Michael G. Crandall

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

Viscosity solutions of Hamilton-Jacobi equations

Publisher Summary This chapter examines viscosity solutions of Hamilton–Jacobi equations. The ability to formulate an existence and uniqueness result for generality requires the ability to discuss non differential solutions of the equation, and this has not been possible before. However, the existence assertions can be proved by expanding on the arguments in the introduction concerning the relationship of the vanishing viscosity method and the notion of viscosity solutions, so users can adapt known methods here. The uniqueness is then the main new point.

Bifurcation, Perturbation of Simple Eigenvalues and Linearized Stability.

Abstract : The eigenvalue of minimum modulus of the Frechet derivative of a nonlinear operator is estimated along a bifurcating curve of zeroes of the operator. This result is applied to the study of a number of differential equations. Parallel results are developed for a class of nonlinear eigenvalue problems of positive type. (Author)

Two approximations of solutions of Hamilton-Jacobi equations

Abstract : Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. The associated initial-value problems almost never have global-time classical solutions, and one must deal with suitable generalized solutions. The correct class of generalized solutions has only recently been established by the authors. This article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates. Analogous results are proved by similar means for the method of vanishing viscosity. (Author)

On a dirichlet problem with a singular nonlinearity

Abstract : Elliptic boundary value problems of the form Lu + g(x,u) in omega and u = 0 on the boundary of omega are studied where g is singular in that g(x,r) goes to infinity uniformly as r goes to zero from above. Existence of classical and generalized solutions is established and an associated nonlinear eigenvalue problem is treated. A detailed study is made of the behaviour of the solutions and their gradients near the boundary of omega. This leads to global estimates for the modulus of continuity of solutions. (Author)

Some Continuation and Variational Methods for Positive Solutions of Nonlinear Elliptic Eigenvalue Problems.

Abstract : Continuation and variational methods are developed to construct positive solutions for nonlinear elliptic eigenvalue problems. The class of equations studied contain in particular models arising in chemical kinetics, nonlinear heat generation, and the gravitational equilibrium of polytropic stars. (Author)

Nonlinear evolution equations in Banach spaces

The evolution problem 0∈du/dt+A(t)u(t),u(s)=x, where theA(t) are nonlinear operators acting in a Banach space, is studied. Evolution operators are constructed from theA(t) under various assumptions. Basic properties of these evolution operators are established and their relationship to the evolution equation is determined. The results obtained extend several known existence theorems and provide generalized solutions of the evolution equation in more general cases.

Stabilization of Solutions of a Degenerate Nonlinear Diffusion Problem

(I) [ 4 = (u% + f(u) in (-L,L) X R+, u(tL,t)=O in R+, u(x, 0) = uo(x) in [-L,L], where m > 1 is a parameter, f is locally Lipschitz continuous, f(0) = 0, and u. is bounded. Problems of this form arise in a number of areas of science; for instance, in models for gas or fluid flow in porous media [2] and for the spread of certain biological populations [13, 161. This paper is divided into two parts. In part I we consider what may be called the motivating example, problem I*, which consists of problem I with the special choice f(u) = U(1 U)(U a) (1) for suitably restricted parameters a. We begin by describing in detail the set % = %(L) of nonnegative equilibrium solutions of problem I*. Clearly 8(L) contains the trivial solution u = 0 for all L > 0. Write 8*(L) = E(L)\(O). In the description of Z*(L) there are two critical parameter values Lo and L1 with 0 < Lo < L1 < + =J. We show that: (i) g*(L) = 4 for 0 < L < LO;

Monotone Difference Approximations for Scalar Conservation Laws.

Abstract : A complete self-contained treatment of the stability and convergence properties of conservation form, monotone difference approximations to scalar conservation laws in several space variables is developed. In particular, the authors prove that general monotone difference schemes always converge and that they converge to the physical weak solution satisfying the entropy condition. Rigorous convergence results follow for dimensional splitting algorithms when each step is approximated by a monotone difference scheme. The results are general enough to include, for instance, Godunovs scheme, the upwind scheme (Differenced through stagnation points), and the Lax-Friedrichs scheme together with appropriate multi-dimensional generalizations. (Author)

A tour of the theory of absolutely minimizing functions

A detailed analysis of the class of absolutely minimizing functions in Euclidean spaces and the relationship to the infinity Laplace equation