Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 268032, 27 pages
doi:10.1155/2011/268032
Research Article
Degenerate Anisotropic Differential Operators
and Applications
Ravi Agarwal,
1
Donal O’Regan,
2
and Veli Shakhmurov
3
1
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
2
Department of Mathematics, National University of Ireland, Galway, Ireland
3
Department of Electronics Engineering and Communication, Okan University, Akfirat,
Tuzla 34959 Istanbul, Turkey
Correspondence should be addressed to Veli Shakhmurov,
Received 2 December 2010; Accepted 18 January 2011
Academic Editor: Gary Lieberman
Copyright q 2011 Ravi Agarwal et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The boundary value problems for degenerate anisotropic differential operator equations with
variable coefficients are studied. Several conditions for the separability and Fredholmness in
Banach-valued L
p
spaces are given. Sharp estimates for resolvent, discreetness of spectrum, and
x
u
x
|
α:l
|
<1
A
α
x
D
α
u
x
f
x
,
1.1
where D
l
p,γ
. This fact allows us to
derive some significant spectral properties of the differential operator. For the exposition of
differential equations with bounded or unbounded operator coefficients in Banach-valued
function spaces, we refer the reader to 8, 15–25.
Let γ γx be a positive measurable weighted function on the region Ω ⊂ R
n
.Let
L
p,γ
Ω; E denote the space of all strongly measurable E-valued functions that are defined on
Ω with the norm
f
p,γ
f
L
p,γ
Ω;E
there is a positive constant C such that
1
|
Q
|
Q
γ
x
dx
1
|
Q
|
Q
γ
−1/p−1
x
dx
p−1
≤ C,
1.3
∪
{
0
}
, 0 ≤ ϕ<π. 1.4
A linear operator A is said to be ϕ-positive in a Banach space E with bound M>0if
DA is dense on E and
A λI
−1
LE
≤ M
1
|
λ
|
−1
,
1.5
for all λ ∈ S
θ
u
p
1/p
, 1 ≤ p<∞, −∞ <θ<∞.
1.6
Let E
1
and E
2
be two Banach spaces. Now, E
1
,E
2
θ,p
,0<θ<1, 1 ≤ p ≤∞will denote
interpolation spaces obtained from {E
1
,E
2
} by the K method 27, Section 1.3.1.
Boundary Value Problems 3
AsetW ⊂ BE
1
,E
j
y
T
j
u
j
E
2
dy ≤ C
1
0
m
j1
r
p,γ
R
n
; E if the map u → Φu F
−1
ΨξFu,
u ∈ SR
n
; E is well defined and extends to a bounded linear operator in L
p,γ
R
n
; E.Theset
of all multipliers in L
p,γ
R
n
; E will denoted by M
p,γ
p,γ
E.
Let
V
n
ξ : ξ
ξ
1
∈
{
0, 1
}
.
1.8
Definition 1.1. A Banach space E is said to be a space satisfying a multiplier condition if, for
any Ψ ∈ C
n
R
n
; BE,theR-boundedness of the set {ξ
β
D
β
ξ
Ψξ : ξ ∈ R
n
\ 0,β ∈ U
n
} implies
that Ψ is a Fourier multiplier in L
p,γ
R
n
; E,thatis,Ψ ∈ M
p,γ
p,γ
E for any p ∈ 1, ∞.
\ 0,β ∈ U
≤ K
1.9
implies that Ψ
h
is a uniform collection of Fourier multipliers.
Definition 1.2. The ϕ-positive operator A is said to be R-positive in a Banach space E if there
exists ϕ ∈ 0,π such that the set {AA ξI
−1
: ξ ∈ S
ϕ
} is R-bounded.
A linear operator Ax is said to be ϕ-positive in E uniformly in x if DAx is
independent of x, DAx is dense in E and AxλI
−1
≤M/1 |λ| for any λ ∈ S
ϕ
,
ϕ ∈ 0,π.
The ϕ-positive operator Ax, x ∈ G is said to be uniformly R-positive in a Banach
space E if there exists ϕ ∈ 0,π such that the set {AxAxξI
−1
: ξ ∈ S
ϕ
} is uniformly
R-bounded; that is,
sup
x∈G
R
denote the space of all compact operators from E
1
to E
2
. For E
1
E
2
E,itis
denoted by σ
∞
E.
4 Boundary Value Problems
For two sequences {a
j
}
∞
1
and {b
j
}
∞
1
of positive numbers, the expression a
j
∼ b
j
means
that there exist positive numbers C
1
∞
E.
Now, s
j
A denotes the approximation numbers of operator A see, e.g., 27,Section
1.16.1.Let
σ
q
E
1
,E
2
⎧
⎨
⎩
A : A ∈ σ
∞
E
1
,E
2
,
∞
j1
Ω; E
0
possessing
generalized derivatives D
l
k
k
u ∂
l
k
u/∂x
l
k
k
such that D
l
k
k
u ∈ L
p,γ
Ω; E with the norm
u
W
l
p,γ
Ω;E
0
,E
i
k
uxγ
k
x
k
∂/∂x
k
i
ux. Consider the following weighted spaces of func-
tions:
W
l
p,γ
G; E
A
,E
u : u ∈ L
p
G; E
A
D
l
k
k
u
L
p
G;E
.
1.14
2. Background
The embedding theorems play a key role in the perturbation theory of DOEs. For estimating
lower order derivatives, we use following embedding theorems from 24.
Theorem A1. Let α α
1
,α
2
, ,α
n
and D
α
D
k1
α
k
l
k
≤ 1, 0 ≤ μ ≤ 1 − κ, 1 <p<∞, 2.1
Boundary Value Problems 5
4Ω ⊂ R
n
is a region such that there exists a bounded linear extension operator from
W
l
p,γ
Ω; EA,E to W
l
p,γ
R
n
; EA,E.
Then, the embedding D
α
W
l
p,γ
Ω; EA,E ⊂ L
p,γ
Ω; EA
1−κ−μ
is continuous. Moreover,
Ω;E
.
2.2
Theorem A2. Suppose that all conditions of Theorem A1 are satisfied. Moreover, let γ ∈ A
p
, Ω be a
bounded region and A
−1
∈ σ
∞
E. Then, the embedding
W
l
p,γ
Ω; E
A
,E
⊂ L
p,γ
Ω; E
2.3
is compact.
Let Sp A denote the closure of the linear span of the root vectors of the linear operator A.
From 18, Theorem 3.4.1,wehavethefollowing.
Let
G
{
x
x
1
,x
2
, ,x
n
:0<x
k
<b
k
}
,γ
x
x
γ
1
1
x
γ
2
2
···x
Let I IW
l
p,β,γ
Ω; EA,E,L
p,γ
Ω; E denote the embedding operator W
l
p,β,γ
Ω; EA,E →
L
p,ν
Ω; E.
From 15, Theorem 2.8, we have the following.
6 Boundary Value Problems
Theorem A4. Let E
0
and E be two Banach spaces possessing bases. Suppose that
0 ≤ γ
k
<p− 1, 0 ≤ β
k
< 1,ν
k
− γ
k
>p
β
k
− 1
− β
k
< 1.
2.7
Then,
s
j
I
W
l
p,β,γ
G; E
0
,E
,L
p,ν
G; E
∼ j
−1/k
0
κ
0
x
|
α:l
|
<1
A
α
x
D
α
u
x
f
x
,
3.1
m
kj
i0
k
,d
k
∈
0,l
k
,
3.2
where
α
α
1
,α
2
, ,α
n
,l
l
1
,l
2
, ,l
n
,
α
1
,α
2
, ,α
n
,
D
α
D
α
1
1
D
α
2
2
···D
α
n
n
,D
i
k
u
k
< 1 −
1
p
,k 1, 2, ,n, G
k0
x
1
,x
2
, ,x
k−1
, 0,x
k1
, ,x
n
,
G
kb
x
1
,x
2
, ,x
k−1
k
j
/
k
0,b
j
,j,k 1, 2, ,n,
3.3
α
jk
, β
jk
, λ are complex numbers, a
k
are complex-valued functions on G, Ax,andA
α
x are
linear operators in E. Moreover, γ
k
and ν
k
are such that
x
k
0
l
p,γ
G; EA,E,L
kj
u 0} and satisfying
3.1 a.e. on G is said to be solution of the problem 3.1-3.2.
We say the problem 3.1-3.2 is L
p
-separable if for all f ∈ L
p
G; E, there exists a
unique solution u ∈ W
l
p,γ
G; EA,E of the problem 3.1-3.2 and a positive constant C
depending only G, p, γ, l, E, A such that the coercive estimate
n
k1
D
l
k
k
u
G; E
A
,E,L
kj
,
Qu
n
k1
a
k
x
D
l
k
k
u A
x
u
|
k
k
b
k
− x
k
−ν
k
dx
k
,k 1, 2, ,n,
3.7
the spaces L
p
G; E and W
l
p,γ
G; EA,E are mapped isomorphically onto the weighted
spaces L
p,γ
G; E and W
l
p,γ
G; EA,E, where
−ν
k
dx
k
.
3.8
Moreover, under the substitution 3.7 the problem 3.1-3.2 reduces to the nondegenerate
BVP
n
k1
a
k
τ
D
l
k
k
u
τ
A
τ
m
kj
i0
α
kji
D
i
k
u
G
k0
0,j 1, 2, ,d
k
,x
k
∈
G
k
,
m
kj
i0
,
3.9
8 Boundary Value Problems
where
G
k0
τ
1
,τ
2
, ,τ
k−1
, 0,τ
k1
, ,τ
n
,
G
kb
τ
1
,τ
2
, ,x
n
τ
,
A
τ
A
a
k
x
1
τ
,x
2
τ
, ,x
n
τ
τ
γ
x
1
τ
,x
2
τ
, ,x
n
τ
.
3.10
By denoting τ,
G,
G
k0
,
x
D
l
k
k
u
x
A
λ
x
u
x
|
α:l
|
<1
A
α
x
k
∈
G
k
,
m
kj
i0
β
kji
D
i
k
u
G
kb
0,x
k
∈ G
k
,j 1, 2, ,l
x
A λ
u
x
f
x
,
L
kj
u f
kj
,j 1, 2, ,d
k
,L
kj
u f
kj
,j 1, 2, ,l
k
− d
k
,
are complex numbers, λ is a complex
parameter, and A is a linear operator in a Banach space E.Letω
k1
,ω
k2
, ,ω
kl
k
be the roots
of the characteristic equations
a
k
ω
l
k
1 0,k 1, 2, ,n.
4.3
Boundary Value Problems 9
Now, let
F
kj
Y
k
,X
k
1−γ
k
,
l
k
l
1
,l
2
, ,l
k−1
,l
k1
, ,l
n
,γ
k
x
γ
1
1
,x
γ
2
2
, ,x
.
4.4
By applying the trace theorem 27, Section 1.8.2,wehavethefollowing.
Theorem A5. Let l
k
and j be integer numbers, 0 ≤ j ≤ l
k
−1, θ
j
1−γ
k
pj 1/pl
k
, x
k0
∈ 0,b
k
.
Then, for any u ∈ W
l
p,γ
G; E
0
,E, the transformations u → D
j
k
uG
kx
0
W
l
p,γ
G;E
0
,E
.
4.5
Proof. It is clear that
W
l
p,γ
G; E
0
,E
W
l
k
p,γ
k
0,b
k
; Y
k
,X
k
k
/
0, and
arg ω
kj
− π
≤
π
2
− ϕ, j 1, 2, ,d
k
,
arg ω
kj
≤
π
2
− ϕ, j d
k
1, ,l
k
4.7
D
i
k
u
L
p
G;E
Au
L
p
G;E
≤ M
f
L
p
G;E
,
D
i
k
u
L
p
G;E
Au
L
p
G;E
≤ M
f
L
p
G;E
ω
l
k
1 0,k 1, 2, ,n.
4.10
Condition 2. Suppose the following conditions are satisfied:
1 a
k
/
0and
arg ω
kj
− π
≤
π
2
− ϕ, j 1, 2, ,d
k
,
arg ω
kj
≤
b
k
− x
k
γ
k
,0≤ γ
k
< 1 − 1/p.
Remark 4.1. Let l 2m
k
and a
k
−1
m
k
b
k
x, where b
k
are real-valued positive functions.
Then, Condition 2 is satisfied for ϕ ∈ 0,π/2.
Consider the inhomogenous BVP 3.1-3.2;thatis,
L λ
u f, L
kj
u f
|
1−i/l
k
D
i
k
u
L
p,γ
G;E
Au
L
p,γ
G;E
≤ C
f
L
,ϕ
2
, ,ϕ
N
be a corresponding
partition of unity; that is, ϕ
j
∈ C
∞
0
, σ
j
supp ϕ
j
⊂ G
j
and
N
j1
ϕ
j
x1. Now, for
u ∈ W
l
p,γ
G; EA,E and u
j
xuxϕ
j
j
x
f
j
x
,L
ki
u
j
Φ
ki
,
4.15
where
f
j
fϕ
j
n
k1
a
k
|
|
α:l
|
<1
ϕ
j
A
α
x
D
α
u
x
,
Φ
ki
ϕ
j
L
ki
u B
ki
ϕ
j
x
A
λ
x
0j
u
j
x
F
j
,
L
ki
u
j
Φ
ki
,i 1, 2, ,l
k
,k 1, 2, ,n,
4.17
where
F
j
f
D
l
k
k
u
j
x
.
4.18
It is clear that γx ∼
n
k1
x
γ
k
k
on neighborhoods of G
j
∩ G
k0
and
γ
x
∼
n
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
j
G
j
,p,γ
Au
j
.
4.20
12 Boundary Value Problems
From the representation of F
j
, Φ
ki
and in view of the boundedness of the coefficients, we get
F
j
G
j
,p,γ
≤
f
j
G
j
,p,γ
x
− a
x
0j
D
l
k
k
u
j
x
G
j
,p,γ
,
Φ
kj
F
≤ M
L
ki
u
F
ki
L
ki
u
F
ki
.
4.21
Now, applying Theorem A1 and by using the smoothness of the coefficients of 4.16, 4.18
and choosing the diameters of σ
j
so small, we see there is an ε>0andCε such that
F
,p,γ
ε
n
k1
D
l
k
k
u
j
x
G
j
,p,γ
≤
fϕ
j
u
j
W
l
p,γ
G
j
;EA,E
≤
fϕ
j
G
j
,p,γ
ε
u
j
W
l
ki
u
F
ki
L
ki
u
F
ki
≤ M
L
ki
u
F
ki
u
0,b
kj
;Y
k
,X
k
≤ ε
u
j
W
l
k
p,γ
k
0,b
kj
;Y
k
,X
k
C
ε
G
j
,p,γ
,
4.24
where
0,b
kj
0,b
k
∩ G
j
. 4.25
Using the above estimates, we get
Φ
kj
F
ki
≤ M
G
j
,p,γ
.
4.26
Consequently, from 4.22–4.26, we have
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
j
G
p,γ
M
ε
u
j
G
j
,p,γ
C
n
k1
l
k
i1
f
ki
F
ki
Au
j
G
j
,p,γ
≤ C
f
G
j
,p,γ
u
j
G
j
,p,γ
k1
x
γ
k
k
b
k
− x
k
ν
k
,0≤ γ
k
, ν
k
< 1 − 1/p.
Consider the problem 3.11. Reasoning as in the proof of Lemma 4.2,weobtain.
Proposition 4.3. Assume Condition 3 hold and suppose that
1 Ax is a uniformly R-positive operator in E for ϕ ∈ 0,π/2, and that a
k
x are
continuous functions on
G, λ ∈ S
ϕ
,
2 AxA
−1
x ∈ CG; BE and A
∞
p,γ
G;E
Au
L
p,γ
G;E
≤ C
f
L
p,γ
G;E
,
4.29
for the solution of problem 3.11.
Let O denote the operator generated by problem 3.11 for λ 0; that is,
D
O
W
l
p,γ
<1
A
α
x
D
α
u.
4.30
Theorem 4.4. Assume that Condition 3 is satisfied and that the following hold:
1 Ax is a uniformly R-positive operator in E, and a
k
x are continuous functions on G,
2 AxA
−1
x ∈ CG; BE, and A
α
A
1−|α:l|−μ
∈ L
∞
G; BE for 0 <μ<1 −|α : l|.
Then, problem 3.11 has a unique solution u ∈ W
l
p,γ
G; EA,E for f ∈ L
p,γ
G; E and
λ ∈ S
L
p,γ
G;E
≤ C
f
L
p,γ
G;E
.
4.31
14 Boundary Value Problems
Proof. By Proposition 4.3 for u ∈ W
l
p,γ
G; EA,E, we have
n
k1
l
k
i0
|
λ
|
p,γ
.
4.32
It is clear that
u
p,γ
1
|
λ
|
L λ
u − Lu
p,γ
≤
1
|
λ
|
u
p,γ
u
W
l
p,γ
G;EA,E
.
4.34
From the above estimate, we have
n
k1
l
k
i0
|
λ
|
1−i/l
k
G; E, there is a unique
solution of the problem 3.11. We consider the smooth functions g
j
g
j
x with respect to a
partition of unity ϕ
j
ϕ
j
y on the region G that equals one on supp ϕ
j
, where supp g
j
⊂ G
j
and |g
j
x| < 1. Let us construct for all j the functions u
j
that are defined on the regions
Ω
j
G ∩ G
j
and satisfying problem 3.11. The problem 3.11 can be expressed as
n
k1
a
f
A
x
0j
− A
x
u
j
n
k1
a
k
x
− a
k
x
0j
4.36
Consider operators O
jλ
in L
p,γ
G
j
; E that are generated by the BVPs 4.17;thatis,
D
O
jλ
W
l
p,γ
G
j
; E
A
,E,L
ki
,i 1, 2, ,l
k
,k 1, 2, ,n,
O
,j 1, ,N.
4.37
Boundary Value Problems 15
By virtue of Theorem A6, the operators O
jλ
have inverses O
−1
jλ
for | arg λ|≤ϕ and
for sufficiently large |λ|. Moreover, the operators O
−1
jλ
are bounded from L
p,γ
G
j
; E to
W
l
p,γ
G
j
; EA,E,andforallf ∈ L
p,γ
G
j
; E, we have
n
AO
−1
jλ
f
L
p,γ
G
j
;E
≤ C
f
L
p,γ
G
j
;E
.
4.38
Extending u
j
to zero outside of supp ϕ
⎧
⎨
⎩
f
A
x
0j
− A
x
O
−1
jλ
n
k1
a
k
x
− a
k
.
4.40
In fact, because of the smoothness of the coefficients of the expression K
jλ
and from
the estimate 4.38,for| arg λ|≤ϕ with sufficiently large |λ|, there is a sufficiently small ε>0
such that
A
x
0j
− A
x
O
−1
jλ
υ
j
L
k
x
0j
D
l
k
k
O
−1
jλ
υ
j
L
p,γ
G
j
;E
≤ ε
υ
j
L
L
p,γ
G
j
;E
≤ ε
υ
j
W
l
p,γ
G
j
;EA,E
C
ε
υ
j
L
I − K
jλ
−1
g
j
f
L
p,γ
G
j
;E
≤
f
L
p,γ
G
j
;E
f
4.44
16 Boundary Value Problems
are solutions of 4.38. Consider the following linear operator U λ in L
p
G; E defined by
D
U λ
W
l
p,γ
G; E
A
,E,L
kj
,j 1, 2, ,l
k
,k 1, 2, ,n,
U λ
f
N
G
j
; EA,E,andfor| arg λ|≤ϕ with sufficiently large
|λ|, we have
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
U
jλ
f
p
to u
N
j1
ϕ
j
U
jλ
f gives
O
λ
u
N
j1
ϕ
j
O
λ
U
jλ
f
Φ
λ
f f
N
j1
l
k
−ν
k
U
jλ
f
|
α:l
|
<1
A
α
n
k1
α
k
ν
k
1
C
α,ν
k
D
sufficiently large |λ|, there is an ε ∈ 0, 1 such that Φ
jλ
<ε. Therefore, there exists a
bounded linear invertible operator I
N
j1
Φ
jλ
−1
; that is, we infer for all f ∈ L
p,γ
G; E
that the BVP 3.11 has a unique solution
u
x
O
−1
λ
f
N
j1
ϕ
j
O
−1
i0
|
λ
|
1−i/l
k
D
i
k
O λ
−1
BL
p,γ
G;E
A
O λ
|
λ
|
1−i/l
k
D
i
k
u
L
p
G;E
Au
L
p
G;E
≤ M
f
D
4
y
u bD
1
x
D
1
y
u a
0
u f,
m
1j
i0
α
1i
D
i
x
u
0,y
0,
m
1j
i0
i0
β
2i
u
i
x, 1
0, 0 ≤ m
2j
≤ 3,
4.52
where
D
i
x
x
α
1
1 − x
α
2
∂
∂x
1
< 0,a
2
> 0.
4.53
Theorem 4.4 implies that for each f ∈ L
p
G, problem 4.52 has a unique solution u ∈
W
l
p
G satisfying the following coercive estimate:
D
2
x
u
L
p
G
D
k
and a
k
b
k
x−1
m
k
, where b
k
are positive continuous function
on G, E C
ν
and Ax is a diagonal matrix-function with continuous components d
m
x > 0.
18 Boundary Value Problems
Then, we obtain the separability of the following BVPs for the system of anisotropic
PDEs with varying coefficients:
n
k1
−1
m
k
b
k
i0
α
kji
D
i
k
u
m
G
k0
0,
m
kj
i0
β
kji
D
i
k
u
m
G
kb
0,
j 1, 2, ,m
u
x
|
α:l
|
<1
A
α
x
D
α
u
x
f
x
,
m
kj
− d
k
,d
k
∈
0,l
k
,
5.1
where
G
{
x
x
1
,x
2
, ,x
n
, 0 <x
k
<b
k
}
,α
x
x
γ
k
k
∂
∂x
k
i
u
x
,
5.2
Consider the operator Q generated by problem 5.1.
Theorem 5.1. Let all the conditions of Theorem 4.4 hold for ν
k
0 and A
−1
∈ σ
∞
E. Then, the
operator Q is Fredholm from W
l
p,γ
κ
n
k1
1
l
k
< 1,s
j
I
E
A
,E
∼ j
−1/ν
,j 1, 2, ,∞,ν>0.
5.3
Then,
a for a sufficiently large positive d
s
j
Q d
W
l
p,γ
G; E
A
,E
,L
p
G; E
5.5
is compact and
s
j
I
W
l
p,γ
G; E
L
p
G; E
,W
l
p,γ
G; E
A
,E
× I
W
l
p,γ
G; E
A
,E
,L
20 Boundary Value Problems
Consider now the operator O in L
p,γ
G; E generated by the nondegenerate BVP
obtained from 5.1 under the mapping 3.7;thatis,
D
O
W
l
p,γ
G; E
A
,E,L
kj
,
Ou
n
k1
a
k
x
l
p,γ
G; EA,E into L
p,γ
G; E.
Result 4. Then,
a for a sufficiently large positive d
s
j
O d
−1
L
p
G; E
∼ j
−1/νκ
, 5.10
b the system of root functions of the differential operator O is complete in L
p,γ
G; E.
6. BVPs for Degenerate Quasielliptic PDE
In this section, maximal regularity properties of degenerate anisotropic differential equations
y
D
β
y
u
x, y
|
α:l
|
<1
v
α
x, y
D
α
y
u
x, y
f
x, y
β
kji
D
i
k
u
G
kb
,y
0,x
k
∈ G
k
,y∈ Ω,
j 1, 2, ,l
k
− d
k
,d
k
∈
0,l
k
,
Boundary Value Problems 21
where D
j
− i∂/∂y
j
, α
kji
, β
kji
are complex number, y y
1
, ,y
μ
∈ Ω ⊂ R
μ
and
G
{
x
x
1
,x
2
, ,x
n
, 0 <x
k
<b
1
,x
2
, ,x
k−1
, 0,x
k1
, ,x
n
,
G
kb
x
1
,x
2
, ,x
k−1
,b
k
,x
k1
, ,x
n
,
0 ≤ m
, ,x
k−1
,x
k1
, ,x
n
,G
k
j
/
k
0,b
j
,
j, k 1, 2, ,n.
6.2
Let
ΩG × Ω, p p
1
,p.Now,L
p
Ω will denote the space of all p-summable scalar-
1
dx
p/p
1
dy
1/p
< ∞.
6.3
Analogously, W
m
p
Ω denotes the Sobolev space with corresponding mixed norm.
Let ω
kj
ω
kj
x, j 1, 2, ,l
k
, k 1, 2, ,ndenote the roots of the equations
a
k
x
ω
l
∈ 1, ∞ and 2m − k>l/r
k
, ν
α
∈ L
∞
,
2 b
jβ
∈ C
2m−m
j
∂Ω for each j, β,m
j
< 2m, γ
n
k1
x
γ
k
k
b
k
− x
k
ν
k
, 0 ≤ γ
6.6
22 Boundary Value Problems
4 for each y
0
∈ ∂Ω, the local BVPs in local coordinates corresponding to y
0
η
|
α
|
2m
a
α
y
0
D
α
ϑ
y
0,
B
j0
ϑ
|
∈ R
m
and for ξ
∈ R
μ−1
with
|ξ
| |η|
/
0,
5 a
k
∈ CG, a
k
x
/
0 and
arg ω
kj
− π
≤
π
2
− ϕ, j 1, 2, ,d
G
,x∈ G.
6.8
Then,
a the following coercive estimate
n
k1
D
l
k
k
u
L
p
Ω
|
f
L
p
Ω
6.9
holds for the solution u ∈ W
l,2m
p,γ
Ω of problem 6.1,
b for λ ∈ Sϕ and for sufficiently large |λ|, there exists a resolvent Q λ
−1
and
n
k1
l
k
i0
|
λ
|
1−i/l
k
k
0 is Fredholm in L
p
Ω,
d the relation with ν
k
0
s
j
Q λ
−1
L
p,q
Ω
∼ j
−1/l
0
κ
0
,l
A
W
2m
p
1
Ω; B
j
u 0
,Au
|
β
|
≤2m
a
β
y
D
β
u
y
.
6.12
|
β
|
≤2m
a
β
y
D
β
u
y
f
y
,
B
j
u
|
β
|
<1
A
α
x
u
L
p
1
≤ C
|
α:l
|
<1
D
α
u
L
p
1
≤ ε
u
2m
p
1
G
,L
p
1
G
∼ j
−1/κ
0
. 6.16
Then, Results 3 and 4 imply assertions c, d, e.
24 Boundary Value Problems
7. Boundary Value Problems for Infinite Systems of Degenerate PDE
Consider the infinity systems of BVP for the degenerate anisotropic PDE
n
k1
a
k
x
D
|
<1
∞
j1
d
αjm
x
D
α
u
m
x
f
m
x
,x∈ G, m 1, 2, ,∞,
m
kj
i0
α
kji
D
0,x
k
∈ G
k
,j 1, 2, ,l
k
− d
k
,
7.1
where d
k
∈ 0,l
k
, a
k
are complex-valued functions, α
kji
, β
kji
are complex numbers. Let
G
{
x
x
1
,x
i
,
G
k0
x
1
,x
2
, ,x
k−1
, 0,x
k1
, ,x
n
,
α
kj
β
kj
x
1
,x
2
, ,x
k−1
,x
k1
, ,x
n
,G
k
j
/
k
0,b
j
,j,k 1, 2, ,n,
D
x
{
d
q
,
u
l
q
D
Du
l
q
∞
m1
|
d
m
u
m
|
q
1/q
< ∞
⎫
b
k
− x
k
ν
k
, 0 ≤ γ
k
, ν
k
< 1 − 1/p, p ∈ 1, ∞, a
k
∈ CG, a
k
x
/
0,
and | arg ω
kj
− π|≤π/2 − ϕ, | arg ω
kj
|≤π/2 − ϕ, j 1, 2, ,l
k
, ϕ ∈ ϕ ∈ 0,π/2, x ∈ G,
d
m
∈ CG, d
αm
∈ L
p
G; l
q
,for| arg λ|≤ϕ and sufficiently large |λ|, the problem
7.1 has a unique solution u {u
m
x}
∞
1
that belongs to the space W
l
p,γ
G, l
q
D,l
q
and
the following coercive estimate holds:
n
k1
⎡
⎣
G
∞
m1
m1
|
d
m
u
m
x
|
q
p/q
dx
⎤
⎦
1/p
≤ C
⎡
⎣
G
∞
m1
f
m
D
j
k
V λ
−1
B
A
V λ
−1
B
≤ M,
7.6
c for ν
k
,m,j 1, 2, ,∞. 7.7
It is clear that the operator A is R-positive in l
q
. The problem 7.1 can be rewritten in
the form 1.1.FromTheorem 4.4, we obtain that problem 7.1 has a unique solution
u ∈ W
l
p
G; l
q
D,l
q
for all f ∈ L
p
G; l
q
and
n
k1
l
k
i0
|
λ
|
1−i/l
k
p
G;l
q
.
7.8
From the above estimate, we obtain assertions a and b. The assertion c is obtained
from Result 4.
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