độ trơn, tính giải tích, tính chính quy gevrey của nghiệm của phương trình nửa tuyến tính elliptic suy biến

G
a,b
k,c
f + ψ

x, y, f,
∂f
∂x
, x
k
∂f
∂y

= 0,
a, b, c ∈ C, k
G
a,b
k,c
=

∂
∂x
− iax
k
∂
∂y

∂
∂x
− ibx

k
∂
∂y

+ icx
k−1
∂
∂y
.
k
G
a,b
k,c
G
a,b
k,c
ψ
G
a,b
k,c
k
G
a,b
k,c
f + ψ

x, y, f,
∂f
∂x
, x

k
∂f
∂y

= 0.
a, b, c ∈ C, k
G
a,b
k,c
=

∂
∂x
− iax
k
∂
∂y

∂
∂x
− ibx
k
∂
∂y

+ icx
k−1
∂
∂y
.

k
G
a,b
k,c
G
a,b
k,c
ψ
G
a,b
k,c
k
G
a,b
k,c
G
a,b
k,c
G
a,b
k,c
k
G
a,b
k,c
k
A(Ω) : Ω,
C
k

(Ω) : k Ω,
D(Ω) : Ω,
D

(Ω) : D(Ω),
C
∞
(Ω) : Ω,
G
s
(Ω) : s Ω,
L
p
loc
: p,
R
n
: n .
¨o
P (D)
P (x, D)
¨o
G
k,λ
=
∂
2
∂x
2
+ x

2k
∂
2
∂y
2
+ iλx
k−1
∂
∂y
,
(x, y) ∈ Ω R
2
, λ ∈ C, i k
G
k,λ
k k
G
k,λ
G
a,b
k,c
= X
2
X
1
+ icx
k−1
∂
∂y
,

a = −1, b = 1.
X
2
=
∂
∂x
− iax
k
∂
∂y
, X
1
=
∂
∂x
− ibx
k
∂
∂y
.
G
a,b
k,c
a, b ab < 0
G
a,b
k,c
k
k
G

a,b
k,c
G
k,λ
G
k,λ
G
a,b
k,c
a, b, c
Re(a) < 0, Re(b) > 0
G
k,λ
f + ψ

x, y, f,
∂f
∂x
, x
k
∂f
∂y

= 0,
k
•
G
k,λ
• G
k,λ

•
ψ
•
ψ
G
a,b
k,c
f + ψ

x, y, f,
∂f
∂x
, x
k
∂f
∂y

= 0.
a = −1, b = 1, c = λ + k G
a,b
k,c
= G
k,λ
.
a = −1, b = 1
k
k k k k
a, b
a = −1, b = 1
a, b, c k

G
a,b
k,c
f + ψ

x, y, f,
∂f
∂x
, x
k
∂f
∂y

= 0.
a, b, c Re(a) < 0, Re(b) > 0 k
(x, y) ∈ Ω R
2
G
a,b
k,c
k
C
2
(
¯
Ω) G
a,b
k,c
G
a,b

k,c
G
m
k,loc
(Ω) S
m
loc
(Ω)
k
k
k
G
a,b
k,c
G
a,b
k,c
k
k
L
1
loc
(R
2
) x, y
G
a,b
k,c
m Ω ⊂ R
n

P (x, D) =

|α|≤m
a
α
(x)D
α
,
x = (x
1
, x
2
, , x
n
) ∈ Ω ⊂ R
n
, α = (α
1
, α
2
, , α
n
)
α
i
∈ N, |α| = α
1
+α
2
+ +α

n
, a
α
(x) ∈ C
∞
(Ω), D
α
=
∂
|α|
i
|α|
∂x
α
1
1
∂x
α
2
2
∂x
α
n
n
.
F (x) ∈ L
1
loc
(Ω)
x

0
∈ Ω P (x, D) F(x)
R
n
P (x, D)F (x) = δ(x − x
0
).
F (x, y) (x, y) ∈ (Ω ×Ω) y ∈ R
n
F (x, y) ∈ L
1
loc
(Ω) x
P (x, D)
P (x, D)F (x, y) = δ(x − y).
P (x, D) =

|α|≤m
a
α
(x)D
α
a
α
(x) ∈ C
∞
(Ω) Ω Ω

 Ω,
M u ∈ C

M
(Ω

) P(x, D)u ∈
C
∞
(Ω

) u ∈ C
∞
(Ω

).
P (x, D) Ω
Ω

 Ω, u ∈ D

(Ω

) P (x, D)u ∈ C
∞
(Ω

) u ∈ C
∞
(Ω

).
P (x, D) a

α
(x) ∈ A(Ω)
Ω Ω

 Ω, u ∈ D

(Ω

) P (x, D)u ∈ A(Ω

)
u ∈ A(Ω

).
P (x, D) a
α
(x) ∈ A(Ω)
Ω Ω

 Ω, u ∈ D

(Ω

) P (x, D)u ∈ G
s
(Ω

)
u ∈ G
s

(Ω

).
G
s
(Ω) 1 ≤ s < ∞
f(x) ∈ C
∞
(Ω) K  Ω C
α x ∈ K
|∂
α
f(x)| ≤ C
|α|+1
(α!)
s
.
(x, τ
α
)
|α|≤m
∈ (Ω

×
˜
Ω) Φ(x, ∂
α
)
|α|≤m
m

Φ(x, ∂
α
)
|α|≤m
: f(x) −→ Φ(x, ∂
α
f(x))
|α|≤m
,
Φ(x, τ
α
)
|α|≤m
∈ C
∞
(Ω ×
˜
Ω).
Φ(x, ∂
α
)
|α|≤m
Ω Ω

 Ω M
f ∈ C
M
(Ω

) Φ(x, ∂

α
f)
|α|≤m
∈ C
∞
(Ω

) f ∈ C
∞
(Ω

)
Φ(x, τ
α
)
|α|≤m
∈ A(Ω ×
˜
Ω)(G
s
(Ω ×
˜
Ω))
Φ(x, ∂
α
)
|α|≤m
Ω
Ω


 Ω, M f ∈
C
M
(Ω

) Φ(x, ∂
α
f)
|α|≤m
∈ A(Ω

)(G
s
(Ω

)) f ∈ A(Ω

)(G
s
(Ω

))
Φ(x, τ
α
)
|α|≤m
∈ A(Ω ×
˜
Ω)(G
s

(Ω ×
˜
Ω)),
Φ(x, ∂
α
)
|α|≤m
Ω Ω

 Ω, f ∈ C
∞
(Ω

)
Φ(x, ∂
α
f)
|α|≤m
∈ A(Ω

)(G
s
(Ω

)) f ∈ A(Ω

)(G
s
(Ω


))
G
a,b
k,c
G
a,b
k,c
G
a,b
k,c
= X
2
X
1
+ icx
k−1
∂
∂y
,
(x, y) ∈ R
2
; a, b, c ∈ C; Re(a) < 0; Re(b) > 0; i =
√
−1 k
X
1
=
∂
∂x
− ibx

k
∂
∂y
, X
2
=
∂
∂x
− iax
k
∂
∂y
.
f(x, y) Ω
∂
α
f(x, y)
∂x
α
,
∂
β
f(x, y)
∂y
β
,
∂
α+β
f(x, y)
∂x

α
∂y
β
,
x
γ
∂
α+β
f(x, y)
∂x
α
∂y
β
,
∂
α
1
f, ∂
β
2
f, ∂
α,β
1,2
f,
γ
∂
α,β
f.
Re(a) < 0 Re(a) > 0
A

+
= −ax
k+1
+ bu
k+1
+ i(k + 1)(y − v)
A
−
= bx
k+1
− au
k+1
− i(k + 1)(y − v)

R = (x
k+1
+ u
k+1
)
2
+ (k + 1)
2
(y − v)
2
R
1
= (x
k+1
− u
k+1

)
2
+ (k + 1)
2
(y − v)
2
R = A
+
A
−
= −ab(x
2k+2
+ u
2k+2
) + (a
2
+ b
2
)(x
k+1
u
k+1
)
+ (k + 1)
2
(y − v)
2
+ i(k + 1)(y − v)(a + b)(x
k+1
− u

k+1
)
p =

(a − b)
2
x
k+1
u
k+1
R
−1
xu = 0,
0 xu = 0,
M = A
−
c
(k+1)(b−a)
+
A
−
k(b−a)−c
(k+1)(b−a)
−
.
k Re(a) < 0 Re(b) > 0
p /∈ (1, +∞)
p = 1 ⇔ y = v, x = ± u u = 0
A, B, C, D ∈ R, C
2

+ D
2
= 0.
p =
A + iB
C + iD
=
(A + iB)(C − iD)
C
2
+ D
2
=
AC + BD + i(−AD + BC)
C
2
+ D
2
·
p AD = BC.
p p = 0 p =
A
C
p =
B
D
p = (a − b)
2
x
k+1

u
k+1

− ab(x
2k+2
+ u
2k+2
) + (a
2
+ b
2
)x
k+1
u
k+1
+ (k + 1)
2
(y − v)
2
+ i(b + a)(x
k+1
− u
k+1
)(k + 1)(y − v)

−1
.
u = 0 p = 0
u = 0 :
y = v

p =
(a
2
− b
2
)X
−ab(X
2
+ 1) + (a
2
+ b
2
)X
X =
x
k+1
u
k+1
a = m+in, m, n ∈ R, m < 0; b = c+id, c, d ∈
R, c > 0,
p =


(m−c)
2
−(n−d)
2

X +2i(m−c)(n−d)X



(−mc+nd)(X
2
+1)
+(m
2
+ c
2
− n
2
− d
2
)X

+ i

(−md − nc)(X
2
+ 1) + 2(mn + cd)X


−1
.
p
p =
[(m − c)
2
− (n − d)
2
]X

(−mc + nd)(X
2
+ 1) + (m
2
+ c
2
− n
2
− d
2
)X
,
p =
2(m − c)(n − d)X
(−md − nc)(X
2
+ 1) + 2(mn + cd)X
.

(m − c)
2
− (n − d)
2

X

(−md − nc)(X
2
+ 1) + 2(mn + cd)X


= 2(m−c)(n−d)X

(−mc+nd)(X
2
+1)+(m
2
+c
2
−n
2
−d
2
)X

.
X = 0 p = 0 ∈ (1, +∞).
X = 0
(md − nc)(c
2
+ d
2
− m
2
− n
2
)(X − 1)
2
= 0,
md −nc = 0 c
2

+ d
2
−m
2
−n
2
= 0 X
2
−2X + 1 = 0.
md − nc = 0.
p =
(a − b)
2
X
−ab(X
2
+ 1) + (a
2
+ b
2
)X
.
(b) > 0 b = 0, p
p =

a
b
− 1

2

X
−
a
b

X
2
+ 1

+

a
b

2
+ 1

X
.
a, b md − nc = 0
a
b
=
m
c
p =

m
c
− 1


2
X
−
m
c

X
2
+ 1

+

m
c

2
+ 1

X
.
X > 0, −
m
c
> 0 p
p ≤

m
c
− 1


2
X
−
m
c
2X +

m
c

2
+ 1

X
= 1.
X = 1 x
k+1
= u
k+1
x = ±u
c
2
+ d
2
−m
2
−n
2
= 0 c

2
+ d
2
= m
2
+ n
2
,
|a| = |b| = r > 0. a, b
a = re
iϕ
1
, b = re
iϕ
2
,
π
2
< ϕ
1
<
3π
2
, −
π
2
< ϕ
2
<
π

2
,
p =
r
2
(e
iϕ
1
− e
iϕ
2
)
2
X
−r
2
e
iϕ
1
+iϕ
2
(X
2
+ 1) + r
2
(e
2iϕ
1
+ e
2iϕ

2
)X
.
p
p =
−2 cos(ϕ
1
+ ϕ
2
)(1 − cos(ϕ
1
+ ϕ
2
))X
−cos(ϕ
1
+ ϕ
2
)((X
2
+ 1) − 2 cos(ϕ
1
− ϕ
2
)X)
,
p =
[sin 2ϕ
1
+ sin 2ϕ

2
− 2 sin(ϕ
1
+ ϕ
2
)]X
−sin(ϕ
1
+ ϕ
2
)(X
2
+ 1) + (sin 2ϕ
1
+ sin 2ϕ
2
)X
.
p =
2[1 − cos(ϕ
1
− ϕ
2
)]X
(X
2
+ 1) − 2 cos(ϕ
1
− ϕ
2

)X
.
X > 0
p ≤
2[1 − cos(ϕ
1
− ϕ
2
)]X
2X − 2 cos(ϕ
1
− ϕ
2
)X
= 1.
X = 1 x = ±u.
y = v p ≤ 1 x = ±u.
X = 1 x = ±u p = 1
y = v.
p =
(a − b)
2
XU
(−ab)(U
2
+ X
2
) + (a
2
+ b

2
)XU + 1 + 2i(a + b)(X − U)
.
X =
x
k+1
(k + 1)(y − v)
, U =
u
k+1
(k + 1)(y − v)
, XU ≥ 0.
a = m + in, b = c + id, p
p =
[(m − c)
2
− (n − d)
2
]XU
(−md+nc)(X
2
+U
2
)+(m
2
+c
2
−n
2
−d

2
)XU −1−(n+d)(X −U)
p =
2(m − c)(n − d)XU
(−md + nc)(X
2
+ U
2
) + 2(mn + cd)XU + (m + c)(X − U)
.
(cn − md)(m
2
+ n
2
− c
2
− d
2
)(X − U)
2
+

(m − c)(m
2
+ n
2
− c
2
− d
2

)
− 2(n − d)(nc − md)

(X − U) − 2(m −c)(n − d) = 0.
(m
2
+ n
2
− c
2
− d
2
)(nc − md) = 0
X − U =
c − m
nc − md
X − U =
2(n − d)
m
2
+ n
2
− c
2
− d
2
.
nc − md = 0, m
2
+ n

2
− c
2
− d
2
= 0,
X − U =
2(n − d)
m
2
+ n
2
− c
2
− d
2
.
m
2
+ n
2
− c
2
− d
2
= 0, nc − md = 0
X − U =
c − m
(nc − md)
.

(nc − md) = m
2
+ n
2
− c
2
− d
2
= 0,
−2(m − c)(n − d) = 0.
−2(m − c)(n − d) = 0 m = c n = d = 0
m = −c a = −b ∈ R.
p =
4b
2
XU
b
2
(X + U)
2
+ 1
< 1.
X − U =
2(n − d)
m
2
+ n
2
− c
2

− d
2
p =
XU
XU +
(m + c)
2
+ (n − d)
2
(m
2
+ n
2
− c
2
− d
2
)
2
≤ 1, XU ≥ 0.
m = −c n = d. m
2
+ n
2
− c
2
− d
2
= 0
X − U =

c − m
nc − md
,
p =
XU
XU +
−mc
(nc − md)
2
.
−mc > 0, XU ≥ 0, p ≤ 1 m = 0
c = 0
p p ≤ 1; p = 1
y = v, x = ±u, u = 0. 
E
a,b
k,c
(x, y, u, v) G
a,b
k,c
.
M = M(x, y, u, v), F (p) = F
a,b
k,c
(p(x, y, u, v)),
E
a,b
k,c
= E
a,b

k,c
(x, y, u, v) = MF (p).
E
a,b
k,c
(x, y, u, v)
G
a,b
k,c
E
a,b
k,c
(x, y, u, v) = δ(x − u, y − v).
E
a,b
k,c
G
a,b
k,c
E
a,b
k,c
= 0
G
a,b
k,c
E
a,b
k,c
(x, y, u, v) = 0

(a − b)
2
u
k+1
x
k+1
R
−2

− ab(u
k+1
− x
k+1
)
2
+
+ (k + 1)
2
(y − v)
2
+ i(k + 1)(y − v)(x
k+1
− u
k+1
)(a + b)

F

(p)
+

1
k + 1

− (a − b)
2
(2k + 1)x
k+1
u
k+1
R
−1
+ k

F

(p)
+
c(c − k(b − a)
(k + 1)
2
(b − a)
2
F (p) = 0.
p(1− p)F

(p) +

k
k + 1
−

2k + 1
k + 1
p

F

(p) +
c(c − k(b − a))
(k + 1)
2
(b − a)
2
F (p) = 0,
F (p)
p(1 − p)F

(p) +

γ −(1 + α + β)p

F

(p) − αβF (p) = 0,
α =
c
(k + 1)(b − a)
, β =
k(b − a) − c
(k + 1)(b − a)
, γ =

k
k + 1
F (p) = C
1
F

c
(k + 1)(b − a)
,
k(b − a) − c
(k + 1)(b − a)
,
k
k + 1
, p

+ C
2
p
1
k+1
F

c + b − a
(k + 1)(b − a)
,
(k + 1)(b − a) −c
(k + 1)(b − a)
,
k + 2

k + 1
, p

:= C
1
F
a,b
k,c;1
(p) + C
2
F
a,b
k,c;2
(p).
F (α, β, γ, p) C
1
, C
2
k F
a,b
k,c
(p)
p /∈ (1, +∞) u = 0 p = 0
G
a,b
k,c
E(x, y, 0, 0) = G
a,b
k,c
C

1
M(x, y, 0, 0) = C
1
G
a,b
k,c
M(x, y, 0, 0)
= −
4(b − a)
1
k+1
πΓ(
1
k+1
)C
1
Γ(
c
(k+1)(b−a)
)Γ(
k(b−a)−c
(k+1)(b−a)
)
δ(x, y),
C
1
= −
Γ(
c
(k+1)(b−a)

)Γ(
k(b−a)−c
(k+1)(b−a)
)
4(b − a)
1
k+1
πΓ(
1
k+1
)
:= C
a,b
k,c
.
u = 0, E
a,b
k,c
(x, y, u, v) F
a,b
k,c
(p)
F
a,b
k,c
(p) p /∈ (1, +∞) p → 1 p → 1
F
a,b
k,c;1
(p) = −

Γ(
1
k+1
)
Γ(
c
(k+1)(b−a)
)Γ(
k(b−a)−c
(k+1)(b−a)
)
log(1 −p) + O(1),
F
a,b
k,c;2
(p) = −
Γ(
k+2
k+1
)
Γ(
c+b−a
(k+1)(b−a)
)Γ(
(k+1)(b−a)−c
(k+1)(b−a)
)
log(1 −p) + O(1).
E
a,b

k,c
(x, y, u, v) x = u, y = v.
p
1
k+1
= ((b −a)
2
R
−1
)
1
k+1
xu → −1 (x, y) → (−u, v)
C
2
= −
Γ(
c+b−a
(k+1)(b−a)
)Γ(
(k+1)(b−a)−c
(k+1)(b−a)
)
4(b − a)
1
k+1
πΓ(
k+2
k+1
)

:= D
a,b
k,c
,
F
a,b
k,c
(p) x = −u, y = v.
c = ±[N(k + 1)(b − a)], c = ±[N(k + 1) + k](b − a),
N |C
a,b
k,c
|, |D
a,b
k,c
| F
a,b
k,c
(p)
u = 0 (x, y) = (u, v).
k a, b, c, k
k a, b, c, k
E
a,b
k,c
(x, y, u, v) = M(C
a,b
k,c
F
a,b

k,c;1
(p) + D
a,b
k,c
F
a,b
k,c;2
(p))
= −
Γ

c
(k+1)(b−a)
)Γ(
k(b−a)−c
(k+1)(b−a)
)F (
c
(k+1)(b−a)
,
k(b−a)−c
(k+1)(b−a)
,
k
k+1
, p

4(b − a)
1
k+1

πΓ(
k
k+1
)A
c
(k+1)(b−a)
+
A
k(b−a)−c
(k+1)(b−a)
−
−
xuΓ(
c+b−a
(k+1)(b−a)
)Γ(
(k+1)(b−a)−c
(k+1)(b−a)
)F (
c+b−a
(k+1)(b−a)
),
(k+1)(b−a)−c
(k+1)(b−a)
,
k+2
k+1
, p)
4(b − a)
−

1
k+1
πΓ(
k+2
k+1
)A
c+b−a
(k+1)(b−a)
+
A
(k+1)(b−a)−c
(k+1)(b−a)
−
G
a,b
k,c
k a, b, c, k
G
a,b
k,c
E
a,b
k,c
(x, y, u, v) = δ(x − u, y − v).
(u, v) ∈ R
2
.
u = 0,
x = u + r cos ϕ; y = v + r sin ϕ.
B

ε
(u, v) =

(x, y) ∈ R
2
: r < ε

,
R
2
ε
(u, v) = R
2
\B
ε
(u, v) =

(x, y) ∈ R
2
: r ≥ ε

.
E
a,b
k,c
∈ L
1
loc
(R
2

(x, y)).
R
2
ε
(u, v), E
a,b
k,c
(x, y, u, v)
E
a,b
k,c
(x, y, u, v) ∈ L
1
(R
2
ε
(u, v)).

B
ε
(u,v)
E
a,b
k,c
(x, y, u, v)dxdy.
(x, y) (u, v) u = 0
x
l
= u
l

+ (lu
l−1
cos ϕ)r +
l(l −1)u
l−2
cos
2
ϕ
2
r
2
+ o(r
2
),
A
+
= (b − a)u
k+1
+ (k + 1)(−au
k
cos ϕ + i sin ϕ)r
−
a(k + 1)ku
k−1
cos
2
ϕ
2
r
2

+ o(r
2
),
A
−
= (b − a)u
k+1
+ (k + 1)(bu
k
cos ϕ − i sin ϕ)r
+
b(k + 1)ku
k−1
cos
2
ϕ
2
r
2
+ o(r
2
),
M = (b −a)
−
k
k+1
u
−k
+ o(1),
X

1
p =
(k + 1)
2
(−bu
k
cos ϕ + i sin ϕ)
(b − a)u
k+2
r + o(r),
R = (b − a)
2
u
2k+2
+ [(b − a)
2
(k + 1)u
2k+1
cos ϕ]r
+

(b − a)
2
(k + 1)ku
2k
cos
2
ϕ
2
+ (k + 1)

2

− abu
2k
cos
2
ϕ + sin
2
ϕ
+ i(b − a)u
k
cos ϕ sin ϕ


r
2
+ o(r
2
),
1 − p =
(k + 1)
2
(−au
k
cos ϕ + i sin ϕ)(bu
k
cos ϕ − i sin ϕ)r
2
(b − a)
2

u
2k+2
+ o(r
2
).

B
ε
(u,v)
E
a,b
k,c
(x, y, u, v)dxdy
=
2π

0
ε

0

(b − a)
−
1
k+1
u
−k
+ o(1)



C
2(b − a)
1
k+1
π
log r + O(1)

rdrdϕ < ∞.

B
ε
(u,v)
E
a,b
k,c
(x, y, u, v)dxdy < ∞.
u = 0 p = 0
F (p) = 1, E
a,b
k,c
(x, y, u, v) = M(x, y, u, v).
(x, y) → (0, v)
x
k+1
= r
k+1
cos
k+1
ϕ,
A

+
= −ar
k+1
cos
k+1
ϕ + i(k + 1)r sin ϕ = r[i(k + 1) sin ϕ −ar
k
cos
k+1
ϕ],
A
−
= br
k+1
cos
k+1
ϕ − i(k + 1)r sin ϕ = r[−i(k + 1) sin ϕ + br
k
cos
k+1
ϕ],
M = r
−
k
k+1


(k + 1)
2
sin

2
ϕ

−
k
k+1
+ o(1)

.

B
ε
(u,v)
E
a,b
k,c
(x, y, u, v)dxdy = C
2π

0
ε

0
r
−
k
k+1

sin
−

k
k+1
ϕ + o(1)

rdrdϕ
= C
+∞

−∞
ε

0
r
−
k
k+1
r

2t
1 + t
2

−
k
k+1
1
1 + t
2
drdt < ∞.


B
ε
(u,v)
E
a,b
k,c
(x, y, u, v)dxdy < ∞.
E
a,b
k,c
(x, y, u, v) ∈ L
1
(R
2
(x, y))
E
a,b
k,c
G
a,b
k,c
.
G
a,b
k,c
E
a,b
k,c
(x, y, u, v) = δ(x − u, y − v),
(G

a,b
k,c
E
a,b
k,c
, ω(x, y)) = ω(u, v) ∀ω(x, y) ∈ C
∞
0
(R
2
).
(G
a,b
k,c
E
a,b
k,c
, ω(x, y)) = (E
a,b
k,c
, G
b,a
k,−c
ω(x, y)).
(E
a,b
k,c
(x, y, u, v), G
b,a
k,−c

ω(x, y)) = lim
ε→0

R
2
ε
(u,v)
E
a,b
k,c
(x, y, u, v)G
b,a
k,−c
ω(x, y)dxdy.

R
2
ε
(u,v)
E
a,b
k,c
(x, y, u, v)G
b,a
k,−c
ω(x, y)dxdy.

R
2
ε

(u,v)
E
a,b
k,c
(x, y, u, v)G
b,a
k,−c
ω(x, y)dxdy
=

R
2
ε
(u,v)
E
a,b
k,c
(x, y, u, v)

X
1
X
2
− icx
k−1
∂
∂y

ω(x, y)dxdy
=


R
2
ε
(u,v)
ω(x, y)G
a,b
k,c
E
a,b
k,c
(x, y, u, v)dxdy
+

r=ε
E
a,b
k,c
(x, y, u, v)

X
2
ω(x, y)(ν
1
− ibx
k
ν
2
) − icx
k−1

ν
2
ω(x, y)

ds
−

r=ε
(ν
1
− iax
k
ν
2
)X
1
E
a,b
k,c
(x, y, u, v)ω(x, y)ds.
B
1
(ω(x, y), a, b, c, k) = X
2
ω(x, y)(ν
1
− ibx
k
ν
2

) − icx
k−1
ν
2
ω(x, y),
B
2
(E
a,b
k,c
(x, y, u, v), a, b, c, k) = (ν
1
− iax
k
ν
2
)X
1
E
a,b
k,c
(x, y, u, v).

R
2
ε
(u,v)
E
a,b
k,c

(x, y, u, v)G
b,a
k,−c
ω(x, y)dxdy =

R
2
ε
(u,v)
ω(x, y)G
a,b
k,c
E
a,b
k,c
(x, y, u, v)dxdy
+

r=ε
E
a,b
k,c
(x, y, u, v)B
1
(ω(x, y), a, b, c, k)ds
−

r=ε
ω(x, y)B
2

(E
a,b
k,c
(x, y, u, v), a, b, c, k)ds
:= I
1
+ I
2
+ I
3
.
G
a,b
k,c
E
a,b
k,c
(x, y, u, v) = 0 R
2
ε
(u, v) I
1
= 0