Anisotropic differential operators with parameters and applications
In the paper, we study the boundary-value problems for parameter-dependent anisotropic differential-operator equations with variable coefficients. Several conditions for the uniform separability and Fredholmness in Banach-valued L p -spaces are given. Sharp uniform estimates for the resolvent are es...
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Shakhmurov, V.B. 2020-02-18T05:03:22Z 2020-02-18T05:03:22Z 2014 Anisotropic differential operators with parameters and applications / V.B. Shakhmurov // Український математичний журнал. — 2014. — Т. 66, № 7. — С. 983–1002. — Бібліогр.: 29 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166056 517.9 In the paper, we study the boundary-value problems for parameter-dependent anisotropic differential-operator equations with variable coefficients. Several conditions for the uniform separability and Fredholmness in Banach-valued L p -spaces are given. Sharp uniform estimates for the resolvent are established. It follows from these estimates that the indicated operator is uniformly positive. Moreover, it is also the generator of an analytic semigroup. As an application, the maximal regularity properties of the parameter-dependent abstract parabolic problem and infinite systems of parabolic equations are established in mixed L p -spaces. Вивчаються граничні задачi для анізотропних диференціально-операторних рівнянь зі змінними коефiцiєнтами, що залежать від параметрів. Наведено кілька умов рівномірної сепарабельності та фредгольмовості в банаховозначних L p -просторах. Встановлено точні рівномірні оцінки для резольвенти, з яких випливає, що вказаний оператор є рівномірно додатним. Більш того, він є також генератором деякої аналітичної напівгрупи. Як застосування, встановлено властивості максимальної регулярності абстрактної параболічної задачі, що залежить від параметра, та нескінченних систем рівнянь параболічного типу в L p -просторах. en Інститут математики НАН України Український математичний журнал Статті Anisotropic differential operators with parameters and applications Анізотропні диференціальні оператори з параметрами та їх застосування Article published earlier |
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Anisotropic differential operators with parameters and applications |
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Anisotropic differential operators with parameters and applications Shakhmurov, V.B. Статті |
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Anisotropic differential operators with parameters and applications |
| title_full |
Anisotropic differential operators with parameters and applications |
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Anisotropic differential operators with parameters and applications |
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Anisotropic differential operators with parameters and applications |
| title_sort |
anisotropic differential operators with parameters and applications |
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Shakhmurov, V.B. |
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Shakhmurov, V.B. |
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Анізотропні диференціальні оператори з параметрами та їх застосування |
| description |
In the paper, we study the boundary-value problems for parameter-dependent anisotropic differential-operator equations with variable coefficients. Several conditions for the uniform separability and Fredholmness in Banach-valued L p -spaces are given. Sharp uniform estimates for the resolvent are established. It follows from these estimates that the indicated operator is uniformly positive. Moreover, it is also the generator of an analytic semigroup. As an application, the maximal regularity properties of the parameter-dependent abstract parabolic problem and infinite systems of parabolic equations are established in mixed L p -spaces.
Вивчаються граничні задачi для анізотропних диференціально-операторних рівнянь зі змінними коефiцiєнтами, що залежать від параметрів. Наведено кілька умов рівномірної сепарабельності та фредгольмовості в банаховозначних L p -просторах. Встановлено точні рівномірні оцінки для резольвенти, з яких випливає, що вказаний оператор є рівномірно додатним. Більш того, він є також генератором деякої аналітичної напівгрупи. Як застосування, встановлено властивості максимальної регулярності абстрактної параболічної задачі, що залежить від параметра, та нескінченних систем рівнянь параболічного типу в L p -просторах.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/166056 |
| citation_txt |
Anisotropic differential operators with parameters and applications / V.B. Shakhmurov // Український математичний журнал. — 2014. — Т. 66, № 7. — С. 983–1002. — Бібліогр.: 29 назв. — англ. |
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2025-11-25T22:29:25Z |
| last_indexed |
2025-11-25T22:29:25Z |
| _version_ |
1850563715080912896 |
| fulltext |
UDC 517.9
V. B. Shakhmurov (Okan Univ., Istanbul, Turkey; Inst. Math. and Mech., Azerbaijan Nat. Acad. Sci.)
ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS
AND APPLICATIONS
АНIЗОТРОПНI ДИФЕРЕНЦIАЛЬНI ОПЕРАТОРИ З ПАРАМЕТРАМИ
ТА ЇХ ЗАСТОСУВАННЯ
In this paper, we study the boundary-value problems for parameter-dependent anisotropic differential-operator equations
with variable coefficients. Several conditions for the uniform separability and Fredholmness in Banach-valued Lp-spaces
are given. Sharp uniform estimates for the resolvent are established. They imply that the indicated operator is uniformly
positive. Moreover, it is also the generator of an analytic semigroup. As an application, the maximal regularity properties
of the parameter-dependent abstract parabolic problem and infinite systems of parabolic equations are derived in mixed
Lp-spaces.
Вивчаються граничнi задачi для анiзотропних диференцiально-операторних рiвнянь зi змiнними коефiцiєнтами, що
залежать вiд параметрiв. Наведено кiлька умов рiвномiрної сепарабельностi та фредгольмовостi в банаховозначних
Lp-просторах. Встановлено точнi рiвномiрнi оцiнки для резольвенти, з яких випливає, що вказаний оператор
є рiвномiрно додатним. Бiльш того, вiн є також генератором деякої аналiтичної напiвгрупи. Як застосування,
встановлено властивостi максимальної регулярностi абстрактної параболiчної задачi, що залежить вiд параметра,
та нескiнченних систем рiвнянь параболiчного типу в Lp-просторах.
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). DOEs have been studied
extensively in the literature (see [1 – 5, 8 – 11, 13 – 24, 26 – 29] and the references therein).
The main aim of the present paper is to discuss the uniform separability properties of BVPs for
the following higher order parameter dependent anisotropic DOE:
n∑
k=1
εkak (x)
∂lku
∂xlkk
+A (x)u+
∑
|α:l|<1
n∏
k=1
ε
αk/lk
k Aα (x)Dαu = f (x) , (1)
where εk are small positive parameters, ak (x) are complex valued continuous functions, A (x) and
Aα (x) are operator valued functions, defined for x ∈ Ω, where Ω is some region in Rn with the
operators A(x) and Aα (x) , acting in a Banach space E, u (x) and f (x) respectively are a E valued
unknown and date functions. The above DOE is a generalized form of the elliptic equation with
parameters. In fact, the special case lk = 2m, k = 1, . . . , n, the equation (1) reduces to elliptic
equation. Note, the principal part of the corresponding differential operator is non self-adjoint.
Nevertheless, the sharp uniform coercive estimate for the resolvent and Fredholmness are established.
Note that, maximal regularity properties for higher order anisotropic DOEs were studied, e.g., in
[3, 5, 21, 23]. Unlike to these, in the present paper, the nonlocal BVP for parameter depended
undegenerate anisotropic equation is studied and uniform separability properties is derived. In
application, the maximal regularity properties of mixed problem for the following parabolic equation:
∂u
∂t
+
n∑
k=1
εkak (x)
∂lku
∂xlk
+A (x)u = f (t, x) (2)
c© V. B. SHAKHMUROV, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7 983
984 V. B. SHAKHMUROV
are obtained. Particularly, the problem (2) occur in atmospheric dispersion of pollutants and evolution
models for phytoremediation of metals from soils. Really, if E = R3, A (x) is a 3-dimensional
functional matrices, i.e., A (x) = [aij (x)] , u = (u1, u2, u3) , i, j = 1, 2, 3, then we get the well
posedeness of the IVP for the system of parabolic PDE with parameters
∂ui
∂t
+
n∑
k=1
(−1)lk εkak (x)
∂2lkui
∂x2lk
+
3∑
j=1
aij (x)uj = fi (t, x)
which arises in phytoremediation process.
Let Lp (Ω;E) denote the space of all strongly measurable E-valued functions that are defined
on the region Ω ⊂ Rn with the norm
‖f‖p = ‖f‖Lp(Ω;E) =
(∫
‖f (x)‖pE dx
)1/p
, 1 ≤ p <∞.
The Banach space E is called a UMD-space if the Hilbert operator (Hf) (x) =
= limε→0
∫
|x−y|>ε
f (y)
x− y
dy is bounded in Lp (R,E) , p ∈ (1,∞) (see., e.g., [6]). UMD-spaces
include e.g. Lp-, lp-spaces and Lorentz spaces Lpq, p, q ∈ (1,∞) .
Let C be the set of complex numbers and
Sϕ = {λ;λ ∈ C, |arg λ| ≤ ϕ} ∪ {0} , 0 ≤ ϕ < π.
Let E1 and E2 be two Banach spaces. B (E1, E2) denotes the space of bounded linear operators
from E1 into E2 endowed with the usual uniform operator topology. For E1 = E2 it denotes by
B (E) . Now (E1, E2)θ,p, 0 < θ < 1, 1 ≤ p ≤ ∞ will denote interpolation spaces defined by the K
method [25] (§ 1.3.1).
A linear operator A is said to be ϕ-positive in a Banach space E with bound M > 0 if D (A) is
dense on E and ∥∥∥(A+ λI)−1
∥∥∥
L(E)
≤M (1 + |λ|)−1
for all λ ∈ Sϕ, ϕ ∈ [0 , π) , I is an identity operator in E. Sometimes A + λI will be written as
A + λ and denoted by Aλ. It is known [25] (§ 1.15.1) that there exists fractional powers Aθ of the
positive operator A. Let E
(
Aθ
)
denote the space D
(
Aθ
)
endowed with graph norm
‖u‖E(Aθ) =
(
‖u‖p +
∥∥∥Aθu∥∥∥p)1/p
, 1 ≤ p <∞, −∞ < θ <∞.
A set W ⊂ B (E1, E2) is called R-bounded (see [6, 8, 26]) if there is a constant C > 0 such that
for all T1, T2, . . . , Tm ∈W and u1,u2, . . . , um ∈ E1, m ∈ N
1∫
0
∥∥∥∥∥∥
m∑
j=1
rj (y)Tjuj
∥∥∥∥∥∥
E2
dy ≤ C
1∫
0
∥∥∥∥∥∥
m∑
j=1
rj (y)uj
∥∥∥∥∥∥
E1
dy,
where {rj} is an arbitrary sequence of independent symmetric {−1, 1}-valued random variables on
[0, 1] .
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS AND APPLICATIONS 985
The smallest C for which the above estimate holds is called a R-bound of the collection W and
is denoted by R (W ) .
Let S (Rn;E) denote the Schwartz class, i.e., the space of all E-valued rapidly decreasing smooth
functions on Rn equipped with its usual topology generated by seminorms. Let Ω be a domain in
Rn. C (Ω;E) and C(m) (Ω;E) will denote the spaces of E-valued bounded uniformly strongly
continuous and m-times continuously differentiable functions on Ω, respectively. Let F denotes the
Fourier transformation. A function Ψ ∈ C (Rn;B (E)) is called a Fourier multiplier in Lp (Rn;E)
if the map u→ Φu = F−1Ψ (ξ)Fu, u ∈ S (Rn;E) is well defined and extends to a bounded linear
operator in Lp (Rn;E) . The set of all multipliers in Lp (Rn;E) will denoted by Mp
p (E) .
Let
Un = {β = (β1, β2, . . . , βn) ∈ Nn : βk ∈ {0, 1}} .
Definition 1. A Banach space E is said to be a space satisfying a multiplier condition if, for any
Ψ ∈ C(n) (Rn;B (E)) the R-boundedness of the set
{
ξβDβ
ξ Ψ (ξ) : ξ ∈ Rn\ {0} , β ∈ Un
}
implies
that Ψ is a Fourier multiplier in Lp (Rn;E) , i.e., Ψ ∈ Mp
p (E) for any p ∈ (1,∞) .
Let Ψh ∈ Mp
p (E) be a multiplier function dependent of the parameter h ∈ Q. The uniform R-
boundedness of the set
{
ξβDβΨh (ξ) : ξ ∈ Rn\ {0} , β ∈ Un
}
, i.e.,
sup
h∈Q
R
({
ξβDβΨh (ξ) : ξ ∈ Rn\ {0} , β ∈ Un
})
≤ K
implies that Ψh is a uniform collection of Fourier multipliers.
Remark 1. Note that, if E is UMD-space then e.g., by virtue of [8] (Theorem 3.25) it satisfies
the multiplier condition.
Definition 2. The ϕ-positive operator A is said to be R-positive in a Banach space E if the set{
A (A+ ξI)−1 : ξ ∈ Sϕ
}
is R-bounded.
An operator function A (x) is said to be ϕ-positive in E uniformly in x if domain D (A (x)) of
the A (x) is independent of x, D (A (x)) is dense in E and
∥∥∥(A (x) + λI)−1
∥∥∥ ≤ M
1 + |λ|
for any
λ ∈ Sϕ, ϕ ∈ [0 , π) .
The ϕ-positive operator A (x) , x ∈ G is said to be uniformly R-positive in a Banach space E if
there exists ϕ ∈ [0 , π) such that the set{
A (x) (A (x) + ξI)−1 : ξ ∈ Sϕ
}
is uniformly R-bounded, i.e.,
sup
x∈G
R
([
A (x) (A (x) + ξI)−1
]
: ξ ∈ Sϕ
)
≤M.
Let σ∞ (E1, E2) denote the space of all compact operators from E1 to E2. For E1 = E2 = E it is
denoted by σ∞ (E) .
Let D (Ω;E) denote the class of all E-valued infinite differentiable functions on domain Ω with
compact supports. For E = C it denotes by D (Ω) .
Let α = (α1, α2, . . . , αn) is a n-tuples of positive integer, Dα = Dα1
1 Dα2
2 . . . Dαn
n and |α| =
=
∑n
k=1
αk.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
986 V. B. SHAKHMUROV
Definition 3. Let f ∈ Lp (Ω;E) . The function (Dαf) : Ω → E is called to be generalized
derivative of f on Ω if the following equality:∫
Ω
Dαf (x)ϕ (x) dx = (−1)|α|
∫
Ω
f (x)Dαϕ (x) dx
holds for all ϕ ∈ D (Ω) .
Let E0 and E be two Banach spaces and E0 is continuously and densely embedded into E and
l = (l1, l2, . . . , ln) .
We let W l
p (Ω;E0, E) denote the space of all functions u ∈ Lp (Ω;E0) possessing generalized
derivatives Dlk
k u =
∂lku
∂xlkk
such that Dlk
k u ∈ Lp (Ω;E) with the norm
‖u‖W l
p(Ω;E0,E) = ‖u‖Lp(Ω;E0) +
n∑
k=1
∥∥∥Dlk
k u
∥∥∥
Lp(Ω;E)
<∞.
Let ε = (ε1, ε2, . . . , εn) . Consider the following parameterized norm:
‖u‖W l
p,ε(Ω;E0,E) = ‖u‖Lp(Ω;E0) +
n∑
k=1
εk
∥∥∥Dlk
k u
∥∥∥
Lp(Ω;E)
<∞.
If G+ = G×R+, p = (p, p1) , Lp (G+;E) will be denote the space of all p-summable E-valued
functions with mixed norm (see, e.g., [7] for E = C), i.e., the space of all measurable E-valued
functions f defined on G for which
‖f‖Lp(G+) =
∫
G
∫
R+
‖f (t, x)‖pE dt
p1/p
dx
1/p1
<∞.
Analogously, W l
p (G+;E) denotes the E-valued anisotropic Sobolev space with corresponding mixed
norm. Let
W l
p (G+;E0, E) = W l
p (G+;E) ∩ Lp (G+;E0)
endowed with norm
‖u‖W l
p(G+;E0,E) = ‖u‖Lp(G+;E0) +
n∑
k=1
∥∥∥Dlk
k u
∥∥∥
Lp(G+;E)
<∞.
2. Background. The embedding theorems in vector valued spaces play a key role in the theory
of DOEs. For estimating lower order derivatives we use following embedding theorems from [24].
Theorem A1. Let α = (α1, α2, . . . , αn) and Dα = Dα1
1 Dα2
2 . . . Dαn
n and suppose the following
conditions are satisfied:
(1) E is a Banach space satisfying the multiplier condition;
(2) A is an R-positive operator in E;
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS AND APPLICATIONS 987
(3) α = (α1, α2, . . . , αn) and l = (l1, l2, . . . , ln) are n-tuples of nonnegative integer such that
κ =
∑n
k=1
αk
lk
≤ 1, 0 ≤ µ ≤ 1− κ, 1 < p <∞, 0 < h ≤ h0, h0 is a fixed positive number and εk
are small positive parameters;
(4) Ω ⊂ Rn is a region such that there exists a bounded linear extension operator from
W l
p (Ω;E (A) , E) to W l
p (Rn;E (A) , E) .
Then the embedding DαW l
p (Ω;E (A) , E) ⊂ Lp
(
Ω;E
(
A1−κ−µ)) is continuous and for all
u ∈W l
p (Ω;E (A) , E) the following uniform estimate holds:
n∏
k=1
ε
αk/lk
k ‖Dαu‖Lp(Ω;E(A1−κ−µ)) ≤ h
µ ‖u‖W l
p,ε(Ω;E(A),E) + h−(1−µ) ‖u‖Lp(Ω;E) .
Remark 2. If Ω ⊂ Rn is a region satisfying the strong l-horn condition (see [7], § 7), E = R,
A = I, then for p ∈ (1,∞) there exists a bounded linear extension operator from W l
p (Ω) =
= W l
p (Ω;R,R) to W l
p (Rn) = W l
p (Rn;R,R) .
Theorem A2. Suppose all conditions of Theorem A1 are satisfied for 0 < µ ≤ 1− κ. Moreover,
let Ω be a bounded region and A−1 ∈ σ∞ (E) . Then the embedding
DαW l
p (Ω;E (A) , E) ⊂ Lp
(
Ω;E
(
A1−κ−µ))
is compact.
Theorem A3. Suppose all conditions of Theorem A1 satisfied. Let 0 < µ ≤ 1 − κ. Then the
embedding
DαW l
p (Ω;E (A) , E) ⊂ Lp
(
Ω; (E (A) , E)κ,p
)
is continuous and there exists a positive constant Cµ such that for all u ∈ W l
p (Ω;E (A) , E) the
uniform estimate holds
n∏
k=1
ε
αk/lk
k ‖Dαu‖Lp(Ω;(E(A),E)κ,p)
≤ Cµ
[
hµ ‖u‖W l
p,ε(Ω;E(A),E) + h−(1−µ) ‖u‖Lp(Ω;E)
]
.
3. Statement of the problem. Consider the nonlocal BVP for the following parameter dependent
anisotropic DOE with variable coefficients:
n∑
k=1
εkak (x)Dlk
k u (x) + [A (x) + λ]u (x) +
∑
|α:l|<1
n∏
k=1
ε
αk/lk
k Aα (x)Dαu (x) = f (x) , (3)
mkj∑
i=0
εσikk
[
αkjiD
i
ku (Gk0) + βkjiD
i
ku (Gkb)
]
= 0, j = 1, 2, . . . , lk, k = 1, 2, . . . , n, (4)
where
σik =
1
lk
(
i+
1
p
)
, α = (α1, α2, . . . , αn) , l = (l1, l2, . . . , ln) , |α : l| =
n∑
k=1
αk
lk
,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
988 V. B. SHAKHMUROV
G = {x = (x1, x2, . . . , xn) , 0 < xk < bk} , Gk0 = (x1, x2, . . . , xk−1, 0, xk+1, . . . , xn) ,
Gkb = (x1, x2, . . . , xk−1, bk, xk+1, . . . , xn) , mkj ∈ {0, 1, . . . , lk − 1} ,
x (k) = (x1, x2, . . . , xk−1, xk+1, . . . , xn) , Gk =
∏
j 6=k
(0, bj) , j, k = 1, 2, . . . , n,
αkji, βkji are complex numbers, λ is a complex and εk are small positive parameters; ak are complex-
valued functions on G, A (x) and Aα (x) are linear operators in E for x ∈ G. We assume that the
domain D (A (x)) of operator valued function A (x) is independent of x. So, it will be denote by
D (A) . The same time, the graphical norm E (A (x)) will be denote by E (A) .
A function u ∈ W l
p (G;E (A) , E, Lkj) =
{
u ∈W l
p (G;E (A) , E) , Lkju = 0
}
satisfying (3)
a. e. on G is said to be solution of the problem (3), (4).
We say the problem (3), (4) is Lp-separable, if for all f ∈ Lp (G;E) there exists a unique
solution u ∈ W l
p (G;E (A) , E) of the problem (3), (4) and a positive constant C depending only on
G, p, l, E,A such that the following uniform coercive estimate holds:
n∑
k=1
εk
∥∥∥Dlk
k u
∥∥∥
Lp(G;E)
+ ‖Au‖Lp(G;E) ≤ C ‖f‖Lp(G;E) .
By applying the trace theorem [25] (§ 1.8.2) we have the following theorem.
Theorem A4. Let m and j be integer numbers, 0 ≤ j ≤ m − 1, θj =
pj + 1
pm
, 0 < ε ≤ 1,
x0 ∈ [0, b] . Then, for u ∈ Wm
p (0, b;E0, E) the transformations u → u(j) (x0) are bounded linear
from Wm
p (0, b;E0, E) onto (E0, E)θj ,p and the following inequality holds:
εθj
∥∥∥u(j) (x0)
∥∥∥
(E0,E)θj ,p
≤ C
(∥∥∥εu(m)
∥∥∥
Lp(0,b;E)
+ ‖u‖Lp (0,b;E0)
)
.
Proof. By virtue of [25] (§ 1.8.2), for u ∈Wm
p (0, b;E0, E) the following inequality holds:∥∥∥u(j) (x0)
∥∥∥
(E0,E)θj ,p
≤ C
(∥∥∥u(m)
∥∥∥
Lp(0,b;E)
+ ‖u‖Lp (0,b;E0)
)
.
Putting ũ (x) = u (µx) for 0 < µ < 1 and by applying the above estimate to ũ (x) we have
µj
∥∥∥u(j) (x0)
∥∥∥
(E0,E)θj ,p
≤
≤ C
µm
b∫
0
∥∥∥u(m) (µx)
∥∥∥p
E
dx
1/p
+
b∫
0
‖u (µx)‖pEo dx
1/p
.
Substituting y = µx, in view of µ < 1 we get
µj
∥∥∥u(j) (x0)
∥∥∥
(E0,E)θj ,p
≤ C
[
µm−1/p
∥∥∥u(m)
∥∥∥
Lp(0,µb;E)
+ µ−1/p ‖u‖Lp (0,µb;E0)
]
≤
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ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS AND APPLICATIONS 989
≤ C
[
µm−1/p
∥∥∥u(m)
∥∥∥
Lp(0,b;E)
+ µ−1/p ‖u‖Lp (0,b;E0)
]
.
By chousing µm = ε we obtain the assertion.
Let
Gkx0 = (x1, x2, . . . , xk−1, x0, xk+1, . . . , xn) , x0 ∈ (0, bk) , k = 1, 2, . . . , n,
Xk = Lp (Gk;E) , Yk = W l(k)
p (Gk;E (A) , E) , l(k) = (l1, l2, . . . , lk−1, lk+1, . . . , ln) .
By virtue of Theorem A4 we obtain the following theorem.
Theorem A5. Let lk and j be integer numbers, θjk =
1 + pj + 1
plk
, xk0 ∈ [0, bk] , j = 0, 1, . . . , lk−
− 1, k = 1, 2, . . . , n. Then, for any u ∈ W l
p (G;E0, E) the transformation u → Dj
ku (Gkx0) is
bounded linear from W l
p (G;E0, E) onto Fkj and the following uniform estimate holds:
ε
θjk
k
∥∥∥Dj
ku (Gkx0)
∥∥∥
Fkj
≤ C
‖u‖Lp(G;E) + εk
∥∥∥Dlk
k u
∥∥∥
Lp(G;E)
+
∑
j 6=k
∥∥∥Dlj
j u
∥∥∥
Lp(G;E)
.
Proof. It is clear that
W l
p (G;E0, E) = W lk
p (0, bk;Yk, Xk) .
Then by applying the Theorem A3 to the space W lk
p (0, bk;Yk, Xk) we obtain the assertion.
4. BVP for partial DOE with parameters. Let us first consider the BVP for the parameter-
dependent DOE with constant coefficients
(Lε + λ)u =
n∑
k=1
εkakD
lk
k u (x) + (A+ λ)u (x) = f (x) , (5)
Lkju =
mkj∑
i=0
εσikk
[
αkjiD
i
ku (Gk0) + βkjiD
i
ku (Gkb)
]
= fkj , (6)
j = 1, 2, . . . , lk, k = 1, 2, . . . , n,
where σik, Gk0 and Gkb are defined by (4), ak are complex numbers, λ is a complex and εk are
small positive parameters and A is a linear operator in a Banach space E. Let ωk1, ωk2, . . . , ωklk be
the roots of the characteristic equations
ak ω
lk + 1 = 0, k = 1, 2, . . . , n.
Let [υknj ] be lk-dimensional matrix, and ηk = |[υknj ]| be determinant of matrix [υkij ] , where
υkij = αkjmj (−ωki)lk , i = 1, 2, . . . , dk, υkij = βkjmjω
lk
ki,
i = dk + 1, dk + 2, . . . , lk, 0 < dk < lk, i, j = 1, 2, . . . , lk.
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990 V. B. SHAKHMUROV
Condition 1. Assume the following conditions are satisfied:
(1) E is a Banach space satisfying the multiplier condition;
(2) A is an R-positive operator in E for ϕ ∈ [0 , π) ;
(3) ak 6= 0,
∣∣αkjmj ∣∣+
∣∣βkjmj ∣∣ > 0, ηk 6= 0 and
|argωkj − π| ≤
π
2
− ϕ, j = 1, 2, . . . , dk, |argωkj | ≤
π
2
− ϕ, j = dk + 1, . . . , lk
for 0 < dk < lk, k = 1, 2, . . . , n.
Consider at first, the homogenous BVP
(Lε + λ)u =
n∑
k=1
εkakD
lk
k u (x) + (A+ λ)u (x) = f (x) , (7)
Lkju = 0, j = 1, 2, . . . , lk. (8)
Let B (ε) denote the operator in Lp (G;E) generated by BVP (7), (8), i. e., the operator defined
as
D (B (ε)) = W l
p (G;E (A) , E, Lkj) =
{
u ∈W l
p (G;E (A) , E) , Lkju = 0,
j = 1, 2, . . . , lk, k = 1, 2, . . . , n, B (ε)u =
n∑
k=1
εkakD
lk
k u+Au
}
.
In a similar way as [5] (Theorem 5.1), [18] and [24] we obtain the following theorem.
Theorem A6. Let Condition 1 be satisfied. Then:
(a) problem (7), (8) for f ∈ Lp (G;E) , λ ∈ Sϕ, ϕ ∈ [0 , π) and sufficiently large |λ| has a unique
solution u that belongs to W l
p (G;E (A) , E) and the following coercive uniform estimate holds:
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥Di
ku
∥∥
Lp(G;E)
+ ‖Au‖Lp(G;E) ≤M ‖f‖Lp(G;E) ; (9)
(b) the operator B (ε) is uniformly R-positive in Lp (G;E) .
Now let
Fkj = (Yk, Xk) 1+pmkj
plk
,p
.
From Theorems A5 and A6 we have the following theorem.
Theorem A7. Suppose Condition 1 is satisfied. Then for sufficiently large |λ| with |arg λ| ≤ ϕ
problem (5), (6) has a unique solution u ∈ W l
p (G;E (A) , E) for all f ∈ Lp (G;E) and fkj ∈ Fkj .
Moreover, the following uniform coercive estimate holds:
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥Di
ku
∥∥
Lp(G;E)
+ ‖Au‖Lp(G;E) ≤
≤M
‖f‖Lp(G;E) +
n∑
k=1
lk∑
j=1
‖fkj‖Fkj
. (10)
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ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS AND APPLICATIONS 991
Consider the BVP (3), (4). Let ωk1 (x) , ωk2 (x) , . . . , ωklk (x) denote the roots of the characteristic
equations
ak (x) ωlk + 1 = 0, k = 1, 2, . . . , n.
Let [υknj ] be lk-dimensional matrix, and ηk (x) = |[υknj ]| be determinant of matrix [υkij ] , where
υkij = αkjmj (−ωki)lk , i = 1, 2, . . . , dk, υkij = βkjmjω
lk
ki,
i = dk + 1, dk + 2, . . . , lk, 0 < dk < lk, i, j = 1, 2, . . . , lk.
Condition 2. Assume:
(1) E is a Banach space satisfying the multiplier condition;
(2) operator valued function A (x) is a uniformly R-positive operator in E for ϕ ∈ [0 , π) ;
(3) ak 6= 0,
∣∣αkjmj ∣∣+
∣∣βkjmj ∣∣ > 0, ηk (x) 6= 0 and
|argωkj − π| ≤
π
2
− ϕ, j = 1, 2, . . . , dk, |argωkj | ≤
π
2
− ϕ, j = dk + 1, . . . , lk,
for x ∈ G, 0 < dk < lk, k = 1, 2, . . . , n.
Remark 3. Let lk = 2mk, k = 1, 2, . . . , n, and ak = (−1)mk bk (x) , where bk are real-valued
positive functions and mk are natural numbers. Then Condition 2 is satisfied for ϕ ∈ [0 , π) .
Theorem 1. Suppose Condition 2 is satisfied and the following hold:
(1) ak (x) are continuous functions on Ḡ, aj (0, x (k)) = aj (bk, x (k)) ;
(2) A (x)A−1 (x̄) ∈ C
(
Ḡ;B (E)
)
, A (0, x (k)) = A (bk, x (k)) ;
(3) Aα (x)A(1−|α:l|−µ) (x) ∈ L∞ (G;B (E)) for 0 < µ < 1− |α : l| .
Then problem (3), (4) has a unique solution u ∈W l
p (G;E (A) , E) for f ∈ Lp (G;E) and λ ∈ Sϕ
with large enough |λ| . Moreover, the following coercive uniform estimate holds:
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥Di
ku
∥∥
Lp(G;E)
+ ‖Au‖Lp(G;E) ≤ C ‖f‖Lp(G;E) . (11)
Proof. First we will show the uniqueness of the solution. For this aim we use microlocal analysis.
Let D1,D2, . . . , DN be rectangular regions with sides parallel to coordinate planes covering G and
let ϕ1, ϕ2, . . . , ϕN be a corresponding partition of unity, i.e., ϕj ∈ C∞0 (G) , σj = supp ϕj ⊂ Dj and∑N
j=1
ϕj (x) = 1, where C∞0 (G) denotes the space of all infinitely differentiable functions on G
with compact support. Now for u ∈ W l
p (G;E (A) , E, Lki) , being solution of the equation (3) and
uj (x) = u (x)ϕj (x) we get
( Lε + λ)uj =
n∑
k=1
εkak (x)Dlk
k uj (x) + (A (x) + λ)uj (x) = fj (x) , Lkiuj = 0, (12)
where
fj (x) = f (x)ϕj (x) +
n∑
k=1
εkak (x)
lk−1∑
i=0
C lki
(
Di
ku (x)
) (
Dlk−i
k ϕj (x)
)
−
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992 V. B. SHAKHMUROV
−
∑
|α:l|<1
n∏
k=1
ε
αk/lk
k ϕj (x)Aα (x)Dαu (x) , i = 1, 2, . . . , lk. (13)
Freezing the coefficients of the equation (12), extending uj (x) in outside of σj we obtain the BVP
n∑
k=1
εkak (x0j)D
lk
k uj (x) + (A (x0j) + λ)uj (x) = Fj (x) , x ∈ Dj , (14)
Lkiuj = 0, i = 1, 2, . . . , lk, k = 1, 2, . . . , n,
where
Fj = fj + [A (x0j)−A (x)]uj +
n∑
k=1
εk [ak (x0j)− ak (x)]Dlk
k uj (x) , (15)
and Cki -are usual coefficients of binomial. By applying Theorem A6 for all u ∈ W l
p (Dj ;E (A) , E)
we obtain the following a priori estimate:
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥Di
kuj
∥∥
Lp(Dj ;E)
+ ‖Auj‖Lp(Dj ;E) ≤ C ‖Fj‖Lp(Dj ;E) (16)
for problems (14) defined on domains Dj containing the boundary points. In a similar way we obtain
the same estimates for domains Dj ⊂ G. By using the representation of Fj , by Theorem A1, in view
of the continuity of coefficients, choosing diameters of supp ϕj sufficiently small we get that for all
small δ there is a positive continuous function C (δ) so that
‖Fj‖Lp(Dj ;E) ≤ ‖f.ϕj‖Lp(Dj ;E) + δ ‖uj‖W l
p,ε(Dj ;E(A),E) + C (δ) ‖uj‖Lp(Dj ;E) . (17)
Consequently, from (15) – (17) we have
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥Di
kuj
∥∥
Lp(Dj ;E)
+ ‖Auj‖Lp(Dj ;E) ≤ C ‖f‖Lp(Dj ;E) +
+δ ‖uj‖W l
p,ε(Dj ;E(A),E) +M (δ) ‖uj‖Lp(Dj ;E) . (18)
Choosing εk < 1 from (18) we obtain
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥Di
kuj
∥∥
Lp(Dj ;E)
+ ‖Auj‖Lp(Dj ;E) ≤
≤ C
[
‖f‖Lp(Dj ;E) + ‖uj‖Lp(Dj ;E)
]
. (19)
Then by using the equality u (x) =
∑N
j=1
uj (x) and (19) we get (11) . Let Oε denote the operator
generated by problem (3), (4) for λ = 0, i.e.,
D (Oε) = W l
p (G;E (A) , E, Lkj) ,
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ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS AND APPLICATIONS 993
Oεu =
n∑
k=1
εkak (x)Dlk
k u+A (x)u+
∑
|α:l|<1
n∏
k=1
εαkk Aα (x)Dαu.
It is clear that
‖u‖Lp(G;E) =
1
|λ|
‖(Oε + λ)u−Oεu‖Lp(G;E) ≤
≤ 1
|λ|
[
‖(Oε + λ)u‖Lp(G;E). + ‖Oεu‖Lp(G;E)
]
.
Hence, by using the definition of W l
p (G;E (A) , E) and applying Theorem A1 we obtain
‖u‖p ≤
C
|λ|
[
‖(Oε + λ)u‖Lp(G;E) + ‖u‖W l
p,ε(G;E(A),E)
]
.
From the above estimate we have
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥Di
ku
∥∥
Lp(G;E)
+ ‖Au‖Lp(G;E) ≤ C ‖(Oε + λ)u‖Lp(G;E) . (20)
The estimate (20) implies that uniqueness of solution of the problem (3), (4). It implies that the
operator Oε + λ has a bounded inverse in its rank space. We need to show that this rank space
coincides with the space Lp (G;E) , i.e., we have to show that for all f ∈ Lp (G;E) there is a unique
solution of the problem (3), (4). We consider the smooth functions gj = gj (x) with respect to a
partition of unity ϕj = ϕj (y) on the region G that equals 1 on supp ϕj , where supp gj ⊂ Dj and
|gj (x)| < 1. Let us construct for all j the functions uj that are defined on the regions Ωj = G ∩Dj
and satisfying problem (3), (4). The problem (3), (4) can be expressed as
n∑
k=1
εkak (x0j)D
lk
k uj (x) +Aλ (x0j)uj (x) = gj
{
f + [A (x0j)−A (x)]uj+
+
n∑
k=1
εk [ak (x0j)− ak (x)]Dlk
k uj −
∑
|α:l|<1
n∏
k=1
ε
αk/lk
k Aα (x)Dαuj
}
, x ∈ Dj , (21)
Lkiuj = 0, j = 1, 2, . . . , N.
Consider operators Ojλ (ε) = Oj (ε) + λ in Lp (Dj ;E) that are generated by BVPs (14), i. e.,
D (Oj (ε)) = W l
p (Dj ;E (A) , E, Lki) , i = 1, 2, . . . , lk, k = 1, 2, . . . , n,
Ojλ (ε)u =
n∑
k=1
εkak (x0j)D
lk
k uj (x) + [A (x0j) + λ]uj (x) , x ∈ Dj , j = 1, . . . , N.
By virtue of Theorem A6, the local operators Ojλ have inverses O−1
jλ for |arg λ| ≤ ϕ and for suffi-
ciently large |λ| . Moreover, the operators O−1
jλ are bounded from Lp (Dj ;E) to W l
p (Dj ;E (A) , E)
and for f ∈ Lp (Dj ;E) we have the following uniform estimate:
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994 V. B. SHAKHMUROV
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥∥Di
kO
−1
jλ f
∥∥∥
Lp(Dj ;E)
+
∥∥∥AO−1
jλ f
∥∥∥
Lp(Dj ;E)
≤ C ‖f‖Lp(Dj ;E) . (22)
Extending solutions uj of problems (21) zero on outside of supp ϕj and using the substitutions
uj = O−1
jλ υj we obtain the operator equations
υj = Kjλυj + gjf, j = 1, 2, . . . , N, (23)
where Kjλ = Kjλ (ε) are bounded linear operators in Lp (Dj ;E) defined by
Kjλ = Kjλ (ε) = gj
{
f + [A (x0j)−A (x)]O−1
jλ +
+
n∑
k=1
εk [ak (x0j)− ak (x)]Dlk
k O
−1
jλ −
∑
|α : l|<1
n∏
k=1
ε
αk/lk
k Aα (x)DαO−1
jλ
}
.
In fact, due to smoothness of the coefficients of the expression Kjλ and in view of the estimate (22),
for sufficiently large |λ| there is a sufficiently small δ > 0 such that∥∥∥[A (x0j)−A (x)]O−1
jλ υj
∥∥∥
Lp(Dj ;E)
≤ δ ‖υj‖Lp(Dj ;E) ,
n∑
k=1
εk
∥∥∥[ak (x0j)− ak (x)]Dlk
k O
−1
jλ υj
∥∥∥
Lp(Dj ;E)
≤ δ ‖υj‖Lp(Dj ;E) .
Moreover, from the assumption (2) and by Theorem A1 we obtain that for all δ > 0 there is a constant
C (δ) > 0 such that
∑
|α:l|<1
n∏
k=1
ε
αk/lk
k
∥∥∥Aα (x)DαO−1
jλ υj
∥∥∥
Lp(Dj ;E)
≤ δ ‖υj‖W l
p(Dj ;E(A),E) + C (δ) ‖υj‖Lp(Dj ;E) .
Hence, for |arg λ| ≤ ϕ with sufficiently large |λ| there is a γ ∈ (0, 1) such that ‖Kjλ‖ < γ.
Consequently, equations (23) for all j have a unique solution υj = [I −Kjλ]−1 gjf . Moreover,
‖υj‖Lp(Dj ;E) =
∥∥∥[I −Kjλ]−1 gjf
∥∥∥
Lp(Dj ;E)
≤ ‖f‖Lp(Dj ;E) .
Thus, [I −Kjλ]−1 gj are bounded linear operators from Lp (G;E) to Lp (Dj ;E) . Thus, the functions
uj = Ujλf = O−1
jλ [I −Kjλ]−1 gjf
are solutions of (21). Consider the following linear operator U = Uε in Lp (G;E) defined by
D (U) = W l
p (G;E (A) , E, Lkj) , j = 1, 2, . . . , lk, k = 1, 2, . . . , n,
Uf =
N∑
j=1
ϕj (y)Ujλf = O−1
jλ [I −Kjλ]−1 gjf, j = 1, 2, . . . , N.
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ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS AND APPLICATIONS 995
It is clear from the constructions Ujλand from the estimate (22) that the operators Ujλ are bounded
linear from Lp (G;E) to W l
p (Dj ;E (A) , E) and for |arg λ| ≤ ϕ with sufficiently large |λ| we have
n∑
k=1
lk∑
i=0
|λ|1−i/lk εk
∥∥Di
kUjλf
∥∥
Lp(Dj ;E)
+ ‖AUjλf‖Lp(Dj ;E) ≤ C ‖f‖Lp(G;E) . (24)
Therefore, U is a bounded linear operator in Lp (G;E) . By contraction of solution operators Ujλ
of local equations (21), acting Oε + λ to u =
∑N
j=1
ϕjUjλf gives
(Oε + λ)u =
N∑
j=1
(Oε + λ) (ϕjUjλf) =
=
N∑
j=1
[ϕj (Oε + λ) (Ujλf) + Φjλf ] =
N∑
j=1
ϕjgjf +
N∑
j=1
Φjλf = f +
N∑
j=1
Φjλf,
where Φjλ = Φjλ (ε) are bounded linear operators defined by
Φjλf =
{
n∑
k=1
εkak
lk−1∑
i=0
C lki
(
Di
kUjλf
)
Dlk−i
k ϕj +
+
∑
|α:l|<1
Aα
n∏
k=1
ε
αk/lk
k
αk−1∑
i=0
Cαki
(
Di
k (Ujλf)
)
Dαk−i
k ϕj
}
.
Indeed, from Theorem A1, the estimate (24) and from the expression Φjλ we obtain that the operators
Φjλ are bounded linear from Lp (G;E) to Lp (G;E) and for |arg λ| ≤ ϕ with sufficiently large |λ|
there is an δ ∈ (0, 1) such that ‖Φjλ‖ < δ. Therefore, there exists a bounded linear invertible operator(
I +
∑N
j=1
Φjλ
)−1
, i.e., we infer for all f ∈ Lp (G;E) that the BVP (3), (4) has a unique solution
u (x) = (Oε + λ)−1 f =
N∑
j=1
ϕjO
−1
jλ [I −Kjλ]−1 gj
I +
N∑
j=1
Φjλ
−1
f.
Result 1. Theorem 1 implies that the resolvent (Oε + λ)−1 satisfies the following anisotropic
type uniform sharp estimate
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥∥Di
k (Oε+λ)−1
∥∥∥
B(Lp(G;E))
+
∥∥∥A (Oε + λ)−1
∥∥∥
B(Lp(G;E))
≤ C
for |arg λ| ≤ ϕ and ϕ ∈ [0 , π) .
Theorem 2. Let all conditions of Theorem 1 hold and A−1 ∈ σ∞ (E) . Then the operator Oε is
Fredholm from W l
p (G;E (A) , E) into Lp (G;E) .
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996 V. B. SHAKHMUROV
Proof. Theorem 1 implies that the operator Oε+λ for sufficiently large |λ| has a bounded inverse
(Oε+λ)−1 from Lp (G;E) to W l
p (G;E (A) , E) , that is the operator Qε+λ is Fredholm from
W l
p (G;E (A) , E) into Lp (G;E) . Then, by Theorem A2 and the perturbation theory of linear
operators we obtain that the operator Oε is Fredholm from W l
p (G;E (A) , E) into Lp (G;E) .
Example 1. Now, let us consider a special case of (3), (4). Let E = C, l1 = 2 and l2 = 4,
n = 2, G = (0, 1)× (0, 1) and A = a (x, y) > 0, i. e., consider the problem
Lεu = −ε1a1
∂2u
∂x2
+ ε2a2
∂4u
∂y4
+ bε
1/2
1 ε
1/4
2
∂2u
∂x∂y
+ au = f (x, y) ,
m1j∑
i=0
εσi11
[
αjiu
(i)
x (0, y) +
m1j∑
i=0
βjiu
(i)
x (1, y)
]
= 0, j = 1, 2, (25)
m2j∑
i=0
εσi22
[
αjiu
(i)
y (0, y) +
m1j∑
i=0
βjiu
(i)
y (1, y)
]
= 0, j = 1, 2, 3, 4,
where ε1 and ε2 are positive parameters, ak = ak (x, y) , k = 1, 2 are real-valued functions on G and
σi1 =
1
2
(
i+
1
p
)
, σi2 =
1
4
(
i+
1
p
)
, m1j ∈ {0, 1} , m2j ∈ {0, 1, 2, 3} ,
ak 6= 0,
∣∣αkjmj ∣∣+
∣∣βkjmj ∣∣ > 0, ηk 6= 0,
a, ak > 0, a, a1, a2 ∈ C
(
Ḡ
)
, b ∈ L∞ (G) , a (0, y) = a (1, y) , a (x, 0) = a (x, 1) ,
ak (0, y) = ak (1, y) , ak (x, 0) = ak (x, 1) , x, y ∈ G, k = 1, 2.
Result 2. Theorem 1 implies that for each f ∈ Lp (G) and sufficiently large a the problem (25)
has a unique solution u ∈W l
p(G) satisfying the uniform coercive estimate
ε1
∥∥D2
xu
∥∥
Lp(G)
+ ε2
∥∥∥D[4]
y u
∥∥∥
Lp(G)
+ ‖u‖Lp(G) ≤ C ‖f‖Lp(G) .
Example 2. Consider the following BVP for the system of anisotropic PDEs with variable
coefficients
n∑
k=1
(−1)mk εkbk (x)D2mk
k um (x) + (dm (x) + λ)um (x) = fm (x) ,
mkj∑
i=0
αkjiε
i
kD
i
kum (Gk0) +
mkj∑
i=0
βkjiε
i
kD
i
kum (Gkb) = 0, k = 1, 2, . . . , n,
j = 1, 2, . . . , 2mk, m = 1, 2, . . . , ν,
where bk are positive continuous function on G, E = Cν , λ is a complex, εk, k = 1, 2, . . . , n, are
positive parameters and dm (x) > 0, m = 1, 2, . . . , ν.
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ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS AND APPLICATIONS 997
Result 3. Let bk, dm ∈ C
(
Ḡ
)
, bk 6= 0,
∣∣αkjmj ∣∣+∣∣βkjmj ∣∣ > 0, ηk 6= 0 and bj (Gk0) = bj (Gkb) ,
dm (Gk0) = dm (Gkb) . Then, Theorem 1 implies that for each f ∈ Lp (G;Cν) and for all λ ∈ S (ϕ)
with sufficiently large |λ| the above problem has a unique solution u ∈ W l
p(G;Cν) satisfying the
uniform coercive estimate
n∑
k=1
2mk∑
i=0
|λ|1−i/2mk εi/2mkk
∥∥Di
ku
∥∥
Lp(G;Cν)
≤ C ‖f‖Lp(G;Cν) .
5. Abstract Cauchy problem for parabolic equation with small parameters. Consider now
mixed BVP for the following parabolic equation with small parameters, i. e.,
∂u
∂t
+
n∑
k=1
εkak (x)Dlk
k u+A (x)u = f (t, x) , (26)
mkj∑
i=0
εσkik
[
αkjiD
i
ku (t, Gk0) + βkjiD
i
ku (t, Gkb)
]
= 0, j = 1, 2, . . . , lk, (27)
u (0, x) = 0, σki =
1
lk
(
i+
1
p
)
, t ∈ R+, x ∈ G,
where A (x) is an operator function in a Banach space E for x ∈ G, ak are complex valued functions,
εk are small positive parameters, G, Gk0 and Gkb are domains defined in the problem (3), (4).
In this section, we obtaın the existence and uniqueness of the maximal regular solution of problem
(26), (27) in mixed Lp-norms.
Let Oε denote differential operator generated by (3), (4) for λ = 0.
Theorem 3. Let all conditions of Theorem 1 are hold for Aα = 0 and ϕ ∈
(π
2
, π
)
. Then:
(a) the operator Oε is an R-positive in Lp (G;E) ;
(b) the operator Oε is a generator of an analytic semigroup.
Proof. Really, by virtue of Theorem 1 we obtain that for f ∈ Lp (G;E) the BVP (3), (4) have a
unique solution expressing in the form
u (x) = (Oε + λ)−1 f =
N∑
j=1
ϕjO
−1
jλ [I −Kjλ]−1 gj
I +
N∑
j=1
Φjλ
−1
f,
where Ojλ = Oj (ε) + λ are local operators generated by BVPs with constant coefficients of type
(7), (8) and Kjλ = Kjλ (ε) ,Φjλ = Φjλ (ε) are uniformly bounded operators defined in the proof of
the Theorem 1. By virtue of Theorem A6 operators Oj (ε) are R-positive. Then by using the above
representation and by virtue of Kahane’s contraction principle, product and additional properties of
the collection of R-bounded operators (see, e.g., [8], Lemma 3.5, Proposition 3.4) we obtain the
assertions.
Theorem 4. Let all conditions of Theorem 3 hold. Then for f ∈ Lp (G+;E) problem (26), (27)
has a unique solution u ∈ W 1,l
p (G+;E (A) , E) and the following uniform coercive estimate holds:∥∥∥∥∂u∂t
∥∥∥∥
Lp(G+;E)
+
n∑
k=1
εk
∥∥∥Dlk
k u
∥∥∥
Lp(G+;E)
+ ‖Au‖Lp(G+;E) ≤ C ‖f‖Lp(G+;E) .
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998 V. B. SHAKHMUROV
Proof. The problem (26), (27) can be expressed as the following Cauchy problem:
du
dt
+Oεu (t) = f (t) , u (0) = 0. (28)
The Theorem 3 implies that the operator Oε is R-positive and also is a generator of an analytic
semigroup in F = Lp (G;E) . Then by virtue of [1] or [26] (Theorem 4.2) we obtain that for all
f ∈ Lp1 ((R+) ;F ) problem (28) has a unique solution u ∈W 1
p1 ((0, 1) ;D (O) , F ) and the following
uniform estimate holds:∥∥∥∥dudt
∥∥∥∥
Lp1 (R+;F )
+ ‖Oεu‖Lp1 (R+;F ) ≤ C ‖f‖Lp1 (R+;F ) . (29)
Since Lp1 (0, 1;F ) = Lp (G+;E) , by Theorem1 we have ‖Oεu‖Lp1 (R+;F ) = D (Oε) . This
relation and the estimate (29) implies the assertion.
6. BVPs for quasielliptic PDE with small parameters. In this section, maximal regularity
properties of anisotropic PDE with small parameters are studied. Maximal regularity properties for
PDEs have been studied, e.g., in [8] for smooth domains and in [12] for nonsmooth domains. In this
section, consider the following BVP with small parameters:
Lu =
n∑
k=1
εkak (x)Dlk
k u (x, y) +
∑
|α|≤2m
aα (y)Dα
y u (x, y) +
+
∑
|β:l|<1
n∏
k=1
ε
αk/lk
k bβ (x, y)Dβ
yu (x, y) + λu (x, y) = f (x, y) , x ∈ G, y ∈ Ω, (30)
Lkju =
mkj∑
i=0
εσkik
[
αkjiD
i
ku (Gk0, y) + βkjiD
i
ku (Gkb, y)
]
= 0, y ∈ Ω, (31)
j = 1, 2, . . . , lk, x (k) ∈ Gk,
Bju =
∑
|β|≤mj
bjβ (y)Dβ
yu (x, y) |y∈∂Ω= 0, x ∈ G, j = 1, 2, . . . ,m, (32)
where Dj = −i ∂
∂yj
, αkji, βkji are complex number, λ is a complex and εk are small positive
parameter, y = (y1, . . . , yµ) ∈ Ω ⊂ Rµ and
σki =
1
lk
(
i+
1
p
)
, G = {x = (x1, x2, . . . , xn) , 0 < xk < bk} ,
Gk0 = (x1, x2, . . . , xk−1, 0, xk+1, . . . , xn) , Gkb = (x1, x2, . . . , xk−1, bk, xk+1, . . . , xn) ,
mkj ∈ {0, 1, . . . , lk − 1} ,
∣∣αkjmj ∣∣+
∣∣βkjmj ∣∣ > 0, j = 1, 2, . . . , lk,
x (k) = (x1, x2, . . . , xk−1, xk+1, . . . , xn) , Gk =
∏
j 6=k
(0, bj) , j, k = 1, 2, . . . , n.
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ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS AND APPLICATIONS 999
Let ωkj = ωkj (x) , j = 1, 2, . . . , lk, k = 1, 2, . . . , n, denote the roots of the equations
ak (x)ωlk + 1 = 0.
Let Qε denote the operator generated by BVP (30) – (33). Let
F = B
(
Lp
(
Ω̃
))
, Ω̃ = G× Ω.
Theorem 5. Let the following conditions be satisfied:
(1) aα ∈ C
(
Ω̄
)
for each |α| = 2m and aα ∈ [L∞ + Lrk ] (Ω) for each |α| = k < 2m with
rk ≥ p1, p1 ∈ (1,∞) , 2m− k > l
rk
and bβ ∈ L∞(Ω̃);
(2) bjβ ∈ C2m−mj (∂Ω) for each j, β, mj < 2m, p ∈ (1,∞) ;
(3) for y ∈ Ω̄, ξ ∈ Rµ, η ∈ S (ϕ1) , ϕ1 ∈
[
0,
π
2
)
, |ξ|+ |η| 6= 0 let
η +
∑
|α|=2m
aα (y) ξα 6= 0;
(4) for each y0 ∈ ∂Ω the local BVPs in local coordinates corresponding to y0
η +
∑
|α|=2m
aα (y0)Dαϑ (y) = 0,
Bj0ϑ =
∑
|β|=mj
bjβ (y0)Dβϑ (y) = hj , j = 1, 2, . . . ,m,
has a unique solution ϑ ∈ C0 (R+) for all h = (h1, h2, . . . , hm) ∈ Rm and for ξp ∈ Rµ−1 with
|ξ p|+ |η| 6= 0;
(5) ak ∈ C
(
Ḡ
)
, ak (x) 6= 0,
∣∣αkjmj ∣∣+
∣∣βkjmj ∣∣ > 0, ηk (x) 6= 0 and
|argωkj − π| ≤
π
2
− ϕ, j = 1, 2, . . . , dk,
|argωkj | ≤
π
2
− ϕ, ϕ ∈
[
0,
π
2
)
,
j = dk + 1, . . . , lk, 0 < dk < lk, k = 1, 2, . . . , n, x ∈ G.
Then:
(a) problem (30) – (33) has a unique solution u ∈ W l,2m
p (Ω̃) for f ∈ Lp(Ω̃) and λ ∈ Sϕ with
large enough |λ| . Moreover, the following coercive uniform estimate holds:
n∑
k=1
lk∑
i=0
|λ|1−i/lk εi/lkk
∥∥∥Dlk
k u
∥∥∥
Lp(Ω̃)
+
∑
|β|=2m
∥∥∥Dβ
yu
∥∥∥
Lp(Ω̃)
+ ‖u‖Lp(Ω̃) ≤ C ‖f‖Lp(Ω̃) ;
(b) for λ ∈ S (ϕ) and for sufficiently large |λ| there exists a resolvent (Qε + λ)−1 and
n∑
k=1
lk∑
i=0
|λ|1−i/lk εik
∥∥∥Di
k (Qε + λ)−1
∥∥∥
F
+
∥∥∥A (Qε + λ)−1
∥∥∥
F
≤M ;
(c) the problem (30) – (33) is Fredholm in Lp(Ω̃) for λ = 0.
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1000 V. B. SHAKHMUROV
Proof. LetE = Lp1 (Ω) . Then by [8] (Theorem 3.6), part (1) of Condition 2 is satisfied. Consider
the operator A which is defined by
D (A) = W 2m
p1 (Ω;Bju = 0) , Au =
∑
|β|≤2m
aβ (y)Dβu (y) .
For x ∈ Ω we also consider operators
Aα (x)u = bα (x, y)Dαu (y) , |α : l| < 1.
The problem (30) – (33) can be rewritten as the form of (3), (4), where u (x) = u (x, .) and f (x) =
= f (x, .) are functions with values in E = Lp1 (Ω) . From [8] (Theorem 8.2) problem
ηu (y) +
∑
|β|≤2m
aβ (y)Dβu (y) = f (y) ,
Bju =
∑
|β|≤mj
bjβ (y)Dβu (y) = 0, j = 1, 2, . . . ,m,
has a unique solution for f ∈ Lp1 (Ω) and arg η ∈ S (ϕ1) , |η| → ∞. Moreover, the operator A is
R-positive in Lp1 , i.e., all conditions of the Theorem 1 hold.
7. Cauchy problem for infinite systems of parabolic equation with small parameters.
Consider the infinity systems of BVP for the anisotropic PDE with parameters
∂u
∂t
+
n∑
k=1
εkak (x)
∂lkum
∂xlkk
+
∞∑
j=1
(dj (x) + λ)um+
+
∑
|α:l|<1
∞∑
j=1
n∏
k=1
ε
αk/lk
k dαjm (x)Dαuj = fm (t, x) , m = 1, 2, . . . ,∞, (33)
mkj∑
i=0
εσkik
[
αkjiD
(i)
k u (t, Gk0) +
mkj∑
i=0
βkjiD
(i)
k u (t, Gkb)
]
= 0, j = 1, 2, . . . , lk, (34)
u (0, x) = 0, x ∈ G, t ∈ (0,∞) , x (k) ∈ Gk, j = 1, 2, . . . , lk,
where ak, dk, dαjm are complex valued functions, εk are small positive parameters and αkji, βkji
are complex numbers. Let
σki =
1
lk
(
i+
1
p
)
, G = {x = (x1, x2, . . . , xn) , 0 < xk < bk} ,
Gk0 = (x1, x2, . . . , xk−1, 0, xk+1, . . . , xn) ,
Gkb = (x1, x2, . . . , xk−1, bk, xk+1, . . . , xn) , mkj ∈ {0, 1, . . . , lk − 1} ,
x (k) = (x1, x2, . . . , xk−1, xk+1, . . . , xn) , Gk =
∏
j 6=k
(0, bj) , j, k = 1, 2, . . . , n,
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ANISOTROPIC DIFFERENTIAL OPERATORS WITH PARAMETERS AND APPLICATIONS 1001
D (x) = {dm (x)} , dm > 0, u = {um} , du = {dmum} , m = 1, 2, . . . ,∞,
lq (D) =
u : u ∈ lq, ‖u‖lq(D) = ‖Du‖lq =
( ∞∑
m=1
|dmum|q
)1/q
<∞
,
j = 1, 2, . . . , lk, k = 1, 2, . . . , n.
Let V = V (ε) denote the operator in Lp (G; lq) generated by problem (34), (35). Let
G+ = (0,∞)×G, B = B (Lp (G; lq)) .
Theorem 6. Let p ∈ (1,∞) , ak ∈ C
(
Ḡ
)
, ai (0, x (k)) = ai (bk, x (k)) , ak (x) 6= 0,
∣∣αkjmj ∣∣+
+
∣∣βkjmj ∣∣ > 0, ηk (x) 6= 0 and |argωkj − π| ≤
π
2
− ϕ, |argωkj | ≤
π
2
− ϕ, j = 1, 2, . . . , lk,
ϕ ∈ ϕ ∈
[
0,
π
2
)
, x ∈ G, dm ∈ C
(
Ḡ
)
, dαjm ∈ L∞ (G) such that
max
α
sup
m
∞∑
j=1
dαjm (x) d
−(1−|α : l|−µ)
j (x) < M for all x ∈ G and 0 < µ < 1− |α : l| .
Then for f (t, x) = {fm (t, x)}∞1 ∈ Lp (G; lq) , |arg λ| ≤ ϕ and sufficiently large |λ| the problem
(34), (35) has a unique solution u = {um (t, x)}∞1 that belongs to the space W 1,l
p (G+, lq (D) , lq)
and the following coercive uniform estimate holds:∥∥∥∥∂u∂t
∥∥∥∥
Lp(G+;lq)
+
n∑
k=1
εk
∥∥∥Dlk
k u
∥∥∥
Lp(G+;lq)
+ ‖Au‖Lp(G+;lq)
≤ C ‖f‖Lp(G+;lq)
.
Proof. Let E = lq, A and Aα (x) be infinite matrices, such that
A = [dmδmj ] , Aα (x) = [dαjm (x)] , m, j = 1, 2, . . . ,∞.
It is clear that the operator A is R-positive in lq. The problem (34), (35) can be rewritten in the form
(26), (27). Then, from Theorem 4 we obtain that the assertion.
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