Hitoshi Ishii

User’s guide to viscosity solutions of second order partial differential equations

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

On lipschitz continuity of the solution mapping to the skorokhod problem, with applications

The solution m the Skorokhoci Problem defines a deieiminisiic mapping of paths that has been found to be useful in several areas of application. Typical uses of the mapping are construction and analysis of deterministic and stochastic processes that are constrained to remain in a given fixed set, such as stochastic differential equations with reflection and stochastic approximation schemes for problems with constraints In this paper we focus on the case where the set is a convex polyhedron and where the directions along which the constraint mechanism is applied arc possibly oblique and multivalued at corner points. Our goal is to characterize as completely as possible those situations in which the solution mapping is Lipschitz continuous. Our approach is geometric in nature, and shows that the Lipschitz continuity holds when a certain convex set, defined in terms of the normal directions to the faces of the polyhedron and the directions of the constraint mechanism, can be shown to exist. All previous inst...

Approximate solutions of the bellman equation of deterministic control theory

We consider an infinite horizon discounted optimal control problem and its time discretized approximation, and study the rate of convergence of the approximate solutions to the value function of the original problem. In particular we prove the rate is of order 1 as the discretization step tends to zero, provided a semiconcavity assumption is satisfied. We also characterize the limit of the optimal controls for the approximate problems within the framework of the theory of relaxed controls.

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

We present a new, direct proof of the uniqueness theorem for a class of Hamilton-Jacobi equations including the eikonal equation in geometric optics. where n G C{Q) and n{x) > 0 on fl. Here the Hamiltonian H{x,u,p) = \p\ — n{x) is independent of u, and so the uniqueness of a viscosity solution of the Dirichlet problem for (0.2) is not a direct consequence of the uniqueness theory mentioned above. It is usually deduced from the uniqueness theory after converting equation (0.2) into equation (0.1) with a strictly increasing Hamiltonian H{x,u,p) in u via a transformation of the unknown (a device of S. N. Kruzkov (9)). Indeed, if u is a viscosity solution of (0.2), then the function v = —e~u is a viscosity solution of n{x)v + \Dv\ — 0. We refer the interested reader to Kruzkov (9), and P. L. Lions (10) and (2) for the details of this approach. The main purpose here is to present a new, direct proof of the uniqueness theorem for a class of Hamilton-Jacobi equations including (0.2) as a special case in the framework of viscosity solutions. §1 contains comparison results which yield the desired uniqueness theorem. An example of application of our comparison theorem is presented in §2. Our techniques are also useful in proving the rate of convergence of the vanishing viscosity method. This subject will be discussed in (7).

Asymptotic stability and blowing up of solutions of some nonlinear equations

in a real Hilbert space H. Here 3~ and a# are subdifferentials of convex lower semicontinuous (I.s.c.) functions 40 and 4 from Hto (-co, + oo] with y, Z,!J + +t& In [l], Brezis studied the existence and uniqueness of solutions of the problem (I.I)-(1.2) when a

A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities

= 0. R A ecently, Otani [8] gave sufficient conditions to ensure existence and uniqueness of global solutions of (1.1)-(1.2). Our purpose is to study asymptotic stability and blowing up (in other words, nonexistence of global solutions) of solutions of (l.l)-( 1.2) assuming 9~ and + to be (nearly) homogeneous functions of degreep and q, 2 < p < q, respectively. The method of proofs employed here is what is called the potential well method (see [7, 9, lo]). Equation (1.1) includes as a special case nonlinear parabolic equations

Viscosity Solutions for a Class of Hamilton-Jacobi Equations in Hilbert Spaces

Abstract We illustrate the effectiveness of viscosity solution methods for Hamilton-Jacobi PDE by demonstrating a new approach to a method of W. Fleming ( [ , [ ) for proving WKB-type representations. We present new proofs of three examples, due originally to Ventcel-Freidlin, Varadhan, and Fleming.

A New Formulation of State Constraint Problems for First-Order PDEs

We study Hamilton-Jacobi equations with an unbounded term in Hilbert spaces. We introduce a new variant of the notion of viscosity solution for a class of Hamilton-Jacobi equations, and obtain comparison and existence results for viscosity solutions. We also examine the usefulness of the notion of viscosity solution in optimal control.

Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients

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.

Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics

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. 12 refs.