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Since u and ∂u are integrable over�1, the the first integral in the latter expression tends to zero,
while the second one tends to
��1
− 〈∂u, ϕ〉dsdt.
As for the last integral, we have
�
〈u, ϕ〉dz = iǫh
∂�ǫ h
∂�ǫ h
� 2π
so by the Cauchy-Schwarz inequality,
�
��
�
� �� 2π
� �
� 〈u, ϕ〉dz�
≤ ǫh |u(ǫhe
� � iθ )| 2 dθ
0
0
≤ √ 2π ǫh
〈u(ǫhe iθ ), ϕ(ǫhe iθ )〉e iθ dθ,
�1/2 �� 2π
�� 2π
0
0
|ϕ(ǫhe iθ )| 2 dθ
|u(ǫhe iθ )| 2 dθ
�1/2
�1/2
�ϕ�∞.
Then (158) implies that the latter quantity tends to zero for h → ∞. Therefore,
��1
��1
〈u, ∂ϕ〉dsdt = − 〈∂u, ϕ〉dsdt,
for every test function ϕ ∈ C ∞ c (�1,�n ). Since ∂u ∈ L p , by the regularity theory of the weak
solutions of ∂ (see Theorem 5.4 (i)), u belongs to W 1,p (�1,�n ). Since p > 2, we conclude that u
is continuous at 0.
Let Q0 and Q1 be closed submanifolds of Q, and assume that there is an isometric embedding
Q ֒→ÊN such that
Q0 = Q ∩ V0, Q1 ∩ V1,
where V0 and V1 are partially orthogonal linear subspaces ofÊN .
6.4. Proposition. Let X :�1 ∩À+ ×T ∗ Q → TT ∗ Q be a smooth vector field such that X(z, q, p)
grows at most polynomially in p, uniformly in (z, q). Let u : Cl(�1 ∩À+ )\{0} → T ∗ Q be a smooth
solution of the equation
such that
If u has finite energy,
∂Ju(z) = X(z, u(z)) ∀z ∈ Cl(�1 ∩À+ ) \ {0}, (159)
u(s) ∈ N ∗ Q0 ∀ s > 0, u(it) ∈ N ∗ Q1 ∀ t > 0.
��1∩À+
then u extends to a continuous map on Cl(�1 ∩À+ ).
|∇u| 2 dsdt < +∞, (160)
Proof. By means of the above isometric embedding, we may regard u as a�N -valued map, satisfying
the equation (159) with ∂J = ∂, the energy estimate (160), and the boundary condition
u(s) ∈ N ∗ V0 ∀ s > 0, u(it) ∈ N ∗ V1 ∀ t > 0.
By the energy estimate (160), u belongs to L p (�1 ∩À+ ,�N ) for every p < +∞: for instance, this
follows from the Poincaré inequality and the Sobolev embedding theorem on�1, after applying a
Schwarz reflection twice and after multiplying by a cut-off function vanishing on ∂�1 and equal
to 1 on a neighborhood of 0. The polynomial growth of X then implies that
X(·, u(·)) ∈ L p (�1 ∩À+ ,�N ) ∀p < +∞. (161)
Therefore, Lemma 6.3 implies that u extends to a continuous map on Cl(�1 ∩À+ ).
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