tóm tắt luận án tiến sĩ độ trơn, tính giải tích, tính chính quy gevrey của nghiệm của phương trình nửa tuyến tính elliptic suy biến

¨o
P (D)
P (x, D)
¨o
¨o
¨o
¨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
,
X
2
=
∂
∂x
− iax
k
∂
∂y
, X
1
=
∂
∂x
− ibx
k

∂
∂y
,
a = −1, b = 1
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
a = −1, b = 1
k

k k
G
a,b
k,c
G
a,b
k,c
f + ψ

x, y, f,
∂f
∂x
, x
k
∂f
∂y

= 0,
a, b, c k
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
.
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 = 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)

−
.
G
a,b
k,c
.
k Re(a) < 0 Re(b) > 0
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 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, +∞)
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
,
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
.
c = ±[N(k + 1)(b − a)], c = ±[N(k + 1) + k](b − a),
N |C
a,b
k,c
|, |D
a,b
k,c
| < ∞.
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)
−
k a, b, c, k
G
a,b
k,c
E
a,b
k,c
(x, y, u, v) = δ(x − u, y − v).
C
2
(Ω)
E
a,b
k,c


X
1
=
∂
∂u
− ibu
k
∂
∂v
,

X
2
=
∂
∂u
− iau
k
∂
∂v
,

G
a,b
k,c
=

X
2


X
1
+ icu
k−1
∂
∂v
Ω ⊂ R
2
f ∈ C
2
(Ω)
f(x, y) =

∂Ω
f(u, v)

B
2
(E
a,b
k,c
(x, y, u, v), a, b, c, k)ds
−

∂Ω
E
a,b
k,c
(x,y,u,v)


B
1
(f(u, v), a, b, c, k)ds +

Ω
E
a,b
k,c
(x, y, u, v)

G
a,b
k,c
f(u, v)dudv(1.10)
Ω

B
1
(f(u, v), a, b, c, k) = (ν
1
− iau
k
ν
2
)

X
1
f(u, v) + icu

k−1
ν
2
f(u, v)

B
2
(E
a,b
k,c
(x, y, u, v), a, b, c, k) = (ν
1
− ibu
k
ν
2
)

X
2
E
a,b
k,c
(x, y, u, v)
ν = (ν
1
, ν
2
) ∂Ω
ψ

k G
a,b
k,c
a, b, c, k
G
m
k,
(Ω) =

f ∈ L
2
(Ω) :

(α,β,γ)∈Ξ
m
k
||
γ
∂
α,β
f||
L
2
(K)
< ∞

,
K Ω
γ
∂

α,β
f := x
γ
∂
α+β
f
∂x
α
∂y
β
Ξ
m
k
=

(α, β, γ) ∈ Z
3
+
: α + β ≤ m, km ≥ γ ≥ α + (1 + k)β − m

.
ψ C
∞
m ≥ 2k + 3
k a, b, c, k G
m
k,
(Ω)
C
∞

(Ω) Ψ
a,b
k,c
f(x, y) Ω R
2
∂
α
f
∂x
α
,
∂
β
f
∂y
β
∂
α
1
f, ∂
β
2
f r
0
= 2k + 2 r ∈ Z
+
Γ
r
(α, β)
Γ

r
= Γ
1
r
∪ Γ
2
r
Γ
1
r
= {(α, β) : α ≤ r
0
, 2α + β ≤ r}
Γ
2
r
= {(α, β) : α ≥ r
0
, α + β ≤ r − r
0
}
|f, Ω|
r
= max
(α,β)∈Γ
r
|∂
α
1
∂

β
2
f, Ω| + max
(α,β)∈Γ
r
α≥1,β≥1
max
(x,y)∈
¯
Ω
|∂
α+2
1
∂
β
2
f|,
|f, Ω| = max
(x,y)∈
¯
Ω

|f| +



∂f
∂x




+



x
k
∂f
∂y




.
k a, b, c, k
ψ ∈ G
s
s ≥ 1 C
∞
(Ω)
G
s
(Ω) Ψ
a,b
k,c
ψ C
∞
(Ω)
Ω Ψ
a,b

k,c
E
a,b
k,c
V
T
, V
T
2T S
V
T
V
T




x
γ
∂
α+β
E
a,b
k,c
(x, y, u, v)
∂x
α
∂y
β





≤ CR
−
1
2
1
, ∀(α, β, γ) ∈ Ξ
1
k
. (1.32)
S
σ
N
(x,y)
(x, y) |x| ≤

2σ
N
(x, y)

1
k+1



γ
∂
α,β


X
2
E
a,b
k,c
(x, y, u, v)



≤
C
σ
k+2
k+1
N
(x, y)
. (1.36)
S
σ
N
(x,y)
(x, y) |x| ≥

2σ
N
(x, y)

1
k+1





γ
∂
α,β

X
2
E
a,b
k,c
(x, y, u, v)
u
k




≤
C
σ
2
N
(x, y)
; ∀(α, β, γ) ∈ Ξ
1
k
.

V
T
δ
V
T
δ (x, y) ∈ V
T
σ
N
(x, y) =
1
N
ρ

(x, y), S

(α, β, γ) ∈ Ξ
1
k
, (α
1
, β
1
) ∈ Γ
N+1
, α
1
≥ 1, β
1
≥ 1

C
61
max
(x,y)∈V
T
δ


γ
∂
α,β
∂
α
1
1
∂
β
1
2
f(x, y)


≤ C
61

T
1
k+1
|f, V
T

δ
|
N+1
+ H
0

H
1
δ

N−r
0
−1
(N − r
0
− 1)!
s

T
1
k+1
+
1
H
1


(α, β, γ) ∈ Ξ
1
k

C
73
max
(x,y)∈V
T
δ


γ
∂
α,β
∂
N+1
2
f(x, y)


≤ C
73

T
1
k+1


f, V
T
δ”



N+1
+
+ H
0

H
1
δ

N−r
0
−1
((N − r
0
− 1)!)
s

T
1
k+1
+
1
H
1

(α, β, γ) ∈ Ξ
1
k
C
98

max
(x,y)∈V
T
δ



γ
∂
α,β
∂
N−r
0
+1
1
f(x, y)



≤ C
98

T
1
k+1


f, V
T
δ




N+1
+
+ H
0

H
1
δ

N−r
0
−1
((N − r
0
− 1)!)
s

T
1
k+1
+
1
H
1


(α

1
, β
1
) ∈ Γ
N+1
\ Γ
N
, α
1
≥ 1, β
1
≥ 1.
C
117
max
(x,y)∈V
T
δ



∂
α
1
+2
1
∂
β
1
2

f(x, y)



≤ C
117

T
1
k+1


f, V
T
δ



N+1
+H
0

H
1
δ

N−r
0
−1


(N − r
0
− 1)!

s

T
1
k+1
+
1
H
1


.
k a, b, c, k
m ≥ 2k + 3
ψ ∈ G
s
(s ≥ 1) G
m
k,loc
(Ω)
G
s
(Ω) Ψ
a,b
k,c
ψ C

m
(Ω)
Ω Ψ
a,b
k,c
G
a,b
k,c
G
a,b
k,c
.
G
a,b
k,c
k
G
a,b
k,c
k
G
a,b
k,c
k 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) = δ(x − u) × δ(y − v)
y,

E
xx
(x, ξ, u, v) + (a + b)x
k
ξ

E
x
(x, ξ, u, v)
+

abx
2k
ξ
2
+ (kb − c)x
k−1
ξ)

E(x, ξ, u, v) = δ(x − u)e
−iξv
. (2.4)
˜
E

xx
(x, ξ, u, v) + (a + b)x
k
ξ
˜
E
x
(x, ξ, u, v)
+

abx
2k
ξ
2
+ (kb − c)x
k−1
ξ

˜
E(x, ξ, u, v) = 0. (2.5)
k ξ > 0
˜
E(x, ξ, u, v) =






































































e
−iξv
e
bξ(u
k+1
−x
k+1
)
k+1

ξ(b−a)
k+1

−1
k+1
(A(a, b, k, c))
−1
×

Γ(
1
k+1
)
Γ(
c+b−a
(k+1)(b−a)
)
Φ(
c
(k+1)(b−a)

,
k
(k+1)
,
(b−a)x
k+1
ξ
k+1
)
+x
Γ(
−1
k+1
)
Γ(
c
(k+1)(b−a)
)

ξ(b−a)
k+1

1
k+1
Φ(
b+c−a
(k+1)
,
k+2
(k+1)

,
(b−a)x
k+1
ξ
k+1
)

×

Γ(
1
k+1
)
Γ(
(k+1)(b−a)−c
(k+1)(b−a)
)
Φ(
k(b−a)−c
(k+1)(b−a)
,
k
(k+1)
,
(a−b)u
k+1
ξ
k+1
)
−u

Γ(
−1
k+1
)
Γ(
k(b−a)−c
(k+1)(b−a)
)

ξ(b−a)
k+1

1
k+1
Φ(
(k+1)(b−a)−c
(k+1)(b−a)
,
k+2
(k+1)
,
(a−b)u
k+1
ξ
k+1
)

x ≥ u,
e
−iξv

e
aξ(u
k+1
−x
k+1
)
k+1

ξ(b−a)
k+1

−1
k+1
(A(a, b, k, c))
−1
×

Γ(
1
k+1
)
Γ(
c+b−a
(k+1)(b−a)
)
Φ(
c
(k+1)(b−a)
,
k

(k+1)
,
(b−a)u
k+1
ξ
k+1
)
+u
Γ(
−1
k+1
)
Γ(
c
(k+1)(b−a)
)

ξ(b−a)
k+1

1
k+1
Φ(
b+c−a
(k+1)
,
k+2
(k+1)
,
(b−a)u

k+1
ξ
k+1
)

×

Γ(
1
k+1
)
Γ(
(k+1)(b−a)−c
(k+1)(b−a)
)
Φ(
k(b−a)−c
(k+1)(b−a)
,
k
(k+1)
,
(a−b)x
k+1
ξ
k+1
)
−x
Γ(
−1

k+1
)
Γ(
k(b−a)−c
(k+1)(b−a)
)

ξ(b−a)
k+1

1
k+1
Φ(
(k+1)(b−a)−c
(k+1)(b−a)
,
k+2
(k+1)
,
(a−b)x
k+1
ξ
k+1
)

x ≤ u.
k ξ < 0
˜
E(x, ξ, u, v) =






































































e
−iξv
e
aξ(u
k+1
−x
k+1
)
k+1

ξ(a−b)
k+1

−1
k+1

A(a, b, k, c)

−1
×

Γ(
1
k+1

)
Γ(
(k+1)(b−a)−c
(k+1)(b−a)
)
Φ(
k(b−a)−c
(k+1)(b−a)
,
k
(k+1)
,
(a−b)x
k+1
ξ
k+1
)
+ x
Γ(
−1
k+1
)
Γ(
k(b−a)−c
(k+1)(b−a)
)

ξ(a−b)
k+1


1
k+1
Φ(
(k+1)(b−a)−c
(k+1)(b−a)
,
k+2
(k+1)
,
(a−b)x
k+1
ξ
k+1
)

×

Γ(
1
k+1
)
Γ(
c+b−a
(k+1)(b−a)
)
Φ(
c
(k+1)(b−a)
,
k

(k+1)
,
(b−a)u
k+1
ξ
k+1
)
− u
Γ(
−1
k+1
)
Γ(
c
(k+1)(b−a)
)

ξ(a−b)
k+1

1
k+1
Φ(
b+c−a
(k+1)
,
k+2
(k+1)
,
(b−a)u

k+1
ξ
k+1
)

x ≥ u,
e
−iξv
e
bξ(u
k+1
−x
k+1
)
k+1

ξ(a−b)
k+1

−1
k+1
(A(a, b, k, c))
−1
×

Γ(
1
k+1
)
Γ(

(k+1)(b−a)−c
(k+1)(b−a)
)
Φ(
k(b−a)−c
(k+1)(b−a)
,
k
(k+1)
,
(a−b)u
k+1
ξ
k+1
)
+ u
Γ(
−1
k+1
)
Γ(
k(b−a)−c
(k+1)(b−a)
)

ξ(a−b)
k+1

1
k+1

Φ(
(k+1)(b−a)−c
(k+1)(b−a)
,
k+2
(k+1)
,
(a−b)u
k+1
ξ
k+1
)

×

Γ(
1
k+1
)
Γ(
c+b−a
(k+1)(b−a)
)
Φ(
c
(k+1)(b−a)
,
k
(k+1)
,

(b−a)x
k+1
ξ
k+1
)
− x
Γ(
−1
k+1
)
Γ(
c
(k+1)(b−a)
)

ξ(a−b)
k+1

1
k+1
Φ(
b+c−a
(k+1)
,
k+2
(k+1)
,
(b−a)x
k+1
ξ

k+1
)

x ≤ u.
E(x, y, u, v) =
1
2π
∞

−∞
e
iyξ
˜
E(x, ξ, u, v)dξ,
G
a,b
k,c
G
a,b
k,c
k
k G
a,b
k,c
G
a,b
k,c
k
G
a,b

k,c
G
a,b
k,c
k
G
a,b
k,c
k
k
a, b, k, c A(a, b, k, c) = 0 k
B
1
(a, b, c, k) = 0 B
2
(a, b, k, c) = 0 k a, b, k, c
(u, v) ∈ R
2
E(u, v) : C
∞
0
(R
2
) → C
E(u, v) : ϕ(x, y) ∈ C
∞
0
(R
2
) −→

1
2π

R
2

E(x, ξ, u, v) ϕ(x, ξ)dξdx ∈ C.
ϕ(x, ξ) =

R
e
iξy
ϕ(x, y)dy
E(u, v)
a, b, k, c (u, v) ∈ R
2
E(u, v) ∈ C
∞
(R
2
\ (u, v)) G
a,b
k,c
E(x, y, u, v) = δ(x − u, y − v).
a, b, k, c
|E(x, y, u, v)| ≤ C

|x
k+1
− u

k+1
|
2
+ |y − v|
2

−
k
2k+2
(2.13)
(x, y, u, v) R
4
E(u, v)
E(x, y, u, v) ∈ L
1
loc
(R
2
(x, y))
a, b, k, c
R
4
, (x, y) = (u, v) (α, β) 0 ≤
α + β ≤ 1
max




x

kβ
∂
α+β
E(x, y, u, v)
∂x
α
∂y
β



,



u
kβ
∂
α+β
E(x, y, u, v)
∂u
α
∂v
β



,




∂
α+β
E(x, y, u, v)
∂x
α
∂u
β



,



x
kβ
∂
α+β
E(x, y, u, v)
∂u
α
∂y
β



,




u
kβ
∂
α+β
E(x, y, u, v)
∂x
α
∂v
β



,



(xu)
kβ
∂
α+β
E(x, y, u, v)
∂y
α
∂v
β





≤ C
α,β

|x
k+1
− u
k+1
| + |y − v|

−
k+α+β
k+1
.
k k
k k
G
a,b
k,c
k
V
T
k
k
k
V
T
a, b, k, c k




x
γ
∂
α+β
E(x, y, u, v)
∂x
α
∂y
β



≤ C

|x
k+1
− u
k+1
|
2
+ (k + 1)
2
(y − v)
2

−
1
2
(α, β, γ) ∈ Ξ
1

k
S
σ
N
(x,y)
(x, y) |x| ≤

2σ
N
(x, y)

1
k+1
|u| ≤ 3

σ
N
(x, y)

1
k+1



x
γ
∂
α+β

X

2
E(x, y, u, v)
∂x
α
∂y
β



≤
C
σ
k+2
k+1
N
(x, y)
∀(α, β, γ) ∈ Ξ
1
k
.
S
σ
N
(x,y)
(x, y) |x| ≤ (2σ
N
(x, y))
1
k+1





x
γ
|u|
k
∂
α+β

X
2
E(x, y, u, v)
∂x
α
∂y
β




S
σ
N
(x,y)
(x,y)
≤
C
σ
2

N
(x, y)
∀(α, β, γ) ∈ Ξ
1
k
.
k a, b, c, k
ψ ∈ G
s
s ≥ 1 C
∞
(Ω)
G
s
(Ω) Ψ
a,b
k,c
ψ C
∞
(Ω)
Ω Ψ
a,b
k,c
k a, b, c, k
m ≥ 2k + 3
ψ ∈ G
s
(s ≥ 1) G
m
k,loc

(Ω)
G
s
(Ω) Ψ
a,b
k,c
ψ C
m
(Ω)
Ω Ψ
a,b
k,c
G
a,b
k,c
=

∂
∂x
− iax
k
∂
∂y

∂
∂x
− ibx
k
∂
∂y


+ icx
k−1
∂
∂y
,
a, b, c Re(a) < 0, Re(b) > 0 k
Ψ
a,b
k,c
f = G
a,b
k,c
f + ψ

x, y, f,
∂f
∂x
, x
k
∂f
∂y

= 0. (1)
Ψ
a,b
k,c
¨o
¨o
¨o