Page 143 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

130 Minimal surfaces

non uniqueness of minimal surfaces. In the final section we come back to the
case of nonparametric surfaces and we give some existence results.
We now briefly discuss the historical background of the problem under con-
sideration. The problem in nonparametric form was formulated and the equation
(E) of minimal surfaces was derived by Lagrange in 1762. It was immediately
understood that the problem was a difficult one. The more general Plateau
problem (the name was given after the theoretical and experimental work of
the physicist Plateau) was solved in 1930 simultaneously and independently by
Douglas and Rado. One of the first two Fields medals was awarded to Douglas
in 1936 for having solved the problem. Before that many mathematicians have
contributed to the study of the problem: Ampère, Beltrami, Bernstein, Bon-
net, Catalan, Darboux, Enneper, Haar, Korn, Legendre, Lie, Meusnier, Monge,
Müntz, Riemann, H.A. Schwarz, Serret, Weierstrass, Weingarten and others.
Immediately after the work of Douglas and Rado, we can quote Courant, Mac
Shane, Morrey, Morse, Tonelli and many others since then. It is still a very
active field.
We conclude this introduction with some comments on the bibliography. We
should firstpointoutthatwegavemanyresults without proofs and the ones that
are given are only sketched. It is therefore indispensable in this chapter, even
more than in the others, to refer to the bibliography. There are several excellent
books but, due to the nature of the subject, they are difficult to read. The most
complete to which we will refer constantly are those of Dierkes-Hildebrandt-
Küster-Wohlrab [39] and Nitsche [78]. As a matter of introduction, interesting
for a general audience, one can consult Hildebrandt-Tromba [58]. We refer also
to the monographs of Almgren [4], Courant [24], Federer [45], Gilbarg-Trudinger
[49] (for the nonparametric surfaces), Giusti [50], Morrey [75], Osserman [80]
and Struwe [91].

5.2 Generalities about surfaces

We now introduce the different types of surfaces that we will consider. We
3
will essentially limit ourselves to surfaces of R , althoughinsomeinstances we
will give some generalizations to R n+1 . Besides the references that we already
mentioned, one can consult books of differential geometry such as that of Hsiung
[61].

3
Definition 5.1 (i) A set Σ ⊂ R will be called a parametric surface (or more
simply a surface) if there exist a domain (i.e. an open and connected set) Ω ⊂ R 2
3
and a (non constant) continuous map v : Ω → R such that
¡ ¢ © 3 ª
Σ = v Ω = v (x, y) ∈ R :(x, y) ∈ Ω .