Table Of Content,.
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EXISTENCE THEOREMS
IN
PARTIAL DIFFERENTIAL
EQUATIONS
• ' II
By Dorothy L. Bernstein
Annals of Mathematics Study
Number 23
ANNALS OF MATHEMATICS STUDIES
Edited by Emil Arlin and Marston Morse
7. Finite Dimensional Vector Spaces, by PAUL R. HALMOS
11. Introduction to Nonlinear Mechanics, by N. 'KRYLOFF and N, BocoLIUBOFF
14. Lectures on Differential Equations, by SOLOMON LEFSCHETZ
15. Topological Methods in the Theory of Functions of a Complex Variable,
by MARSTON MORSE
16. Transcendental Numbers, by CARL LUDWIG SIEGEL
17. Probleme General de la Stabilite du Mouve~ent, by M. A. LIAPOUNOFF
18. A Unified Theory of Special. Functions, by C. A. TRUESDELL
19. Fourier Transforms, by S. BOCHNER and K. CHANDRASEKHARAN
2.0. Contributions to the Theory of Nonlinear Oscillations, edited by
S. LEFSCHETZ
21. Functional Operators, Vol. I, by JoHN VON NEUMANN
22. Functional Operators, Vol. II, by JoHN VON NEUMANN
23. Existence Theorems in Partial Differential Equations, by DOROTHY L.
BERNSTEIN
24. Contributions to the Theory of Games, edited by A. W. TucKER
25. Contributions to Fourier Analysis, by A. ZYGMUNi>, W. TRANSUE, M. MoRsE,
A. P. CALDERON, and S. BocHNER
26. A Theory of Cross-Spaces, by ROBERT ScHATTEN
27. Isoperimetric Inequalities in Mathematical Physics, by G. PoLYA and
G. SzEco
EXISTENCE THEOREMS
IN
PARTIAL DIFFERENTIAL
EQUATIONS
By Dorothy L. Bernstein
Princeton, New Jersey
Princeton University Press
1950
Copyright, 1950, by Princeton University Press
London: Geoffrey Cumberlege, Oxford University Press
Printed in the United States of America
This study was prepared as part of a survey of large-scale
computational problems conducted by Engineering Re
search Associates, Inc., under contract with the Office of
Naval Research, contract N6onr-240. Reproduction in whole
or in part will be permitted for any purpose of the United
States Government.
FOREWORD
This report, " Existence Theorems in Partial Differential
Equations" has been prepared for the Office of Naval Research by
Engineering Research Associates, Inc.i under Contract N6onr-240.
Work under this contract included an analysis of information con
cerning problems involving extensive computations, particularly in
connection with military research, in order to determine optimum
formulation of problems for solution by computing machines. This
list of existen.ce theorems has been compiled to assist those branches
of the military which are faced with computational problems involvi1;1;;
partial differential equations.
Dr. C. B. Tompkins was in charge of the work under the
contract and arranged with Professor Dorothy L. Bernstein of the
University of Rochester for the preparation of this report.
H. T. Engstrom
Vice President
27 June 1950
PREFACE
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 calcu
lating 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 compu
tational extent, and they naturally examined the power of these machines to
solve their problems. The Office of Naval Research had an obligation to ad
vise 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.
My task was frequently one of pointing out that the machines will
not run themselves. However, I tried to make this advice as constructive as
possible, and to do this I frequently tried to estimate the size of machine
which, if produced, might be useful in the outstanding problems presented.
This, in turn, demanded that we assume a method of solutiou.
Many of the problems were problems involving partial differential
equations. The solution, in many cases, was to be brought about (according
to the vociferous engineers) by:
(1) buying a machine;
(2) replacing the differential equation by a similar difference
equation with a fine but otherwise arbitrary grid;
(3) closing the eyes, mumbling something about truncation
error and round off error; and
(4) pressing the button to stare the machine.
One of the simplest problems and one of the J>est known in which
this procedure requires either luck, prayer or further thought is an equa
tion for heat flow:
au=~ , 0 ~ t, 0 ~ x ~ 1,
ot ox2
with initial and boundary conditions:
u(O,x) f(x),
u(t,O) go ( t),
u(t,1) gi( t).
The straightforward difference equation procedure is known to
work well on this problem if the grid is properly c:hcsen; however, that
arbitrary fineness of the grid does not suffice has long been known.
Shortly and verbally, the differential equation defines die ·following
ef feet: for two adjacent values of t the convexity or the cone a vi ty
vi.
of u, considered as a function of x for these values of t, is decreased
during the passage from the earlier t to the later. The rate of flat
tening depends continuously on the convexity. However, for the obvious
difference equation treatment, the flattening for a whole t-step depends
on t.he convexity at the begiiming, and unless the t-step is short. a con
vexity may be flfittened and the modification continued until a sharp
concavity is introduced. The length of the t-step required to do this
depends on the square of the lengtl1 of the x-step, so that an arbitrary
refinement of the grid may lead from a good approximi>.t e solution to a
bad one.
Since these elementary facts seemed to be surprisingly gener
ally unknown, and since neither time nor energy nor budget nor knowledge
perruitted n,e to give to each of tht' Navy's partial different.i1•l equations
the attention that would be required to apply most of the know} f:dge al
ready 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 compilatiou, Professor Bernstein guided herself by con
sidering 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
cons tr-ucti ve existence theorems contained in the l i tera tu re whicl, is
reasonably easily available. They have been asst:r~hled with judgment and
analysis; in particular Professor Bernstein believes that t.h· l i :;t.ed
theorems have been correctly proved and that they may be appli"uble or
there is some considen-1.ble probability that they can be modified so as
to be applicable to numerical methods of calculation.
I have been somewhnt disrespectful to the optimists. Now, let
me ,warn against extravagant or harmful pessimism. The classical mathema
tical approach to the study of diffe1·cnt.iul equations is one of proving
that a problem has at least one solution, and sometime" of proving unique
ness. There are many times when eitl1er an applied mathew1t.:ilian or a pure
mathernatician may well seek a numerical solution even though he is not
certain it is obtainable in the manner chosen (or at all); the purpose may
be the application or it may be one of experimental ari t.hmetic - a calcu
lation which it is hoped will help in an effort tc guess a correct state
ment of a proposition which may later be proved. If he believes that there
is a considerable positive probability of success and if he can tell whether
his final 1·esult i~ an adequate approximation to a solution, there may be no
reason for him to demand a firm proof; however, he is obviously foolish if
he does not make use of any published material relevant to his problerr. The
purpose of this collection is to make this relevant material eFtsily avail
able in outline form.
The mission outlined above was not a routine one of copying
theorems. The material was drE>wn fr-on. many sources, and not all authors
were uniform or careful in defining the pro~;J ems they attacked. Dr.
Bernstein tried to furnish this U11i forr.1i ty by stating what is meant by a
solution and by a concise formulation of the various boundary problems
