Details of a Researcher - ISHII, Hitoshi

写真a

ISHII, Hitoshi

Affiliation

Faculty of Education and Integrated Arts and Sciences

Job title

Professor Emeritus

Homepage URL

http://www.f.waseda.jp/hitoshi.ishii/

Education

　

-

1975

Waseda University Graduate School, Division of Science and Engineering
　

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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

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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

Applied mathematics and statistics
Basic mathematics
Mathematical analysis
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

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Papers

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

Ann. Inst. H. Poincaré Anal. Non Linéaire 35 ( 2 ) 417 - 441 2018 [Refereed]

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 > 0 converges to a solution of the ergodic problem as lambda -> 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

Indiana Univ. Math. J. 66 ( 1 ) 315 - 367 2017 [Refereed]
On the Langevin equation with variable friction

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

Calc. Var. Partial Differential Equations 56 ( 6 ) 2017 [Refereed]
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 > 0, we establish the convergence of the whole family {u(lambda)} lambda > 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]

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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]

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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]

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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]

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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]

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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]

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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 -> 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]

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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
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 -> 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

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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

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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 -> 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

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We show that the convergence, as p --> 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)) <= 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
Simultaneous Effects of Homogenization and Vanishing Viscosity in Fully Nonlinear Elliptic Equations

Hitoshi Ishii

Funkcialaj Ekvacioj 46 ( 1 ) 63 - 88 2003

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Relaxation of Hamilton-Jacobi equations

H. Ishii, P. Loreti

Arch. Ration. Mech. Anal. 169 ( 4 ) 264 - 305 2003

DOI
Relaxation in an L^∞-optimization problem

H. Ishii, P. Loreti

Proc. Roy. Soc. Edinburgh Sect. A 133 ( 3 ) 599 - 615 2003
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 of Hamilton-Jacobi equations

H. Ishii, P. Loreti

Arch. Ration. Mech. Anal. 169 ( 4 ) 264 - 305 2003

DOI
Relaxation in an L^∞-optimization problem

H. Ishii, P. Loreti

Proc. Roy. Soc. Edinburgh Sect. A 133 ( 3 ) 599 - 615 2003
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

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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 > 0 or by the condition that X-t is an element of G for all t > 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 > 0 or by the condition that X-t is an element of G for all t > 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 --> 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

H. Ishii

Proc. Roy. Soc. Edinburgh Sect. A 131 ( 1 ) 137 - 154 2001
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 --> 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
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.
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 characterization of the existence of solutions for Hamilton-Jacobi equations

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

Appl. Math. Optim. 42 ( 1 ) 35 - 50 2000

DOI
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 characterization of the existence of solutions for Hamilton-Jacobi equations

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

Appl. Math. Optim. 42 ( 1 ) 35 - 50 2000

DOI
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
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 υ_t-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 υ_t-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

H. Ishii

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

H ISHII

JOURNAL OF FUNCTIONAL ANALYSIS 105 ( 2 ) 301 - 341 1992.05
User's guide to viscosity solutions of second order partial differential equations

M. G. Crandall, H. Ishii, P.-L. Lions

Bull. Amer. Math. Soc. (N. S.) 27 ( 1 ) 1 - 67 1992
Perron's method for monotone systems of second-order elliptic partial differential equations

H. Ishii

Differential Integral Equations 5 ( 1 ) 1 - 24 1992
User's guide to viscosity solutions of second order partial differential equations

M. G. Crandall, H. Ishii, P.-L. Lions

Bull. Amer. Math. Soc. (N. S.) 27 ( 1 ) 1 - 67 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
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 second-order elliptic PDEs

Hitoshi Ishii, Shigeaki Koike

Comm. 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 second-order elliptic PDEs

Hitoshi Ishii, Shigeaki Koike

Comm. 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 PDEs

Hitoshi Ishii

Duke Math. J. 62 ( 3 ) 633 - 661 1991
On oblique derivative problems for fully nonlinear second-order elliptic partial differential equations on nonsmooth domains

p. Dupuis, H. Ishii

Nonlinear Anal. 15 ( 12 ) 1123 - 1138 1990
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 second-order elliptic partial differential equations

Hitoshi Ishii, P.-L. Lions

J. Differential Equations 83 ( 1 ) 26 - 78 1990
On oblique derivative problems for fully nonlinear second-order elliptic partial differential equations on nonsmooth domains

p. Dupuis, H. Ishii

Nonlinear Anal. 15 ( 12 ) 1123 - 1138 1990
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 second-order elliptic partial differential equations

Hitoshi Ishii, P.-L. Lions

J. Differential Equations 83 ( 1 ) 26 - 78 1990
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 second-order elliptic PDEs

Hitoshi Ishii

Comm. Pure Appl. Math. 42 ( 1 ) 15 - 45 1989
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
Representation of solutions of Hamilton-Jacobi equations

Hitoshi Ishii

Nonlinear Anal. 12 ( 2 ) 121 - 146 1988
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
Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisited

M. G. Crandall, H. Ishii, P.-L. Lions

J. Math. Soc. Japan 39 ( 4 ) 581 - 596 1987
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
Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisited

M. G. Crandall, H. Ishii, P.-L. Lions

J. Math. Soc. Japan 39 ( 4 ) 581 - 596 1987
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 representations of solutions of Hamilton-Jacobi equations with convex Hamiltonians

H. Ishii

Recent topics in nonlinear PDE, North-Holland Math. Stud. 1985
Differential games and nonlinear first order PDE on bounded domains

L. C. Evans, H. Ishii

Manuscripta Math. 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 first order PDE on bounded domains

L. C. Evans, H. Ishii

Manuscripta Math. 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 L^p 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 L^p 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 W^{p, α} to W^{q, β} for the solutions of the Petrovskii well posed Cauchy problems.

Hitoshi Ishii

Proc. Japan Acad. 49 705 - 710 1973
Estimates from W^{p, α} to W^{q, β} for the solutions of the Petrovskii well posed Cauchy problems

Hitoshi Ishii

Proc. Japan Acad. 49 705 - 710 1973

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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

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

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
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
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
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
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

Bulletin of Mathematical Sciences 編集委員
2011

-

Now

Journal de Mathematiques Pures et Appliquees 編集委員
2011

-

Now

Bulletin of Mathematical Sciences Editor
2011

-

Now

Journal de Mathematiques Pures et Appliquees Editor
2008

-

Now

Advances in Calculus of Variations 編集委員
2008

-

Now

Advances in Calculus of Variations Editor
2000

-

Now

Nonlinear Differential Equations and Applications 編集委員
2000

-

Now

Nonlinear Differential Equations and Applications Editor
2018.07

-

2019.07

New trends in Hamilton-Jacobi: PDE, Control, Dynamical Systems and Geometry Scientific Committee.
2016.06

-

2017.08

Eighth International Conference on Differential and Functional Differential Equations Program Committee
2008.10

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2014.09

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