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SOME CHARACTERIZATION, UNIQUENESS AND EXISTENCE RESULTS 181
assume that n is such that D 1 n contains G1 ∪ . . . ∪ Gm. Let E1, . . . , E k(n)
be the closed domains of R 2 \Ω which are contained in D 1 n, set ai = ∂Ei,
αn = a1 ∪ . . . ∪ a k(n).
Let Ln be the catenoid tangent to the cylinder H = C 1 n × R, C 1 n = ∂D 1 n,
along the circle C 1 n. Assume that Ln ∩ {z ≥ 0} is the graph of the function
vn in R2 \D1 n. We choose Rn ≥ n2 such that |∇vn| ≤ s/2 at the circle C2 n
centered at the origin with radius Rn. Let N be the unit normal vector to
the catenoid that points to the rotational axis. Set In = Hn ∩ Ln ∩ {z ≥ 0},
where Hn := C2 n× R. Set Ωn = D2 n\(E1 ∪ . . . ∪ Ek(n)), where D2 n is the disk
bounded by C2 n.
We set
Tn = t ≥ 0 | ∃ut ∈ C ∞ (Ωn), such that Q0(ut) = 0,
sup |∇ut| ≤ s, and ut|αn = 0, ut| C2 = t n .
Ωn
We have Tn = ∅ since 0 ∈ Tn and obviously sup Tn < +∞. Set tn = sup Tn.
We prove that tn ∈ Tn and that sup αn |∇utn| = s. Given t ∈ Tn, we first
observe that sup C 2 n |∇ut| ≤ s/2. In fact: Let η denote the interior unit
normal vector to C 2 n, and denote also by η its extension to R 3 \{z − axis} by
radial translation in each plane z = c, and let Nt be the unit normal vector
to the graph Gt of ut pointing upwards. Since Gt is contained in the convex
hull of its boundary, we have 〈Nt, η〉 ≥ 0 at In.
Moving Ln down if necessary, we have that Ln ∩ Gt = ∅. Going up with
Ln until it touches the circle C 2 n,t centered at the z−axis, with radius Rn,
contained in the plane z = t, we will obtain, from the maximum principle,
that 〈Nt, η〉 < 〈N, η〉 < 1 at In (recall that any vertical translation of Ln does
not intersect any curve αi). Since, by construction, Ln is given as a graph
of a function vn such that |∇vn| ≤ s/2 at C 2 n it follows that |∇ut| ≤ s/2 at
C 2 n.
Let {sm} ⊂ Tn be a sequence converging to tn as m → ∞. Since the
functions usm are uniformly bounded having uniformly bounded gradient,
standard C k estimates guarantee us the existence of a subsequence of {usm}
converging uniformly C ∞ on Ωn and to a solution w ∈ C ∞ (Ωn) of Q0 = 0
in Ωn. One of course has sup Ωn |∇w| ≤ s, w| C 2 n = tn and w|αn = 0, so that
tn ∈ Tn and w = utn.
Suppose that sup Ωn |∇utn| < s. Applying the implicit function theorem,
one can guarantee the existence of a solution u ′ ∈ C ∞ (Ωn) to Q0 = 0
vanishing at αn and taking on a value tn+ɛ on C 2 n, ɛ > 0. For ɛ small enough,
one still has sup Ωn |∇u ′ | < s. It follows that tn + ɛ ∈ T, a contradiction!
Therefore, sup Ωn |∇utn| = s. Since sup C 2 n |∇utn| ≤ s/2, we obtain, from the
gradient maximum principle, sup αn |∇utn| = s.