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The map ϕǫ is a homeomorphism of Σ + onto itself, it is biholomophic in the interior, it preserves
the upper part of the boundary i +Ê+ , while it slides the left part i[0, 1] and the lower partÊ+
by moving the corner point 0 into the real positive number
α(ǫ) := − i
arccos(1 + 2ǫ).
π
The function ǫ ↦→ α(ǫ) is invertible, and we denote by α ↦→ ǫ(α) its inverse. Moreover, ϕǫ converges
to the identity uniformly on compact subsets of Σ + for ǫ → 0. An explicit computation shows
that
If u : Σ + → T ∗Ê2n and α > 0, we define
ϕ ′ ǫ − 1 → 0 in Lp (Σ + ), if 1 < p < 4. (185)
v(z) := u(ϕ ǫ(α)(z)).
Since ϕǫ is holomorphic, ∂(u ◦ ϕǫ) = ϕ ′ ǫ · ∂u ◦ ϕǫ. Therefore, u solves the equation (181) if and
only if v solves the equation
Given 2 < p < 4, we set
∂v(z) + ϕ ′ ǫ(α) (z)f(λ, ϕ ǫ(α)(z), v(z)) = 0. (186)
W 1,p
∗ (Σ + , T ∗Ê2n ) = � v ∈ W 1,p (Σ + , T ∗Ê2n ) | v(s, 0) ∈ N ∗ ∆Ên ∀s ≥ 0,
and we consider the operator
v(s, 1) ∈ N ∗ (0) ∀s ≥ 0, v(0, t) ∈ N ∗ (0) ∀t ∈ [0, 1] � ,
F : [0, +∞[×Êk × W 1,p
∗ (Σ + , T ∗Ê2n ) → L p (Σ + , T ∗Ê2n ), F(α, λ, v) = ∂v + ϕ ′ ǫ(α) f(λ, ϕ ǫ(α)(·), v),
where ϕ0 = id. The problem of finding (α, ũ) in � M K Υ (γ1, γ2; x) with ũ close to ũ 0 is equivalent
to finding zeroes of the operator F of the form (α, λ, v) with α > 0. By (185), the operator
F is continuous, and its differential D (λ,v)F with respect to the variables (λ, v) is continuous.
The transversality assumption that ũ 0 is a non-degenerate solution of problem � M Ω K (γ1, γ2; x) is
translated into the fact that D (λ,v)F(0, 0, 0) is an isomorphism. Then the parametric inverse
maping theorem implies that there is a unique curve α ↦→ (λ(α), v(α)), 0 < α < α0, converging to
(0, 0) for α → 0, and such that (λ(α), v(α)) is the unique zero of F(α, ·, ·) in a neighborhood of
(0, 0). This concludes the proof of statement (ii).
6.5 Proof of Proposition 4.7
The setting. We recall the setting of section 4.4. Let γ1 ∈ P(L1), γ2 ∈ P(L2), and x ∈
P Θ (H1 ⊕H2). If α ≥ 0, M K α (γ1, γ2; x) is the space of solutions u : [0, +∞[×[0, 1] → T ∗ M 2 of the
equation
satisfying the boundary conditions
∂H1⊕H2,J(u) = 0, (187)
π ◦ u(0, ·) ∈ W u (γ1; −grad g1ËΛ L1 ) × W u (γ2; −grad g2ËΛ L2 ), (188)
(u(s, 0), −u(s, 1)) ∈ N ∗ ∆M×M if 0 ≤ s < α, (189)
(u(s, 0), −u(s, 1)) ∈ N ∗ ∆ Θ M if s ≥ α, (190)
99
lim u(s, ·) = x. (191)
s→+∞