Page 153 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

140 Minimal surfaces

Theorem 5.19 Under the above hypotheses there exists Σ 0 ∈ S so that

Area (Σ 0 ) ≤ Area (Σ) , ∀ Σ ∈ S .
¡ ¢
Moreover there exists v satisfying (S1), (S2) and (S3), such that Σ 0 = v Ω and
¡ ¢
(i) v ∈ C ∞ Ω; R 3 with ∆v =0 in Ω,
2 2
(ii) E = |v x | = G = |v y | and F = hv x ; v y i =0.
(iii) v maps the boundary ∂Ω topologically onto the Jordan curve Γ.
Remark 5.20 (i) The theorem asserts that Σ 0 is of minimal area. To solve
completely Plateau problem, we still must prove that Σ 0 is a regular surface (i.e.
v x ×v y 6=0 everywhere); we will then be able to apply Theorem 5.12 to conclude.
We will mention in the next section some results concerning this problem. We
also have a regularity result, namely that v is C ∞ and harmonic, as well as a
choice of isothermal coordinates (E = G and F =0).
(ii) The proof uses properties of conformal mappings in a significant way and
hence cannot be generalized as such to R n+1 , n ≥ 2. The results of De Giorgi,
Federer, Fleming, Morrey, Reifenberg (cf. Giusti [50], Morrey [75]) and others
deal with such a problem
Proof. We will only give the main ideas of the proof. It is divided into four
steps.
Step 1.Let Σ ∈ S and define
ZZ
Area (Σ)= J (v)= |v x × v y | dxdy
Ω
ZZ ³ ´
1 2 2
D (v) ≡ |v x | + |v y | dxdy .
2
Ω
We then trivially have
J (v) ≤ D (v) (5.7)
since we know that

p 1 1 ³ 2 2 ´
|v x × v y | = EG − F ≤ (E + G)= |v x | + |v y | . (5.8)
2
2 2
Furthermore, we have equality in (5.8) (and hence in (5.7)) if and only if E = G
and F =0 (i.e., the parametrization is given by isothermal coordinates). We
then consider the minimization problems

(D) d =inf {D (v): v satisfies (S1), (S2), (S3)}
(A) a =inf {Area (Σ): Σ ∈ S} .