Page 18 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Model problem and some examples 5

Solutions of this problem, when they exist, are catenoids.More precisely the
minimizer is given, λ> 0 and µ denoting some constants, by
x + µ
u (x)= λ cosh .
λ
Example: Mechanical system. Consider a mechanical system with M
particles whose respective masses are m i and positions at time t are u i (t)=
3
(x i (t) ,y i (t) ,z i (t)) ∈ R , 1 ≤ i ≤ M.Let
M M
1 X 0 2 1 X ¡ 02 02 02 ¢
0
T (u )= m i |u | = m i x + y + z i
i
i
i
2 2
i=1 i=1
be the kinetic energy and denote the potential energy with U = U (t, u). Finally
let
f (t, u, ξ)= T (ξ) − U (t, u)
be the Lagrangian. In our formalism we have n =1 and N =3M.
Example: Dirichlet integral. This is the most celebrated problem of the
calculus of variations. We have here n> 1, N =1 and
½ Z ¾
1 2
(P) inf I (u)= |∇u (x)| dx : u = u 0 on ∂Ω .
2 Ω
As for every variational problem we associate a differential equation which is
nothing other than Laplace equation,namely ∆u =0.

Example: Minimal surfaces. This problem is almost as famous as the
3
preceding one. The question is 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
closed curve, one that is of minimal area. A variant of this problem is known
as Plateau problem. One can realize experimentally such surfaces by dipping a
wire into a soapy water; the surface obtained when pulling the wire out from
the water is then a minimal surface.
The precise formulation of the problem depends on the kind of surfaces that
we areconsidering. Wehaveseenabove howtowrite theproblem forminimal
surfaces of revolution. We now formulate the problem for more general surfaces.

Case 1: Nonparametric surfaces. We consider (hyper) surfaces 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. These surfaces are
therefore graphs of functions. The fact that ∂Σ is a preassigned curve Γ,reads