of nonlinear evolution equation and its related topics

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

  • Advanced Analysis on Evolving Patterns in Nonlinear Phenomena Driven by Singular Structure

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Presentations

  • Developments of the theory of viscosity solutions for nonlinear partial differential equations

    HITOSHI ISHII  [Invited]

    第1回日本数学会賞小平邦彦賞 授賞式および受賞講演会 

    Presentation date: 2019.09

  • The vanishing discount problem for weakly coupled systems of Hamilton-Jacobi equations

    HITOSHI ISHII  [Invited]

    4th Swiss-Japanese PDE seminar 

    Presentation date: 2019.09

  • The vanishing discount problem for weakly coupled systems of Hamilton-Jacobi equations

    HITOSHI ISHII  [Invited]

    New trends in Hamilton-Jacobi: PDE, Control, Dynamical Systems and Geometry 

    Presentation date: 2019.07

  • The Dirichlet problem for truncated Laplacians

    HITOSHI ISHII  [Invited]

    The Peoples' Friendship University of Russia, 

    Presentation date: 2019.04

  • Asymptotic problems for the Langevin equation with variable friction

    HITOSHI ISHII  [Invited]

    The Peoples' Friendship University of Russia, 

    Presentation date: 2019.04

  • The vanishing discount problem for Hamilton-Jacobi equations

    HITOSHI ISHII  [Invited]

    応用解析研究会 

    Presentation date: 2019.04

  • The vanishing discount problem for Hamilton-Jacobi equations in Euclidean n space

    HITOSHI ISHII  [Invited]

    PDEs at Valparaiso, a conference in honor of Patricio Felmer's 60th birthday 

    Presentation date: 2018.12

  • The vanishing discount problem for Hamilton-Jacobi equations in Euclidean n space

    HITOSHI ISHII  [Invited]

    From Optimal Control to Maximum Principle 

    Presentation date: 2018.09

  • Two asymptotic problems concerning the Langevin equation with variable friction

    HITOSHI ISHII  [Invited]

    The tenth meeting on Probability and PDE, Tsuda University, 

    Presentation date: 2018.08

  • The Langevin equation with variable friction and Smoluchowski-Kramers approximation

    HITOSHI ISHII  [Invited]

    12th AIMS Conference NTU 

    Presentation date: 2018.07

  • The vanishing discount problem for fully nonlinear degenerate elliptic PDEs

    HITOSHI ISHII

    12th AIMS Conference NTU 

    Presentation date: 2018.07

  • The vanishing discount problem for fully nonlinear degenerate elliptic PDEs

    HITOSHI ISHII  [Invited]

    Nanjing University 

    Presentation date: 2018.06

  • The vanishing discount problem for fully nonlinear degenerate elliptic PDEs

    HITOSHI ISHII  [Invited]

    Seminari di Analisi Matematica, Universita di Bologna 

    Presentation date: 2018.05

  • Asymptotic problems for the Langevin equation with variable friction

    HITOSHI ISHII  [Invited]

    Seminario di Analisi Matematica, Sapienza Universita di Roma 

    Presentation date: 2018.05

  • Asymptotic problems for Hamilton-Jacobi equations and weak KAM theory

    HITOSHI ISHII  [Invited]

    Wolfgang Wasow Lectures, University of Wisconsin-Madison 

    Presentation date: 2018.04

  • The Langevin equation with variable friction and Smoluchowski-Kramers approximation

    HITOSHI ISHII  [Invited]

    81st Midwest PDE Seminar 

    Presentation date: 2018.04

  • The Langevin equation with variable friction and Smoluchowski-Kramers approximation

    HITOSHI ISHII  [Invited]

    Royal Institue of Technology, Sweden 

    Presentation date: 2017.09

  • The Langevin equation and Smoluchowski-Kramers approximation with variable friction

    HITOSHI ISHII  [Invited]

    Seminar at Fudan University 

    Presentation date: 2017.08

  • The Langevin equation and Smoluchowski-Kramers approximation with variable friction

    HITOSHI ISHII  [Invited]

    Viscosity solution approach to asymptotic problems in front propagation, dynamical system and related topics, RIMS 

    Presentation date: 2017.07

  • The vanishing discount problem for fully nonlinear degenerate elliptic PDEs

    HITOSHI ISHII  [Invited]

    Mostly Maximum Principle at BIRS , Canada 

    Presentation date: 2017.03

  • The vanishing discount problem for fully nonlinear degenerate elliptic PDEs

    HITOSHI ISHII  [Invited]

    Beyond Hamilton-Jacobi, Last call to Bordeaux 

    Presentation date: 2017.01

  • Viscosity solutions and asymptotic problrms

    HITOSHI ISHII  [Invited]

    Presentation date: 2016.09

  • The vanishing discount problem and generalized Mather measures

    HITOSHI ISHII  [Invited]

    AMCS Seminar at KAUST 

    Presentation date: 2016.08

  • A boundary value problem of the Neumann type for elliptic equations on the positive orthant

    HITOSHI ISHII  [Invited]

    Mostly Maximum Principle at Agropoli, Italy 

    Presentation date: 2015.08

  • Metastability for parabolic equations with drift

    HITOSHI ISHII  [Invited]

    Nonlinear Elliptic PDEs at the End of the World, Chile 

    Presentation date: 2015.05

  • Metastability for parabolic equations with drift

    HITOSHI ISHII  [Invited]

    Beyond Hamilton-Jacobi in Avignon, Palais des Papes, 

    Presentation date: 2014.04

  • Large time behavior of solutions of Hamilton-Jacobi equations with Neumann type BC

    HITOSHI ISHII  [Invited]

    Dynamical Optimization in PDE and Geometry, Universite Bordeaux 1 

    Presentation date: 2011.12

  • Small stochastic perturbations of Hamiltonian flows: a PDE approach

    HITOSHI ISHII  [Invited]

    DFDE 2011, The Peoples' Friendship University of Russia, 

    Presentation date: 2011.08

  • Stochastic Perturbations to Hamiltonian Flows: a PDE approach

    HITOSHI ISHII  [Invited]

    14th Riviere-Fabes symposium, University of Minessotta 

    Presentation date: 2011.04

  • Long-time behavior of solutions of Hamilton-Jacobi equations with Neumann type boundary conditions

    HITOSHI ISHII  [Invited]

    14th Riviere-Fabes symposium, University of Minessotta 

    Presentation date: 2011.04

  • 格子転位モデルに現れる 非局所ハミルトン・ヤコビ方程式について

    HITOSHI ISHII

    日本数学会秋季総合分科会(函数方程式分科会) 大阪大学 

    Presentation date: 2009.09

  • Nonlocal Hamilton-Jacobi Equations Arising in Dislocation Dynamics

    HITOSHI ISHII  [Invited]

    Nonlinear Analysis Workshop at ANU, Australia 

    Presentation date: 2009.03

  • Nonlinear singular integral equations and approximation of p-Laplace equations

    HITOSHI ISHII  [Invited]

    2nd International Conference on Reaction-Di usion Systems & Viscosity Solutions at Providence University, Taiwan 

    Presentation date: 2008.07

  • Asymptotic solutions of Hamilton–Jacobi equations for large time and related topics

    HITOSHI ISHII  [Invited]

    ICIAM 

    Presentation date: 2007.07

  • Asymptotic solutions for large time of Hamilton-Jacobi equations

    HITOSHI ISHII  [Invited]

    ICM 

    Presentation date: 2006.08

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Specific Research

  • 完全非線形楕円型方程式の主固有値に関する研究

    2005  

     View Summary

    P.-L. Lions による論文,Bifurcation and optimal stochastic control, Nonlinear Analysis, Vol. 7, 1983年,pp. 177-207,において得られた非線形2階楕円型方程式に対する主固有値問題に関する結果を考察し,この論文における主要な証明の方法である確率制御の方法を通常の偏微分方程式理論の解析的な方法に置き換える可能性を第一に探った.さらに,その結果を確率制御の方法では扱えないアイザックス型の非線形2階楕円型方程式に応用することを研究した.このために,一般の非線形2階楕円型方程式に対する強最大値原理の確立,H. IshiiとP.-L. Lions の論文,Viscosity solutions of fully nonlinear second-order elliptic differential equations, J. Differential Equations, 83巻,1990年,pp. 26-78,で得られている解のヘルダー連続性の評価の精密化,方程式の未知関数への単調依存性がない場合の連続な解の存在定理の確立を行った.これらの結果を応用して,半固有値の存在を証明し,その性質を研究した.特に,半固有値に対する固有関数の存在の確立,半固有値と正値解の一意性との関係の確立,解の一意性とそれを保障する半固有値の定義とその存在の確立などを行った.

  • 粘性解とその応用

    2001  

     View Summary

    退化楕円型編微分方程式に対する状態拘束問題に対する粘性解の存在、一意性、解の連続性について研究し、存在と一意性のための十分条件を与えた。さらに、解の連続度についての一般的評価を与えた。一方で、確率制御に関して、状態拘束問題を考え、この問題の値関数が対応するハミルトン・ヤコビ・ベルマン方程式に対する状態拘束問題の解になっていることを証明した。この結果については、第35回中華民国数学会年会での招待講演において発表した。ガウス曲率流の一般化として、石の磨耗のモデルを考察して、石が必ずしも凸でない場合に対応する曲率流を研究した。まず、石の境界面がグラフとして記述される場合に、対応する編微分方程式の粘性解の存在と比較について比較的一般的な結果を得た。その後、レベル・セットアプローチによる石がコンパクトな場合のこの曲率流を考察し、難題であったレベル・セット法における編微分方程式の粘性解の比較定理の証明に成功し、それに基づき粘性解の存在を証明した。ボルツマン方程式の線形化方程式の漸近問題を考慮に入れ、1階の編微分方程式系(無限連立系)の漸近問題を研究した。この問題は、ランダム発展過程の制御問題と関連する。この問題について、一般的な粘性解の存在定理を単調関数族の考えを用いる斬新な方法で証明し、さらに、初期遷移層の発現する場合も込めて、漸近問題の収束、極限方程式の同定を行い、これに成功した。ペロン・フロベニウスの定理、リース・シャウダーの定理に基づく方法で、極限方程式の導出が行われた。以上が研究成果の概要である。

 

Committee Memberships

  • 2011
    -
    Now

    Journal de Mathematiques Pures et Appliquees  編集委員

  • 2008
    -
    Now

    Advances in Calculus of Variations  編集委員

  • 2000
    -
    Now

    Nonlinear Differential Equations and Applications  編集委員

  • 2018.07
    -
    2019.07

    New trends in Hamilton-Jacobi: PDE, Control, Dynamical Systems and Geometry  Scientific Committee.

  • 2011
    -
    2018

    Bulletin of Mathematical Sciences  編集委員

  • 2016.06
    -
    2017.08

    Eighth International Conference on Differential and Functional Differential Equations  Program Committee

  • 2008.10
    -
    2014.09

    日本学術会議  連携会員

  • 1996.07
    -
    2000.06

    日本数学会  国際交流委員会

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