Agnes Scott College

Dorothy L. Bernstein

Existence Theorems in Partial Differential Equations
Princeton University Press, 1950

Preface (by C. Tompkins)

I asked Dr. Bernstein to collect this set of existence theorems during the parlous times just after the war when it was apparent to a large and vociferous set of engineers that the electronic digital calculating machines they were then developing would replace all mathematicians and, indeed, all thinkers except the engineers building more machines.

At that time many navy activities faced problems of great computational extent, and they naturally examined the power of these machines to solve their problems. The Office of Naval Research had an obligation to advise these activities, and I had an obligation to advise the office of Naval Research under the terms of contract N6 onr-240 between the Office of Naval Research and Engineering Research Associates, Incorporated. In effect, much of my advice was directly to the activities desiring the equipment.


Since these elementary facts seemed to be surprisingly generally unknown, and since neither time nor energy nor budget nor knowledge permitted me to give to each of the navy's partial differential equations the attention that would be required to apply most of the knowledge already published, this list of existence theorems collected with an eye to computing was obviously desirable to present to the proprietors of problems involving partial differential equations.

In the compilation, Professor Bernstein guided herself by considering the possibility of application to calculation; however, she did not limit herself to theorems that had been proved to be reliable for use on digital computers. This consideration led straightforwardly to the constructive existence theorems contained in the literature which is reasonably easily available. They have been assembled with judgment and analysis; in particular Professor Bernstein believes that the listed theorems have been correctly proved and that they may be applicable or there is some considerable probability that they can be modified so as to be applicable to numerical methods of calculation.


The mission outlined above was not a routine one of copying theorems. The material was drawn from many sources, and not all authors were uniform or careful in defining the problems they attacked. Dr. Bernstein tried to furnish this uniformity by stating what is meant by a solution and by a concise formulation of the various boundary problems encountered. She clarified the hypotheses under which solutions are established, and she carefully defined the regions in which the solutions are valid. She formulated the lemmas connecting various problems and indicated transformations which permit the application of the solution of a given problem to solve related problems. Finally, she included a few definitions which should add greatly to the convenience of most users by eliminating the need for reference to other works.

Table of Contents

Chapter I - Introduction

Chapter II - The Initial Value Problem and the Problem of Cauchy for First Order Differential Equations

  1. Formulation of the Problem
  2. The Initial value Problem for Equations of Normal Type
  3. Solutions of Problem of Cauchy and Initial Value Problem; Equation not in Normal Form
  4. Linear and Quasi-Linear Equations
  5. Systems of First Order Equations

Chapter III - Second Order Differential Equations

  1. Definitions, Classifications, Characteristic Equations, Transformations
  2. The Quasi-Linear Hyperbolic Equation
  3. The Linear Hyperbolic Equation
  4. General Hyperbolic Equation
  5. Systems of Hyperbolic Equations in Two Variables
  6. Hyperbolic Equations in More than Two Variables
  7. Parabolic Equations
  8. Elliptic Equations
  9. General Second Order Partial Differential Equations

Chapter IV - Partial Differential Equations of Order n > 2