Page 15 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

2 Introduction

Euler and Lagrange who found a systematic way of dealing with problems in this
field by introducing what is now known as the Euler-Lagrange equation. This
work was then extended in many ways by Bliss, Bolza, Carathéodory, Clebsch,
Hahn, Hamilton, Hilbert, Kneser, Jacobi, Legendre, Mayer, Weierstrass, just to
quote a few. For an interesting historical book on the one dimensional problems
of the calculus of variations, see Goldstine [52].
In the nineteenth century and in parallel to some of the work that was men-
tioned above, probably, the most celebrated problem of the calculus of variations
emerged, namely the study of the Dirichlet integral; a problem of multiple in-
tegrals. The importance of this problem was motivated by its relationship with
the Laplace equation. Many important contributions were made by Dirichlet,
Gauss, Thompson and Riemann among others. It was Hilbert who, at the turn
of the twentieth century, solved the problem and was immediately after imitated
by Lebesgue and then Tonelli. Their methods for solving the problem were,
essentially, what are now known as the direct methods of the calculus of vari-
ations. We should also emphasize that the problem was very important in the
development of analysis in general and more notably functional analysis, mea-
sure theory, distribution theory, Sobolev spaces or partial differential equations.
This influence is studied in the book by Monna [73].
The problem of minimal surfaces has also had, almost at the same time as
the previous one, a strong influence on the calculus of variations. The problem
was formulated by Lagrange in 1762. Many attempts to solve the problem were
made by Ampère, Beltrami, Bernstein, Bonnet, Catalan, Darboux, Enneper,
Haar, Korn, Legendre, Lie, Meusnier, Monge, Müntz, Riemann, H.A. Schwarz,
Serret, Weierstrass, Weingarten and others. Douglas and Rado in 1930 gave,
simultaneously and independently, the first complete proof. One of the first two
Fields medals was awarded to Douglas in 1936 for having solved the problem.
Immediately after the results of Douglas and Rado, many generalizations and
improvements were made by Courant, Leray, Mac Shane, Morrey, Morse, Tonelli
and many others since then. We refer for historical notes to Dierkes-Hildebrandt-
Küster-Wohlrab [39] and Nitsche [78].
In 1900 at the International Congress of Mathematicians in Paris, Hilbert
formulated 23 problems that he considered to be important for the development
of mathematics in the twentieth century. Three of them (the 19th, 20th and
23rd) were devoted to the calculus of variations. These “predictions” of Hilbert
have been amply justified all along the twentieth century and the field is at the
turn of the twenty first one as active as in the previous century.
Finally we should mention that we will not speak of many important topics
of the calculus of variations such as Morse or Liusternik-Schnirelman theories.
The interested reader is referred to Ekeland [40], Mawhin-Willem [72], Struwe
[92] or Zeidler [99].