Page 140 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 5

Minimal surfaces

5.1 Introduction

We start by explaining informally the problem under consideration. We want
3
to find among all surfaces Σ ⊂ R (or more generally in R n+1 , n ≥ 2)with
prescribed boundary, ∂Σ = Γ,where Γ is a Jordan curve, one that is of minimal
area.
Unfortunately the formulation of the problem in more precise terms is del-
icate. It depends on the kind of surfaces we are considering. We will consider
two types of surfaces: parametric and nonparametric surfaces. The second ones
are less general but simpler from the analytical point of view.
We start with the formulation for nonparametric (hyper)surfaces (this case
is easy to generalize to R n+1 ). These are of the form
© n+1 ª
Σ = v (x)= (x, u (x)) ∈ R : x ∈ Ω
n
with u : Ω → R and where Ω ⊂ R is a bounded domain. The surface Σ is
therefore the graph of the function u.The fact that ∂Σ is prescribed reads now
as u = u 0 on ∂Ω,where u 0 is a given function. The area of such 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
1,1
(P) inf I (u)= f (∇u (x)) dx : u ∈ u 0 + W (Ω) .
0
Ω
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