Page 19 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

6 Introduction

now as u = u 0 on ∂Ω,where u 0 is a given function. The area of such a surface
is given by
Z
Area (Σ)= I (u)= f (∇u (x)) dx
Ω
n
where, for ξ ∈ R ,we haveset
q
2
f (ξ)= 1+ |ξ| .
The problem is then written in the usual form
½ Z ¾
(P) inf I (u)= f (∇u (x)) dx : u = u 0 on ∂Ω .
Ω
Associated with (P) we have the so called minimal surface equation
n
³ ´ X
2
u u
(E) Mu ≡ 1+ |∇u| ∆u − u x i x j x i x j =0
i,j=1
which is the equation that any minimizer u of (P) should satisfy. In geometrical
terms this equation just expresses the fact that the corresponding surface Σ has
its mean curvature that vanishes everywhere.
Case 2: Parametric surfaces. Nonparametric surfaces are clearly too restric-
tive from the geometrical point of view and one is lead to consider parametric
n
surfaces.These are sets Σ ⊂ R n+1 so that there exist a domain Ω ⊂ R and a
map v : Ω → R n+1 such that
¡ ¢ © ª
Σ = v Ω = v (x): x ∈ Ω .
3
For example, when n =2 and v = v (x, y) ∈ R , ifwedenoteby v x × v y the
normal to the surface (where a × b stands for the vectorial product of a, b ∈ R 3
and v x = ∂v/∂x, v y = ∂v/∂y)we find that the area is given by
ZZ
Area (Σ)= J (v)= |v x × v y | dxdy .
Ω
In terms of the notations introduced at the beginning of the present section we
have n =2 and N =3.
2
Example: Isoperimetric inequality.Let A ⊂ R be a bounded open set
whose boundary, ∂A,is a sufficiently regular simple closed curve. Denote by
L (∂A) the length of the boundary and by M (A) the measure (the area) of A.
The isoperimetric inequality states that
2
[L (∂A)] − 4πM (A) ≥ 0 .