Page 142 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Introduction 129

More generally if n ≥ 2,wedefine (cf. Theorem 4.4.10 in Morrey [75])

⎡ 2 ⎤ 1/2
n+1 Ã ¡ 1 i−1 i+1 n+1 ¢ !
X ∂ v , ..., v ,v , ..., v
g (∇v)= ⎣ ⎦
∂ (x 1 , ..., x n )
i=1
¡ ¢
1
where ∂ u , ..., u n /∂ (x 1 , ..., x n ) stands for the determinant of the n × n ma-
¡ ¢
i
trix ∂u /∂x j . In the terminology of Section 3.5 such a function g is
1≤i,j≤n
polyconvex but not convex. The area for such a surface is therefore given by
Z
Area (Σ)= J (v)= g (∇v (x)) dx .
Ω
The problem is then, given Γ,to find a parametric surface that minimizes
(Q) inf {Area (Σ): ∂Σ = Γ} .
It is clear that problem (Q) is more general than (P). It is however a more
complicated problem than (P) for several reasons besides the geometrical ones.
Contrary to (P) it is a vectorial problem of the calculus of variations and the
Euler-Lagrange equations associated to (Q) form now a system of (n +1) partial
differential equations. Moreover, although, as for (P), any minimizer is a solution
of these equations, it is not true in general, contrary to what happens with (P),
that every solution of the Euler-Lagrange equations is necessarily a minimizer
of (Q). Finally uniqueness is also lost for (Q) in contrast with what happens for
(P).
We now come to the definition of minimal surfaces. A minimal surface will
be a solution of the Euler-Lagrange equations associated to (Q), it will turn
out that it has (see Section 5.2) zero mean curvature. We should draw the
attention to the misleading terminology (this confusion is not present in the
case of nonparametric surfaces): a minimal surface is not necessarily a surface
of minimal area, while the converse is true, namely, a surface of minimal area is
a minimal surface.
The problem of finding a minimal surface with prescribed boundary is known
as Plateau problem.
We now describe the content of the present chapter. In most part we will
only consider the case n =2. In Section 5.2 we will recall some basic facts
about surfaces, mean curvature and isothermal coordinates. We will then give
several examples of minimal surfaces. In Section 5.3 we will outline some of the
main ideas of the method of Douglas, as revised by Courant and Tonelli, for
solving Plateau problem. This method is valid only when n =2, since it uses
strongly the notion and properties of conformal mappings. In Section 5.4 we
briefly, and without proofs, mention some results of regularity, uniqueness and