Page 12 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Preface to the French

Edition

The present book is a result of a graduate course that I gave at the Ecole
Polytechnique F´ed´erale of Lausanne during the winter semester of 1990–1991.
The calculus of variations is one of the classical subjects in mathematics.
Several outstanding mathematicians have contributed, over several centuries,
to its development. It is still a very alive and evolving subject. Besides its
mathematical importance and its links with other branches of mathematics, such
as geometry or differential equations, it is widely used in physics, engineering,
economics and biology. I have decided, in order to remain as unified and concise
as possible, not to speak of any applications other than mathematical ones.
Every interested reader, whether physicist, engineer or biologist, will easily see
where, in his own subject, the results of the present monograph are used. This
fact is clearly asserted by the numerous engineers and physicists that followed
the course that resulted in the present book.
Let us now examine the content of the monograph. It should first be em-
phasized that it is not a reference book. Every individual chapter can be, on its
own, the subject of a book, For example, I have written one that, essentially,
covers the subject of Chapter 3. Furthermore several aspects of the calculus
of variations are not discussed here. One of the aims is to serve as a guide in
the extensive existing literature. However, the main purpose is to help the non
specialist, whether mathematician, physicist, engineer, student or researcher, to
discover the most important problems, results and techniques of the subject.
Despite the aim of addressing the non specialists, I have tried not to sacrifice
the mathematical rigor. Most of the theorems are either fully proved or proved
under stronger, but significant, assumptions than stated.
The different chapters may be read more or less independently. In Chapter
p
1, I have recalled some standard results on spaces of functions (continuous, L or
Sobolev spaces) and on convex analysis. The reader, familiar or not with these
subjects, can, at first reading, omit this chapter and refer to it when needed in
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