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4 continuous tone images, Responses to survey mailed by the University of Michigan Alumnae Council to women who had attended the University in the period 1870-1924. The survey included information on activities, memories of faculty and college life, descriptions of influence of University on their lives, occupations, public services, etc., (dlps) 8730.0628.001, (voicesdb) 629, https://www.lib.umich.edu/about-us/policies/copyright-policy

M. G. Crandall has been working on several problems: existence questions for abstract evolution equations, existence and uniqueness for certain classes of parabolic, Hamilton-Jacobi, 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 post-doctoral 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 Hamilton-Jacobi equation.

This paper is the second in a series by the authors concerned with the theory of viscosity solutions Hamilton-Jacobi equations in infinite dimensional spaces. The first paper introduced a notion of viscosity solution appropriate for the study of Hamilton-Jacobi equations in spaces with the so-called Radon-Nikodym property and obtained uniqueness theorems under assumptions paralleling the finite dimensional theory. The main results of the current paper concern existence of solutions of stationary and time-dependent Hamilton-Jacobi equations. In order to establish these results it is necessary to overcome the difficulties associated with the fact that bounded sets are not precompact in infinite dimensions and this is done by sharp constructive estimates coupled with the use of differential games to solve regularized problems. Interest in this subject arises on the abstract side from the desire to contribute to the theory of linear partial differential equations in infinite dimensional spaces to treat natural questions raised by the finite dimensional theory. Interest also arises from potential applications to the theory of control of partial differential equations. However, the results herein do not apply directly to problems of the form arising in the control of partial differential equations, a question which wil be treated in the next paper of the series. Additional keywords: Banach spaces, Existence theory. (Author)

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 Hamilton-Jacobi 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.

The recent introduction of the theory of viscosity solutions of nonlinear first-order partial differential equations - which we will call Hamilton-Jacobi equations or HJE's here - has stimulated a very strong development of the existence and uniqueness theory of HJE's as well as a revitalization and perfection of the theory concerning the interaction between HJE's and the diverse areas in which they arise. The areas of application include the calculus of variations, control theory and differential games. This paper is the first of a series by the authors concerning the theoretical foundations of a corresponding program in infinite dimensional spaces. The basic question of what the appropriate notion of a viscosity solution should be in an infinite dimensional space is answered in spaces with the Radon-Nikodym property by observing that the finite dimensional characterization may be used essentially unchanged. Technical difficulties which arise in attempting to work with this definition because bounded continuous functions on balls in infinite dimensional spaces need not have maxima are dispatched with the aid of the variational principle which states that maxima do exist upon perturbation by an arbitrarily small linear functional.

The theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years. This development occurred because the authors established uniqueness criteria for generalized solutions - called viscosity solutions - which correctly identify the solutions sought in areas of application, including control theory, differential games and the calculus of variations. The concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept. A feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results. The current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs. It is also felt that the presentation of the comparison results, which imply uniqueness, continuous dependence, and are used to estimate moduli of continuity, has something to offer over earlier presentations in special cases.(Author)

This is a review paper which outlines the main points of the theory of nonlinear semigroups and evolution governed by accretive operators. The subject is now rather mature, so most of the principal ideas and results are not new. However, the presentation here is organized differently from that in other sources and does touch upon recent results. An attempt has been made to make this paper a pleasant route to a certain view of the subject. This manuscript represents the author's contribution to the proceedings of the Symposium on Nonlinear Functional Analysis and Applications held in Berkely in July 1983. (Author)

The very successful theory of quasilinear evolution equations, which applies to many problems of mathematical physics, has been developed by T. Kato. The theory obtains solutions of quasilinear problems via contraction mappings which are defined by means of a theory of linear evolution equations also developed by Kato. In the current work we show how the existence and continuous dependence theorems obtained by Kato can be proved by discretization in time. As opposed to earlier work in this direction, the current results are much sharper concerning the continuity properties of the solutions of the discretized problem and the strength of the norms in which they converge.

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)

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 time-dependent problems with forcing terms to the simpler study of related problems without forcing terms.

An m-accretive operator is an abstration which covers many nonlinear differential operators arising in applications. This paper shows how certain problems involving evolution equations with m-accretive operators and delay effects in the time dependence can be discussed within the existing abstract theory. Indeed, a (theoretically) simple iterative procedure is shown to converge to the desired solution. (Author), Also prepared under Grant NSC-MCS-75-21868.

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)

The nonlinear Volterra equation was studied as well as the corresponding problem with infinite delay. Under various assumptions on the nonlinear operators A, B and on the given functions a, F, f, h existence theorems are obtained for nonlinear volterra equation and nonlinear volterra equation with infinite delay, followed by results concerning boundedness and asymptotic behavior of solutions on (0 less than or equal to t less than infinity); two applications of the theory to problems of nonlinear heat flow with infinite memory are also discussed.

results are given which apply to semilinear parabolic equations. Simple examples are provided by initial-value problems for semilinear equation like u sub t = delta u + f(u). As applied to this situation, the results show that for suitable f and t greater than 0 the solution u(t,phi) of the problem with the initial-value phi varies continuously together with its space derivatives as phi varied in a weaker sense - e.g. small perturbations of phi in some L sub P-norms cause small changes in u(t,phi) in the space on continuously differentiable functions. (Author)

The theory of scalar first-order fully nonlinear partial differential equations has recently enjoyed a strong development. One major step was a proof by M. Crandall and P.-L. Lions of the uniqueness of certain generalized solutions-called 'viscosity solutions'- of problems involving such equations with the scope to accommodate applications to, for example, differential games. Following this event there has been a continuous stream of work concerning the existence, approximation and representation of viscosity solutions of Hamilton-Jacobi equations as well as interaction of the theory of viscosity solutions and areas of application (primarily control theory and differential games), and refined uniqueness results. This survey paper, which corresponds to an invited address by the first author at an international symposium on differential equations held in March 1983 at the University of Alabama-Birmingham, introduces the relevant concepts and describes the major results up to, roughly, July 1983. (Author)

Recently M. G. Crandall and P. L. Lions introduced the notion of 'viscosity solutions' of scalar nonlinear first order partial differential equations. Viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics. The value of this concept is established by the fact that very general existence, uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application. The notion of a 'viscosity solution' admits several equivalent formulations. Here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner. Moreover, by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper. (Author)

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)

The authors continue a discussion of a problem posed by Hille (1951) in a paper titled, 'On the Generation of Semigroups and the Theory of Conjugate Functions.'

DTNH2217D00031/693JJ920F000082, Interest in how drugs other than alcohol affect roadway safety has increased. Studies conducted by NHTSA and others gave insights, but a knowledge gap exists on drug prevalence among drivers, pedestrians, and bicyclists seriously or fatally injured in crashes. This study sought to fill this gap by examining drug prevalence among a large sample (N = 7,279) of seriously injured roadway users at seven trauma centers and fatally injured crash victims at four medical examiners. Overall, 55.8% of the injured or killed tested positive for one or more drugs including alcohol, and, overall, 19.9% tested positive for two or more categories of drugs. The results are a first look at drug prevalence among a large sample of seriously or fatally injured roadway users.