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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
completeness of root elements of the corresponding differential operators are obtained. In the last
section, some applications of the main results are given.
1. Introduction and Notations
It is well known that many classes of PDEs, pseudo-Des, and integro-DEs can be expressed
as differential-operator equations DOEs. As a result, many authors investigated PDEs as
a result of single DOEs. DOEs in H-valued Hilbert space valued function spaces have
been studied extensively in the literature see 1–14 and the references therein. Maximal
regularity properties for higher-order degenerate anisotropic DOEs with constant coefficients
and nondegenerate equations with variable coefficients were studied in 15, 16.
The main aim of the present paper is to discuss the separability properties of BVPs for
higher-order degenerate DOEs; that is,
n
k1
a
k
x
D
l
k
k
u
x
A
x
u
x
|
α:l
|
<1
A
α
x
D
α
u
x
f
x
,
1.1
where D
i
k
uxγ
k
x
k
∂/∂x
k
i
ux, γ
k
are weighted functions, A and A
α
are linear
operators in a Banach Space E. The above DOE is a generalized form of an elliptic equation.
In fact, the special case l
k
2m, k 1, ,nreduces 1.1 to elliptic form.
2 Boundary Value Problems
Note, the principal part of the corresponding differential operator is nonself-
adjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent, Fredholmness,
discreetness of the spectrum, and completeness of root elements of this operator are
established.
We prove that the corresponding differential operator is separable in L
p
; that is, it has
a bounded inverse from L
p
to the anisotropic weighted space W
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
f
x
p
E
γ
x
dx
1/p
, 1 ≤ p<∞.
1.2
For γx ≡ 1, the space L
p,γ
Ω; E will be denoted by L
p
Ω; E.
The weight γ we will consider satisfies an A
p
condition; that is, γ ∈ A
p
,1<p<∞ if
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
for all cubes Q ⊂ R
n
.
The Banach space E is called a UMD space if the Hilbert operator Hfx
lim
ε → 0
|x−y|>ε
fy/x−ydy is bounded in L
p
R, E, p ∈ 1, ∞see, e.g., 26. UMD spaces
include, for example, L
p
, l
p
spaces, and Lorentz spaces L
pq
, p, q ∈ 1, ∞.
Let C be the set of complex numbers and
S
ϕ
λ; λ ∈ C,
arg λ
≤ ϕ
∪
{
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
ϕ
,ϕ∈ 0,π, I is an identity operator in E,andBE is the space of bounded linear
operators in E. Sometimes A λI will be written as A λ anddenotedbyA
λ
. It is known 27,
Section 1.15.1 that there exists fractional powers A
θ
of the sectorial operator A.LetEA
θ
denote the space DA
θ
with graphical norm
u
EA
θ
u
p
A
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
2
is called R-bounded see 3, 25, 26 if there is a constant C>0
such that for all T
1
,T
2
, ,T
m
∈ W and u
1,
u
2
, ,u
m
∈ E
1
, m ∈ N
1
0
m
j1
r
j
y
T
j
u
j
E
2
dy ≤ C
1
0
m
j1
r
j
y
u
j
E
1
dy,
1.7
where {r
j
} is a sequence of independent symmetric {−1, 1}-valued random variables on 0, 1.
The smallest C for which the above estimate holds is called an R-bound of the
collection W andisdenotedbyRW.
Let SR
n
; E denote the Schwartz class, that is, the space of all E-valued rapidly
decreasing smooth functions on R
n
.LetF be the Fourier transformation. A function Ψ ∈
CR
n
; BE is called a Fourier multiplier in L
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
,ξ
2
, ,ξ
n
∈ R
n
,ξ
j
/
0
,
U
n
β
β
1
,β
2
, ,β
n
∈ N
n
: β
k
∈
{
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, ∞.
Let Ψ
h
∈ M
p,γ
p,γ
E be a multiplier function dependent on the parameter h ∈ Q.The
uniform R-boundedness of the set {ξ
β
D
β
Ψ
h
ξ : ξ ∈ R
n
\ 0,β ∈ U};thatis,
sup
h∈Q
R
ξ
β
D
β
Ψ
h
ξ
: ξ ∈ R
n
\ 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
ξ
β
D
β
A
x
A
x
ξI
−1
: ξ ∈ R
n
\ 0,β ∈ U
≤ M.
1.10
Let σ
∞
E
1
,E
2
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
and C
2
such that
C
1
a
j
≤ b
j
≤ C
2
a
j
. 1.11
Let σ
∞
E
1
,E
2
denote the space of all compact operators from E
1
to E
2
. For E
1
E
2
E,
it is denoted by σ
∞
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
s
q
j
A
< ∞, 1 ≤ q<∞
⎫
⎬
⎭
. 1.12
Let E
0
and E be two Banach spaces and E
0
continuously and densely embedded into
E and l l
1
,l
2
, ,l
n
.
We let W
l
p,γ
Ω; E
0
,E denote the space of all functions u ∈ L
p,γ
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
u
L
p,γ
Ω;E
0
n
k1
D
l
k
k
u
L
p,γ
Ω;E
< ∞.
1.13
Let D
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
,
u
W
l
p,γ
G;EA,E
u
L
p
G;EA
n
k1
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
α
1
1
D
α
2
2
···D
α
n
n
and suppose that the following con-
ditions are satisfied:
1 E is a Banach space satisfying the multiplier condition with respect to p and γ,
2 A is an R-positive operator in E,
3 α α
1
,α
2
, ,α
n
and l l
1
,l
2
, ,l
n
are n-tuples of nonnegative integer such that
κ
n
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,
for all positive number h<∞ and u ∈ W
l
p,γ
Ω; EA,E, the following estimate holds
D
α
u
L
p,γ
Ω;EA
1−κ −μ
≤ h
μ
u
W
l
p,γ
Ω;EA,E
h
−1−μ
u
L
p,γ
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.
Theorem A3. Assume that
1 E is an UMD space and A is an operator in σ
p
E, p ∈ 1, ∞,
2 μ
1
,μ
2
, ,μ
s
are non overlapping, differentiable arcs in the complex plane starting at the
origin. Suppose that each of the s regions into which the planes are divided by these arcs is
contained in an angular sector of opening less then π/p,
3 m>0 is an integer so that the resolvent of A satisfies the inequality
R
λ, A
O
|
λ
|
−1
, 2.4
as λ → 0 along any of the arcs μ.
Then, the subspace Sp A contains the space E.
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
γ
n
n
.
2.5
Let
β
k
x
β
k
k
,ν
n
k1
x
ν
k
k
,γ
n
k1
x
γ
k
k
.
2.6
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
, 1 <p<∞,
s
j
I
E
0
,E
∼ j
−1/k
0
,k
0
> 0,j 1, 2, ,∞, κ
0
n
k1
γ
k
− ν
k
p
l
k
− 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
. 2.8
3. Statement of the Problem
Consider the BVPs for the degenerate anisotropic DOE
n
k1
a
k
x
D
l
k
k
u
x
A
x
λ
u
x
|
α:l
|
<1
A
α
x
D
α
u
x
f
x
,
3.1
m
kj
i0
α
kji
D
i
k
u
G
k0
0,j 1, 2, ,d
k
,
m
kj
i0
β
kji
D
i
k
u
G
kb
0,j 1, 2, ,l
k
− d
k
,d
k
∈
0,l
k
,
3.2
where
α
α
1
,α
2
, ,α
n
,l
l
1
,l
2
, ,l
n
,
|
α : l
|
n
k1
α
k
l
k
,
G
{
x
x
1
,x
2
, ,x
n
, 0 <x
k
<b
k
,
}
,α
α
1
,α
2
, ,α
n
,
D
α
D
α
1
1
D
α
2
2
···D
α
n
n
,D
i
k
u
x
x
γ
k
k
b
k
− x
k
ν
k
∂
∂x
k
i
u
x
,
0 ≤ γ
k
,ν
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
,b
k
,x
k1
, ,x
n
, 0 ≤ m
kj
≤ l
k
− 1,
x
k
x
1
,x
2
, ,x
k−1
,x
k1
, ,x
n
,G
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
x
−γ
k
k
b
k
− x
k
−ν
k
dx
k
< ∞,x
k
∈
0,b
k
,k 1, 2, ,n.
3.4
Boundary Value Problems 7
A function u ∈ W
l
p,γ
G; EA,E,L
kj
{u ∈ W
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
L
p
G;E
Au
L
p
G;E
≤ C
f
L
p
G;E
3.5
holds.
Let Q be a differential operator generated by problem 3.1-3.2 with λ 0; that is,
D
Q
W
l
p,γ
G; E
A
,E,L
kj
,
Qu
n
k1
a
k
x
D
l
k
k
u A
x
u
|
α:l
|
<1
A
α
x
D
α
u.
3.6
We say the problem 3.1-3.2 is Fredholm in L
p
G; E if dimKer Q dimKer Q
∗
< ∞,
where Q
∗
is a conjugate of Q.
Remark 3.1. Under the substitutions
τ
k
x
k
0
x
−γ
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
G
n
k1
0,
b
k
,
b
k
b
k
0
x
−γ
k
k
b
k
− x
k
−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
λ
u
τ
|
α:l
|
<1
A
α
τ
D
α
u
τ
f
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
β
kji
D
i
k
u
G
kb
0,x
k
∈
G
k
,j 1, 2, ,l
k
− d
k
,d
k
∈
0,l
k
,
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
, ,τ
k−1
,
b
k
,τ
k1
, ,τ
n
,
a
k
τ
a
k
x
1
τ
,x
2
τ
, ,x
n
τ
,
A
τ
A
a
k
x
1
τ
,x
2
τ
, ,x
n
τ
,
A
k
τ
A
k
a
k
x
1
τ
,x
2
τ
, ,x
n
τ
, γ
τ
γ
x
1
τ
,x
2
τ
, ,x
n
τ
.
3.10
By denoting τ,
G,
G
k0
,
G
kb
, a
k
τ,
Aτ,
A
k
y, γ
k
τ again by x, G, G
k0
, G
kb
, a
k
x, Ax,
A
k
x, γ
k
, respectively, we get
n
k1
a
k
x
D
l
k
k
u
x
A
λ
x
u
x
|
α:l
|
<1
A
α
x
D
α
u
x
f
x
,
m
kj
i0
α
kji
D
i
k
u
G
kb
0,j 1, 2, ,l
k
− d
k
,x
k
∈
G
k
,
m
kj
i0
β
kji
D
i
k
u
G
kb
0,x
k
∈ G
k
,j 1, 2, ,l
k
− d
k
,d
k
∈
0,l
k
.
3.11
4. BVPs for Partial DOE
Let us first consider the BVP for the anisotropic type DOE with constant coefficients
L λ
u
n
k1
a
k
D
l
k
k
u
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
,
4.1
where
D
i
k
u
x
x
γ
k
k
∂
∂x
k
i
u
x
,
4.2
L
kj
are boundary conditions defined by 3.2, a
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
pm
kj
/pl
k
,p
,X
k
L
p
G
k
; E
,Y
k
W
l
k
p,γ
k
G
k
; E
A
,E
,
l
k
l
1
,l
2
, ,l
k−1
,l
k1
, ,l
n
,γ
k
x
γ
1
1
,x
γ
2
2
, ,x
γ
k−1
k−1
,x
γ
k1
k1
, ,x
l
n
n
,
G
kx
0
x
1
,x
2
, ,x
k−1
,x
k0
,x
k1
, ,x
n
.
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
are bounded linear from
W
l
p,γ
G; E
0
,E onto F
kj
, and the following inequality holds:
D
j
k
u
G
kx
0
F
kj
≤ C
u
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
.
4.6
Then, by applying the trace theorem 27, Section 1.8.2 to the space W
l
k
p,γ
k
0,b
k
; Y
k
,X
k
,
we obtain the assertion.
Condition 1. Assume that the following conditions are satisfied:
1 E is a Banach space satisfying the multiplier condition with respect to p ∈ 1, ∞
and the weight function γ
n
k1
x
γ
k
k
,0≤ γ
k
< 1 − 1/p;
2 A is an R-positive operator in E for ϕ ∈ 0,π/2;
3 a
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
for 0 <d
k
<l
k
, k 1, 2, ,n.
Let B denote the operator in L
p
G; E generated by BVP 4.1.In15, Theorem 5.1 the
following result is proved.
Theorem A6. Let Condition 1 be satisfied. Then,
a the problem 4.1 for f ∈ L
p
G; E and | arg λ|≤ϕ with sufficiently large |λ| has a unique
solution u that belongs to W
l
p
G; EA,E and the following coercive uniform estimate
holds:
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
L
p
G;E
Au
L
p
G;E
≤ M
f
L
p
G;E
,
4.8
b the operator B is R-positive in L
p
G; E.
10 Boundary Value Problems
From Theorems A5 and A6 we have.
Theorem A7. Suppose that Condition 1 is satisfied. Then, for sufficiently large |λ| with | arg λ|≤ϕ
the problem 4.1 has a unique solution u ∈ W
l
p,γ
G; EA,E for all f ∈ L
p
G; E and f
kj
∈ F
kj
.
Moreover, the following uniform coercive estimate holds:
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
L
p
G;E
Au
L
p
G;E
≤ M
f
L
p
G;E
n
k1
l
k
j1
f
kj
F
kj
.
4.9
Consider BVP 3.11.Letω
k1
x,ω
k2
x, ,ω
kl
k
x be roots of the characteristic equations
a
k
x
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
≤
π
2
− ϕ, j d
k
1, ,l
k
,
4.11
for
0 <d
k
<l
k
,k 1, 2, ,n, ϕ∈
0,
π
2
,
4.12
2 E is a Banach space satisfying the multiplier condition with respect to p ∈ 1, ∞
and the weighted function γ
n
k1
x
γ
k
k
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
kj
. 4.13
Lemma 4.2. Assume that Condition 2 is satisfied and the following hold:
1 Ax is a uniformly R-positive operator in E for ϕ ∈ 0,π/2, and a
k
x are continuous
functions on
G, λ ∈ S
ϕ
,
2 AxA
−1
x ∈ CG; BE and A
∞
A
1−|α:l|−μ
∈ L
∞
G; BE for 0 <μ<1 −|α : l|.
Then, for all λ ∈ Sϕ and for sufficiently large |λ| the following coercive uniform estimate
holds:
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
L
p,γ
G;E
Au
L
p,γ
G;E
≤ C
f
L
p,γ
G;E
n
k1
l
k
j1
f
kj
F
kj
,
4.14
for the solution of problem 4.13.
Boundary Value Problems 11
Proof. Let G
1,
G
2
, ,G
N
be regions covering G and let ϕ
1
,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
x,weget
L λ
u
j
n
k1
a
k
x
D
l
k
k
u
j
x
A
λ
x
u
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
A
α
x
n
k1
α
k
−1
i0
C
αik
D
α
k
−i
k
ϕ
j
D
i
k
u −
|
α:l
|
<1
ϕ
j
A
α
x
D
α
u
x
,
Φ
ki
ϕ
j
L
ki
u B
ki
ϕ
j
L
ki
u,
4.16
here, L
ki
and B
ki
are boundary operators which orders less than m
ki
− 1. Freezing the
coefficients of 4.15 , we have
n
k1
a
k
x
0j
D
l
k
k
u
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
j
A
x
0j
− A
x
u
j
n
k1
a
k
x
− a
x
0j
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
k1
b
k
− x
k
ν
k
,
4.19
on neighborhoods of G
j
∩ G
kb
and γx ∼ C
j
on other parts of the domains G
j
, where C
j
are positive constants. Hence, the problems 4.17 are generated locally only on parts of the
boundary. Then, by Theorem A7 problem 4.17 has a unique solution u
j
and for | arg λ|≤ϕ
the following coercive estimate holds:
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
j
G
j
,p,γ
Au
j
G
j
,p,γ
≤ C
F
j
G
j
,p,γ
n
k1
l
k
i1
Φ
kj
F
ki
.
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
A
x
0j
− A
x
u
j
G
j
,p,γ
n
k1
a
k
x
− a
x
0j
D
l
k
k
u
j
x
G
j
,p,γ
,
Φ
kj
F
ki
≤
ϕ
j
L
ki
u
F
ki
B
j
ϕ
j
L
ki
u
F
ki
≤ 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
j
G
j
,p,γ
≤
f
j
G
j
,p,γ
ε
A
x
0j
u
j
G
j
,p,γ
ε
n
k1
D
l
k
k
u
j
x
G
j
,p,γ
≤
fϕ
j
G
j
,p,γ
M
|
α:l
|
<1
A
α
x
D
α
u
j
x
G
j
,p,γ
ε
u
j
W
l
p,γ
G
j
;EA,E
≤
fϕ
j
G
j
,p,γ
ε
u
j
W
l
p,γ
G
j
;EA,E
C
ε
u
j
G
j
,p,γ
.
4.22
Then, using Theorem A5 and using the smoothness of the coefficients of 4.16, 4.18,weget
Φ
ki
F
kj
≤ M
L
ki
u
F
ki
L
ki
u
F
ki
≤ M
L
ki
u
F
ki
u
j
W
l
k
−1
p,γ
0,b
kj
;Y
k
,X
k
. 4.23
Now, using Theorem A1, we get that there is an ε>0andCε such that
u
j
W
l
k
−1
p,γ
k
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
u
j
L
p,γ
k
≤ ε
u
j
W
l
p,γ
G
j
;EA,E
C
ε
u
j
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
L
ki
u
F
ki
ε
u
j
W
l
p,γ
G
j
;EA,E
C
ε
u
j
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
j
,p,γ
Au
j
G
j
,p,γ
≤ C
f
G
j
,p,γ
ε
u
j
W
2
p,γ
M
ε
u
j
G
j
,p,γ
C
n
k1
l
k
i1
f
ki
F
ki
.
4.27
Boundary Value Problems 13
Choosing ε<1 from the above inequality, we obtain
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
j
G
j
,p,γ
Au
j
G
j
,p,γ
≤ C
f
G
j
,p,γ
u
j
G
j
,p,γ
n
k1
l
k
i1
f
ki
F
ki
.
4.28
Then, by using the equality ux
N
j1
u
j
x and the above estimates, we get 4.14.
Condition 3. Suppose that part 1.1 of Condition 1 is satisfied and that E is a Banach space
satisfying the multiplier condition with respect to p ∈ 1, ∞ and the weighted function γ
n
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
∞
A
1−|α:l|−μ
∈ L
∞
G; BE for 0 <μ<1 −|α : l|.
Then, for all λ ∈ Sϕ and for sufficiently large |λ|, the following coercive uniform estimate
holds
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
L
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
G; E
A
,E,L
kj
,
Ou
n
k1
a
k
x
D
l
k
k
u A
x
u
|
α:l
|
<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
ϕ
with large enough |λ|. Moreover, the following coercive uniform estimate holds:
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
L
p,γ
G;E
Au
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
|
1−i/l
k
D
i
k
u
p,γ
Au
p,γ
≤ C
L λ
u
p,γ
u
p,γ
.
4.32
It is clear that
u
p,γ
1
|
λ
|
L λ
u − Lu
p,γ
≤
1
|
λ
|
L λ
u
p,γ
Lu
p,γ
.
4.33
Hence, by using the definition of W
l
p,γ
G; EA,E and applying Theorem A1, we obtain
u
p,γ
≤
1
|
λ
|
L 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
D
i
k
u
p,γ
Au
p,γ
≤ C
L λ
u
p,γ
.
4.35
The estimate 4.35 implies that problem 3.11 has a unique solution and that the operator
O λ has a bounded inverse in its rank space. We need to show that this rank space coincides
with the space L
p,γ
G; E; that is, we have to show that for all f ∈ L
p,γ
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
k
x
0j
D
l
k
k
u
j
x
A
λ
x
0j
u
j
x
g
j
⎧
⎨
⎩
f
A
x
0j
− A
x
u
j
n
k1
a
k
x
− a
k
x
0j
D
l
k
k
u
j
−
|
α:l
|
<1
A
α
x
D
α
u
j
⎫
⎬
⎭
,
L
ki
u
j
0,j 1, 2, ,N.
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
jλ
u
n
k1
a
k
x
0j
D
l
k
k
u
j
x
A
λ
x
0j
u
j
x
,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
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
O
−1
jλ
f
L
p,γ
G
j
;E
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
j
in the equalities 4.36, and using the
substitutions u
j
O
−1
jλ
υ
j
, we obtain the operator equations
υ
j
K
jλ
υ
j
g
j
f, j 1, 2, ,N, 4.39
where K
jλ
are bounded linear operators in L
p
G
j
; E defined by
K
jλ
g
j
⎧
⎨
⎩
f
A
x
0j
− A
x
O
−1
jλ
n
k1
a
k
x
− a
k
x
0j
D
l
k
k
O
−1
jλ
−
|
α:l
|
<1
A
α
x
D
α
O
−1
jλ
⎫
⎬
⎭
.
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
p,γ
G
j
;E
≤ ε
υ
j
L
p,γ
G
j
;E
,
n
k1
a
k
x
− a
k
x
0j
D
l
k
k
O
−1
jλ
υ
j
L
p,γ
G
j
;E
≤ ε
υ
j
L
p,γ
G
j
;E
.
4.41
Moreover, from assumption 2.2 of Theorem 4.4 and Theorem A1 for ε>0, there is a constant
Cε > 0 such that
|
α:l
|
<1
A
α
x
D
α
O
−1
jλ
υ
j
L
p,γ
G
j
;E
≤ ε
υ
j
W
l
p,γ
G
j
;EA,E
C
ε
υ
j
L
p,γ
G
j
;E
.
4.42
Hence, for | arg λ|≤ϕ with sufficiently large |λ|, there is a δ ∈ 0, 1 such that K
jλ
<δ.
Consequently, 4.39 for all j have a unique solution υ
j
I − K
jλ
−1
g
j
f. Moreover,
υ
j
L
p,γ
G
j
;E
I − K
jλ
−1
g
j
f
L
p,γ
G
j
;E
≤
f
L
p,γ
G
j
;E
.
4.43
Thus, I − K
jλ
−1
g
j
are bounded linear operators from L
p,γ
G; E to L
p,γ
G
j
; E.Thus,the
functions
u
j
U
jλ
f O
−1
jλ
I − K
jλ
−1
g
j
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
j1
ϕ
j
y
U
jλ
f O
−1
jλ
I − K
jλ
−1
g
j
f.
4.45
It is clear from the constructions U
j
and from the estimate 4.39 that the operators U
jλ
are
bounded linear from L
p,γ
G; E to W
l
p,γ
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
AU
jλ
f
p
≤ C
f
p
.
4.46
Therefore, U λ is a bounded linear operator in L
p,γ
G; E. Since the operators
U
jλ
coincide with the inverse of the operator O
λ
in L
p,γ
G
j
; E, then acting on O
λ
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
Φ
jλ
f,
4.47
where Φ
jλ
are bounded linear operators defined by
Φ
jλ
f
⎧
⎨
⎩
n
k1
a
k
l
k
ν1
C
kν
D
ν
k
ϕ
j
D
l
k
−ν
k
U
jλ
f
|
α:l
|
<1
A
α
n
k1
α
k
ν
k
1
C
α,ν
k
D
ν
k
k
ϕ
j
D
α
k
−ν
k
k
U
jλ
f
⎫
⎬
⎭
. 4.48
Indeed, from Theorem A1 and estimate 4.46 and from the expression Φ
jλ
,weobtainthat
the operators Φ
jλ
are bounded linear from L
p
G; E to L
p
G; E,andfor| arg λ|≤ϕ with
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
jλ
I − K
jλ
−1
g
j
⎛
⎝
I
N
j1
Φ
jλ
⎞
⎠
−1
f.
4.49
Result 1. Theorem 4.4 implies that the resolvent O λ
−1
satisfies the following anisotropic
type sharp estimate:
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
O λ
−1
BL
p,γ
G;E
A
O λ
−1
BL
p,γ
G;E
≤ C,
4.50
for | arg λ|≤ϕ, ϕ ∈ 0,π/2.
Boundary Value Problems 17
Let Q denote the operator generated by BVP 3.1-3.2.FromTheorem 4.4 and
Remark 3.1, we get the following.
Result 2. Assume all the conditions of Theorem 4.4 hold. Then,
a the problem 3.1-3.2 for f ∈ L
p
G; E, | arg λ|≤ϕ and for sufficiently large |λ|
has a unique solution u ∈ W
l
p,γ
G; EA,E, and the following coercive uniform
estimate holds
n
k1
l
k
i0
|
λ
|
1−i/l
k
D
i
k
u
L
p
G;E
Au
L
p
G;E
≤ M
f
L
p
G;E
,
4.51
b if A
−1
∈ σ
∞
E, then the operator O is Fredholm from W
l
p,γ
G; EA,E into
L
p
G; E.
Example 4.5. Now, let us consider a special case of 3.1-3.2.LetE C, l
1
2andl
2
4,
n 2, G 0, 1 × 0, 1 and A q; that is, consider the problem
Lu a
1
D
2
x
u a
2
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
β
1i
u
i
1,y
0,m
1j
∈
{
0, 1
}
,
m
2j
i0
α
2i
D
i
y
u
x, 0
0,
m
2j
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
i
,D
i
y
y
β
1
1 − y
β
2
∂
∂y
i
,
u u
x, y
,a
k
∈ C
G
,a
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
4
y
u
L
p
G
u
L
p
G
≤ C
f
L
p
G
.
4.54
Example 4.6. Let l
k
2m
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
x
D
2m
k
k
u
m
x
d
m
x
u
m
x
f
m
x
,
m
kj
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
k
,m 1, 2, ,ν,
4.55
in the vector-valued space L
p,γ
G; C
ν
.
5. The Spectral Properties of Anisotropic Differential Operators
Consider the following degenerated BVP:
n
k1
a
k
x
D
l
k
k
u
x
A
x
u
x
|
α:l
|
<1
A
α
x
D
α
u
x
f
x
,
m
kj
i0
α
kji
D
i
k
u
G
k0
0,j 1, 2, ,d
k
,
m
kj
i0
β
kji
D
i
k
u
G
kb
0,j 1, 2, ,l
k
− d
k
,d
k
∈
0,l
k
,
5.1
where
G
{
x
x
1
,x
2
, ,x
n
, 0 <x
k
<b
k
}
,α
α
1
,α
2
, ,α
n
,
D
α
D
α
1
1
D
α
2
2
···D
α
n
n
,D
i
k
u
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
G; EA,E into L
p
G; E.
Proof. Theorem 4.4 implies that the operator Q λ for sufficiently large |λ| has a bounded
inverse O λ
−1
from L
p
G; E to W
l
p
G; EA,E; that is, the operator Q λ is Fredholm
from W
l
p
G; EA,E into L
p
G; E. Then, from Theorem A2 and the perturbation theory
of linear operators, we obtain that the operator Q is Fredholm from W
l
p,γ
G; EA,E into
L
p
G; E.
Boundary Value Problems 19
Theorem 5.2. Suppose that all the conditions of Theorem 5.1 are satisfied with ν
k
0. Assume that
E is a Banach space with a basis and
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
−1
L
p
G; E
∼ j
−1/νκ
, 5.4
b the system of root functions of the di fferential operator Q is complete in L
p
G; E.
Proof. Let IE
0
,E denote the embedding operator from E
0
to E.FromResult2, there exists a
resolvent operator Q d
−1
which is bounded from L
p
G; E to W
l
p,γ
G; EA,E. Moreover,
from Theorem A4 and Remark 3.1, we get that the embedding operator
I
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
A
,E
,L
p
G; E
∼ j
−1/νκ
. 5.6
It is clear that
Q d
−1
L
p
G; E
Q d
−1
L
p
G; E
,W
l
p,γ
G; E
A
,E
× I
W
l
p,γ
G; E
A
,E
,L
p
G; E
.
5.7
Hence, from relations 5.6 and 5.7,weobtain5.4.Now,Result1 implies that the
operator Q d is positive in L
p
G; E and
Q d
−1
∈ σ
q
L
p
G; E
,q>
1
ν κ
.
5.8
Then, from 4.52 and 5.6, we obtain assertion b.
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
D
l
k
k
u A
x
u
x
|
α:l
|
<1
A
α
x
D
α
u.
5.9
From Theorem 5.2 and Remark 3.1, we get the following.
Result 3. Let all the conditions of Theorem 5.1 hold. Then, the operator O is Fredholm from
W
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
are studied. Maximal regularity properties for PDEs have been studied, for example, in 3
for smooth domains and in 28 for nonsmooth domains.
Consider the BVP
Lu
n
k1
a
k
x
D
l
k
k
u
x, y
|
β
|
≤2m
a
β
y
D
β
y
u
x, y
|
α:l
|
<1
v
α
x, y
D
α
y
u
x, y
f
x, y
,x∈ G, y ∈ Ω,
m
kj
i0
α
kji
D
i
k
u
G
k0
,y
0,x
k
∈ G
k
,y∈ Ω,j 1, 2, ,d
k
,
m
kj
i0
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
,
B
j
u
|
β
|
m
j
b
jβ
y
D
β
y
u
x, y
y∈∂Ω
0,x∈ G, j 1, 2, ,m,
6.1
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
k
}
,D
i
k
x
γ
k
k
b
k
− x
k
ν
k
∂
∂x
k
i
,
G
k0
x
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
kj
≤ l
k
− 1,
α
kj
β
kj
> 0,j 1, 2, ,l
k
.
x
k
x
1
,x
2
, ,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-
valued functions with mixed norm see, e.g., 29, Section 1, page 6, that is, the space of all
measurable functions f defined on
Ω, for which
f
L
p
Ω
G
Ω
f
x, y
p
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
k
1 0.
6.4
Let Q denote the operator generated by BVP 6.1.Let
F B
L
p
Ω
. 6.5
Theorem 6.1. Let the following conditions be satisfied:
1 a
α
∈ CΩ for each |α| 2m and a
α
∈ L
∞
L
r
k
Ω for each |α| k<2m with r
k
≥ p
1
,
p
1
∈ 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
k
, ν
k
< 1 − 1/p,
p ∈ 1, ∞,
3 for y ∈
Ω, ξ ∈ R
μ
, η ∈ Sϕ
1
, ϕ
1
∈ 0,π/2, |ξ| |η|
/
0 let
η
|
α
|
2m
a
α
y
ξ
α
/
0,
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
|
β
|
m
j
b
jβ
y
0
D
β
ϑ
y
h
j
,j 1, 2, ,m
6.7
has a unique solution ϑ ∈ C
0
R
for all h h
1
,h
2
, ,h
m
∈ 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
k
,
arg ω
kj
≤
π
2
− ϕ, ϕ ∈
0,
π
2
,
j d
k
1, ,l
k
, 0 <d
k
<l
k
,k 1, 2, ,n, v
α
∈ L
∞
G
,x∈ G.
6.8
Then,
a the following coercive estimate
n
k1
D
l
k
k
u
L
p
Ω
|
β
|
2m
D
β
y
u
L
p
Ω
u
L
p
Ω
≤ C
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
D
i
k
Q λ
−1
F
A
Q λ
−1
F
≤ M,
6.10
c the problem 6.1 for ν
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
0
n
k1
1
l
k
, κ
0
μ
2m
6.11
holds,
e for ν
k
0 the system of root functions of the BVP 6.1 is complete in L
p
Ω.
Boundary Value Problems 23
Proof. Let E L
p
1
Ω. Then, from 3, Theorem 3.6,part1.1 of Condition 1 is satisfied.
Consider the operator A which is defined by
D
A
W
2m
p
1
Ω; B
j
u 0
,Au
|
β
|
≤2m
a
β
y
D
β
u
y
.
6.12
For x ∈ Ω, we also consider operators
A
α
x
u v
α
x, y
D
α
u
y
,
|
α : l
|
< 1. 6.13
The problem 6.1 can be rewritten as the form of 3.1-3.2, where uxux, · and fx
fx, · are functions with values in E L
p
1
Ω.From3, Theorem 8.2 problem
ηu
y
|
β
|
≤2m
a
β
y
D
β
u
y
f
y
,
B
j
u
|
β
|
≤m
j
b
jβ
y
D
β
u
y
0,j 1, 2, ,m
6.14
has a unique solution for f ∈ L
p
1
Ω and arg η ∈ Sϕ
1
, |η|→∞. Moreover, the operator
A, generated by 5.8 is R-positive in L
p
1
; that is, part 2.2 of Condition 1 holds. From 2.2,
3.7,andby29, Section 18, we have
|
α:l
|
<1
A
α
x
u
L
p
1
≤ C
|
α:l
|
<1
D
α
u
L
p
1
≤ ε
u
B
2m1−|α:l|
p
1
,1
u
L
p
1
,
6.15
that is, all the conditions of Theorem 5.2 and Result 4 are fulfilled. As a result, we obtain
assertion a and b of the theorem. Also, it is known e.g., 27, Theorem 3.2.5, Section 4.10
that the embedding W
2m
p
1
Ω ⊂ L
p
1
Ω is compact and
s
j
I
W
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
l
k
k
u
m
x
∞
j1
d
j
x
λ
u
m
x
|
α:l
|
<1
∞
j1
d
αjm
x
D
α
u
m
x
f
m
x
,x∈ G, m 1, 2, ,∞,
m
kj
i0
α
kji
D
i
k
u
G
k0
0,x
k
∈ G
k
,j 1, 2, ,d
k
,
m
kj
i0
β
kji
D
i
k
u
G
kb
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
2
, ,x
n
, 0 <x
k
<b
k
}
,D
i
k
x
γ
k
k
b
k
− x
k
ν
k
∂
∂x
k
i
,
G
k0
x
1
,x
2
, ,x
k−1
, 0,x
k1
, ,x
n
,
α
kj
β
kj
> 0,
G
kb
x
1
,x
2
, ,x
k−1
,b
k
,x
k1
, ,x
n
, 0 ≤ m
kj
≤ l
k
− 1,
x
k
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
m
x
}
,d
m
> 0,u
{
u
m
}
,du
{
d
m
u
m
}
,m 1, 2, ···∞,
l
q
D
⎧
⎨
⎩
u : u ∈ l
q
,
u
l
q
D
Du
l
q
∞
m1
|
d
m
u
m
|
q
1/q
< ∞
⎫
⎬
⎭
,
j 1, 2, ,l
k
,k 1, 2, ,n.
7.2
Let V denote the operator in L
p
G; l
q
generated by problem 7.1.Let
B B
L
p
G; l
q
. 7.3
Boundary Value Problems 25
Theorem 7.1. Let γ
n
k1
x
γ
k
k
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
∞
G such that
max
α
sup
m
∞
j1
d
αjm
x
d
−1−|α:l|−μ
j
<M, ∀x ∈ G, 0 <μ<1 −
|
α : l
|
.
7.4
Then,
a for all fx{f
m
x}
∞
1
∈ 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
D
l
k
k
u
m
x
q
p/q
dx
⎤
⎦
1/p
⎡
⎣
G
∞
m1
|
d
m
u
m
x
|
q
p/q
dx
⎤
⎦
1/p
≤ C
⎡
⎣
G
∞
m1
f
m
x
q
p/q
dx
⎤
⎦
1/p
,
7.5
b there exists a resolvent V λ
−1
of the operator V and
n
k1
l
k
j0
|
λ
|
1−j/l
k
D
j
k
V λ
−1
B
A
V λ
−1
B
≤ M,
7.6
c for ν
k
0, the system of root functions of the BVP 7.1 is complete in L
p
G; l
q
.
Proof. Let E l
q
, A and A
α
x be infinite matrices such that
A
d
m
δ
mj
,A
α
x
d
αjm
x
,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
D
i
k
u
L
p
G;l
q
Au
L
p
G;l
q
≤ M
f
L
p
G;l
q
.
7.8
From the above estimate, we obtain assertions a and b. The assertion c is obtained
from Result 4.
