Page 141 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

128 Minimal surfaces

As already seen in Chapter 3, even though the function f is strictly convex
p
and f (ξ) ≥ |ξ| with p =1, we cannot use the direct methods of the calculus
of variations, since we are lead to work, because of the coercivity condition
f (ξ) ≥ |ξ|, inanonreflexive space W 1,1 (Ω). In fact, in general, there is no
1,1
minimizer of (P) in u 0 + W 0 (Ω). We therefore need a different approach to
deal with this problem.
Before going further we write the associated Euler-Lagrange equation to (P)

⎡ ⎤ ⎡ ⎤
n
∇u X ∂ u x i
(E) div q ⎦ = ⎣ q ⎦ =0
⎣
2 ∂x i 2
1+ |∇u| i=1 1+ |∇u|
or equivalently
n
³ ´ X
2
u u
(E) Mu ≡ 1+ |∇u| ∆u − u x i x j x i x j =0 .
i,j=1
The last equation is known as the minimal surface equation.If n =2 and
u = u (x, y),itreads as
¡ 2 ¢ ¡ 2 ¢
Mu = 1+ u y u xx − 2u x u y u xy + 1+ u x u yy =0 .
¡ ¢
Therefore any C 2 Ω minimizer of (P) should satisfy the equation (E) and
conversely, since the integrand f is convex. Moreover, since f is strictly convex,
the minimizer, if it exists, is unique.
The equation (E) is equivalent (see Section 5.2) to the fact that the mean
curvature of Σ, denoted by H, vanishes everywhere.
It is clear that the above problem is, geometrically, too restrictive. Indeed
if any surface can be locally represented as a graph of a function (i.e., a non-
parametric surface), it is not the case globally. We are therefore lead to consider
more general ones known as the parametric surfaces.These are sets Σ ⊂ R n+1
n
so that there exist a domain (i.e. an open and connected set) 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