Page 149 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

136 Minimal surfaces

and hence
¯ ZZ
d ¯ u x ϕ + u y ϕ y
x
I (u + ϕ) ¯ = q dxdy =0, ∀ϕ ∈ C ∞ (Ω) .
0
d ¯ 2 2
=0 Ω 1+ u + u
x y
Since u ∈ C 2 ¡ ¢
Ω we have, after integration by parts and using the fundamental
lemma of the calculus of variations (Theorem 1.24),
⎡ ⎤ ⎡ ⎤
∂ u x ∂ u y
⎦ + ⎦ =0 in Ω (5.2)
⎣ q ⎣ q
∂x 2 2 ∂y 2 2
1+ u + u 1+ u + u
x y x y
or equivalently
¡ 2 ¢ ¡ 2 ¢
Mu = 1+ u u xx − 2u x u y u xy + 1+ u u yy =0 in Ω . (5.3)
y x
© ª
This just asserts that H =0 and hence Σ u = (x, y, u (x, y)) : (x, y) ∈ Ω is a
minimal surface.
q
2
(i) ⇒ (ii). We start by noting that the function ξ → f (ξ)= 1+ |ξ| ,
© ª
2
where ξ ∈ R , is strictly convex. So let Σ u = (x, y, u (x, y)) : (x, y) ∈ Ω be
a minimal surface. Since H =0, wehavethat u satisfies (5.2) or (5.3). Let
© ª ¡ ¢
Σ u = (x, y, u (x, y)) : (x, y) ∈ Ω with u ∈ C 2 Ω and u = u on ∂Ω.We want
to show that I (u) ≤ I (u).Since f is convex, we have
f (ξ) ≥ f (η)+ h∇f (η); ξ − ηi , ∀ξ, η ∈ R 2
and hence
1
f (u x ,u y ) ≥ f (u x , u y )+ q h(u x , u y );(u x − u x ,u y − u y )i .
2
1+ u + u 2 y
x
Integrating the above inequality and appealing to (5.2) and to the fact that
u = u on ∂Ω we readily obtain the result.
The uniqueness follows from the strict convexity of f.
We next introduce the notion of isothermal coordinates (sometimes also called
conformal parameters). This notion will help us to understand the method of
Douglas that we will discuss in the next section.
Let us start with an informal presentation. Among all the parametrizations of
a given curve the arc length plays a special role; for a given surface the isothermal
2 2
coordinates play a similar role. They are given by E = |v x | = G = |v y | and
F = hv x ; v y i =0, which means that the tangent vectors are orthogonal and have
equal norms. In general and contrary to what happens for curves, we can only
locally find such a system of coordinates (i.e., with E = G and F =0), according
to the result of Korn, Lichtenstein and Chern [21].