Crandall, Michael G.Overview
Publication Timeline
Most widely held works by
Michael G Crandall
Nonlinear evolution equations : proceedings of a symposium
by University of Wisconsin  Madison, 1977 Symposium on Nonlinear Evolution Equations
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Book
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21 editions published between 1977 and 1978 in English and Undetermined and held by 325 WorldCat member libraries worldwide
Semigroups, theory and applications
by H Brézis
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Book
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11 editions published in 1986 in English and held by 257 WorldCat member libraries worldwide
Directions in partial differential equations
by University of WisconsinMadison
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Book
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12 editions published in 1987 in English and Undetermined and held by 253 WorldCat member libraries worldwide
Calculus of Variations and Nonlinear Partial Differential Equations With a historical overview by Elvira Mascolo
by Luigi Ambrosio
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2 editions published in 2008 in English and held by 21 WorldCat member libraries worldwide
Semigroups, theory and applications
by H Brézis
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Book
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6 editions published in 1986 in English and held by 17 WorldCat member libraries worldwide
Semigroups, theory and applications
(
Book
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4 editions published in 1986 in English and held by 10 WorldCat member libraries worldwide
Viscosity solutions of partial differential equations
by Michael G Crandall
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Visual
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3 editions published between 1990 and 1991 in English and held by 10 WorldCat member libraries worldwide Live recording of a lecture by Michael Crandall at which he presents an expository introduction to the theory of viscosity solutions of parital differential equations
Regularizing Effects for ut + A phi(u) = 0 in L1
by Michael G Crandall
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Book
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6 editions published between 1980 and 1981 in English and held by 6 WorldCat member libraries worldwide
On the relation of the operator [delta]/[delta]s + [delta]/[delta]t to evolution governed by accretive operators
by Michael G Crandall
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Book
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4 editions published in 1975 in English and held by 6 WorldCat member libraries worldwide Existence theorems for the equation du/dt + Au = f(t), where A is an accretive operator in a general Banach space X, are typically proved by showing that limits of solutions of discrete approximations to the equation exist. Here the estimates required to show this convergence are exhibited as special cases of estimates relating solutions of difference schemes for partial derivative with respect to s + partial derivative with respect to tau to exact solutions
Regularizing effects of homogeneous evolution equations
by Michael G Crandall
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Book
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4 editions published in 1980 in English and held by 6 WorldCat member libraries worldwide It is wellknown that solving the initialvalue problem for the heat equation forward in time takes a 'rough' initial temperature into a temperature which is smooth at later times t> greater than 0. One aspect of this is the validity of certain estimates on tut when u is a solution of the heat equation. In this paper we prove related estimates on nonlinear evolution equations which are governed by homogeneous nonlinearities. The results apply to classes of nonlinear diffusion equations and to conservation laws. The results are interesting from the point of view of identifying a new 'regularization' mechanism and the estimates themselves cast new light on the nature of the solutions of some initialvalue problems with rough initial data
Some relations between nonexpansive and order preserving mappings
by Michael G Crandall
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Book
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3 editions published in 1979 in English and held by 5 WorldCat member libraries worldwide It is shown that nonlinear operators which preserve the integral are order preserving if and only if they are nonexpansive in L(1) and that those wich commute with translation by a constant are order preserving if and only if they are nonexpansive in L(infinity). Examples are presented and involving partial differential equations, difference approximations and rearrangements. (Author)
Viscosity Solutions of HamiltonJacobi Equations
by Mathematics Research Center (United States. Army)
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Book
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6 editions published between 1981 and 1984 in English and held by 5 WorldCat member libraries worldwide At the classical level, when one considers boundary value problems for nonlinear scalar first order partial differential equations there are parts of the boundary where one does not expect to be able to prescribe boundary data. Likewise, uniqueness theorems can be proved for solutions which are prescribed only on parts of the boundary. However, globally defined classical solutions of first order nonlinear problems are rare, owing to the formation of shocks. This theoretical difficulty has recently been overcome for equations of HamiltonJacobi type via the development of the theory of viscosity solutions, a sort of generalized solution for which good existence and uniqueness theorems hold. This note is concerned, in the context of viscosity solutions, with identifying parts of the boundary which are irrelevant for a given equation from the point of view of requiring data in order to prove uniqueness. This involves knowing when a viscosity solution of an equation (in the viscosity sense) in the interior of the domain may be extended by continuity to a solution in the viscosity sense to points on the boundary. The results obtained to this effect are supplemented by examples delimiting their sharpness
Bifurcation, perturbation of simple eigenvalues and linearized stability
by Michael G Crandall
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Book
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4 editions published between 1972 and 1973 in English and held by 5 WorldCat member libraries worldwide 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)
Remarks on Generators of Analytic Semigroups
by Michael G Crandall
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Book
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4 editions published in 1979 in English and held by 4 WorldCat member libraries worldwide This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups in L superscript p, 1 <p <infinity, while the sufficient condition in the other characterization is meaningful in the case of nonlinear operators. (Author)
Monotone Difference Approximations for Scalar Conservation Laws
by Michael G Crandall
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Book
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4 editions published in 1979 in English and held by 4 WorldCat member libraries worldwide A complete selfcontained 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, Godunov's scheme, the upwind scheme (Differenced through stagnation points), and the LaxFriedrichs scheme together with appropriate multidimensional generalizations. (Author)
An Approximation of Integrable Functions by Step Functions with an Application
by Michael G Crandall
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Book
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4 editions published between 1978 and 1979 in English and held by 4 WorldCat member libraries worldwide A method of approximating functions which are integrable on (0, infinity) by piecewise constant functions is presented and studied in this paper. The method used and the properties established for it allow one to reduce the study of the convergence of a difference method of theoretical interest for nonlinear timedependent problems with forcing terms to the simpler study of related problems without forcing terms
Some Problems in Nonlinear Analysis
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Book
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4 editions published between 1988 and 1991 in English and held by 4 WorldCat member libraries worldwide M.G. Crandall has been working on several problems: existence questions for abstract evolution equations, existence and uniqueness for certain classes of parabolic, HamiltonJacobi, and degenerate elliptic equations, and questions related to the control of partial differential equations. P.H. Rabinowitz has been studying the existence of periodic and connecting orbits for certain nonlinear pendulum type equations. He has obtained the existence of periodic and subharmonic solutions for families of singular Hamiltonian systems. A postdoctoral fellow, S. Angenent has developed a new approach to a class of maps of interest in the study of dynamical systems. He is also working on nonlinear parabolic equations such as arise in modelling the melting of solids and on problems in population dynamics. Four predoctoral students are treating problems on periodic solutions of finite and infinite dimensional Hamiltonian systems, on variational methods to treat ordinary and partial differential equations, and on the relationship between differential games and viscosity solutions of the HamiltonJacobi equation. (KR)
Nonlinear Evolution Equations
by Michael G Crandall
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Book
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4 editions published between 1972 and 1978 in English and held by 4 WorldCat member libraries worldwide The evolution problem 0 belong to du/dt + A(t)u(t), u(s) = x, where the A(t) are nonlinear operators acting in a Banach space, is studied. Evolution operators are constructed from the A(t) under various assumptions. Basic properties of these evolution operators are established and their relationship to the evolution equation is studied. The results obtained extend several known existence theorems and provide generalized solutions of the evolution equation in more general cases. (Author)
A singular semilinear equation in L 1 (R)
by Michael G Crandall
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Book
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2 editions published in 1975 in English and held by 3 WorldCat member libraries worldwide
Viscosity solutions of partial differential equations Columbus, OH, August 1990
by Michael G Crandall
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Visual
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1 edition published in 1991 in English and held by 3 WorldCat member libraries worldwide more
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Associated Subjects
Analytic mappings Approximation theory Banach spaces Bifurcation theory Canada Convergence Difference equations Differential equations Differential equations, Elliptic Differential equations, Nonlinear Differential equations, Parabolic Differential equations, Partial Differential equations, PartialNumerical solutions Eigenvalues Evolution equations Evolution equations, Nonlinear Existence theorems Functions HamiltonJacobi equations Inequalities (Mathematics) Initial value problems Mathematical optimization Mathematics Nonlinear operators Scientists Semigroups Semigroups of operators United States Viscosity

Alternative Names
Crandall, M. G.
Crandall, M. G. (Michael G.)
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