Page 158 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Regularity, uniqueness and non uniqueness 145

5.3.1 Exercises
2
Exercise 5.3.1 Let Ω ⊂ R be a bounded smooth domain and
ϕ (x, y)= (λ (x, y) ,µ (x, y))
¡ ¢
be a conformal mapping from Ω onto B.Let v ∈ C 1 B; R 3 and w = v ◦ ϕ;
show that
ZZ ZZ
h i h i
2 2 2 2
|w x | + |w y | dxdy = |v λ | + |v µ | dλdµ .
Ω B
5.4 Regularity, uniqueness and non uniqueness

We are now going to give some results without proofs. The first ones concern
the regularity of the solution found in the previous section.
We have seen how to find a minimal surface with a C ∞ parametrization. We
have seen, and we will see it again below, that several minimal surfaces may
exist. The next result gives a regularity result for all such surfaces with given
boundary (see Nitsche [78], page 274).

Theorem 5.21 If Γ is a Jordan curve of class C k,α (with k ≥ 1 an integer
and 0 <α < 1) then every solution of Plateau problem (i.e., a minimal surface
¡ ¢
whose boundary is Γ)admits a C k,α Ω parametrization.
However the most important regularity result concerns the existence of a
regular surface (i.e., with v x × v y 6=0) which solves Plateau problem? We have
seen in Section 5.3, that the method of Douglas does not answer this question.
A result in this direction is the following (see Nitsche [78], page 334).

Theorem 5.22 (i) If Γ is an analytical Jordan curve and if its total curvature
does not exceed 4π then any solution of Plateau problem is a regular minimal
surface.

(ii) If a solution of Plateau problem is of minimal area then the result remains
true without any hypothesis on the total curvature of Γ.

Remark 5.23 The second part of the theorem allows, a posteriori, to assert that
the solution found in Section 5.3 is a regular minimal surface (i.e., v x ×v y 6=0),
provided Γ is analytical.

We now turn our attention to the problem of uniqueness of minimal sur-
faces. Recall first (Theorem 5.12) that we have uniqueness when restricted to
nonparametric surfaces. For general surfaces we have the following uniqueness
result (see Nitsche [78], page 351).