Page 154 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The Douglas-Courant-Tonelli method 141

We will show, in Step 2, that there exists a minimizer v of (D), whose components
are harmonic functions, which means that ∆v =0.Moreover, this v verifies
¡ ¢
E = G and F =0 (cf. Step 3) and hence, according to Theorem 5.17, Σ 0 = v Ω
solves Plateau problem (up to the condition v x × v y 6=0). Finally we will show,
in Step 4, that in fact a = d = D (v) and thus, since E = G, F =0 and (5.7)
holds, we will have found that Σ 0 is also of minimal area.
Step 2. We now show that (D) has a minimizer. This does not follow from
the results of the previous chapters; it would be so if we had chosen a fixed
parametrization of the boundary Γ.Since S 6= ∅,we can find a minimizing
sequence {v ν } so that
D (v ν ) → d. (5.9)

Any such sequence {v ν } will not, in general, converge. The idea is to replace v ν
v
by a harmonic function e ν such that v ν = e ν on ∂Ω.More precisely, we define
v
e v ν as the minimizer of
v
D (e ν )= min {D (v): v = v ν on ∂Ω} . (5.10)
Such a e ν exists and its components are harmonic (cf. Chapter 2). Combining
v
(5.9) and (5.10), we still have
v
D (e ν ) → d.
Without the hypotheses (S2), (S3), this new sequence {e ν } does not converge
v
either. The condition (S3) is important, since (see Exercise 5.3.1) Dirichlet
integral is invariant under any conformal transformation from Ω onto Ω;(S3)
allows to select a unique one. The hypothesis (S2) and the Courant-Lebesgue
v
lemma imply that {e ν } is a sequence of equicontinuous functions (see Courant
[24] page 103, Dierkes-Hildebrandt-Küster-Wohlrab [39] pages 235-237 or Nitsche
[78], page 257). It follows from Ascoli-Arzela theorem (Theorem 1.3) that, up
to a subsequence,
→ v uniformly.
e v ν k
Harnack theorem (see, for example, Gilbarg-Trudinger [49], page 21), a classical
property of harmonic functions, implies that v is harmonic, satisfies (S1), (S2),
(S3) and
D (v)= d.
2 2
Step 3. We next show that this map v verifies also E = G (i.e., |v x | = |v y | )
and F =0 (i.e., hv x ; v y i =0). We will use, in order to establish this fact,
the technique of variations of the independent variables that we have already
encountered in Section 2.3, when deriving the second form of the Euler-Lagrange
equation.
Since the proof of this step is a little long, we subdivide it into three substeps.