Degenerate Anisotropic Differential Operators and Applications

Boundary Value Problems

Volume 2011, Article ID 268032,27pages doi:10.1155/2011/268032

Research Article

Degenerate Anisotropic Differential Operators and Applications

Ravi Agarwal,

Donal O’Regan,

and Veli Shakhmurov

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

2Department of Mathematics, National University of Ireland, Galway, Ireland

3Department of Electronics Engineering and Communication, Okan University, Akfirat, Tuzla 34959 Istanbul, Turkey

Correspondence should be addressed to Veli Shakhmurov,veli.sahmurov@okan.edu.tr Received 2 December 2010; Accepted 18 January 2011

Academic Editor: Gary Lieberman

Copyrightq2011 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-valuedLpspaces 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 equationsDOEs. 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 literaturesee1–14and the references therein. Maximal regularity properties for higher-order degenerate anisotropic DOEs with constant coefficients and nondegenerate equations with variable coefficients were studied in15,16.

The main aim of the present paper is to discuss the separability properties of BVPs for higher-order degenerate DOEs; that is,

n k1

akxD_k^lkux Axux

|α:l|<1

AαxD^αux fx, 1.1

where Dⁱ_k ux γkxk∂/∂xkⁱux, γ_k are weighted functions, A and A_α are linear operators in a Banach SpaceE. The above DOE is a generalized form of an elliptic equation.

In fact, the special casel_k2m,k1, . . . , nreduces1.1to elliptic form.

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 inLp; that is, it has a bounded inverse fromLp to the anisotropic weighted space W_p,γ^l. 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 to8,15–25.

Letγ γxbe a positive measurable weighted function on the regionΩ ⊂ Rⁿ. Let L_p,γΩ;Edenote the space of all strongly measurableE-valued functions that are defined on Ωwith the norm

f

p,γ f

Lp,γΩ;E fx^p

Eγxdx

_1/p

, 1≤p <∞. 1.2 Forγx≡1, the spaceL_p,γΩ;Ewill be denoted byL_pΩ;E.

The weightγwe will consider satisfies anA_pcondition; that is,γ ∈ A_p, 1< p < ∞if there is a positive constantCsuch that

1

|Q|

Q

γxdx 1

|Q|

Q

γ^−1/p−1xdx _p−1

≤C, 1.3

for all cubesQ⊂Rⁿ.

The Banach space E is called a UMD space if the Hilbert operator Hfx limε→0 |x−y|>εfy/x−ydyis bounded inLpR, E,p∈1,∞ see, e.g.,26. UMD spaces include, for example,L_p,l_pspaces, and Lorentz spacesL_pq,p,q∈1,∞.

LetC be the set of complex numbers and S_ϕ

λ; λ∈C,argλ≤ϕ

∪ {0}, 0≤ϕ < π. 1.4

A linear operatorAis said to beϕ-positive in a Banach spaceEwith boundM >0 if DAis dense onEand

A λI⁻¹

LE≤M1 |λ|⁻¹, 1.5

for allλ∈Sϕ, ϕ∈0, π,Iis an identity operator inE, andBEis the space of bounded linear operators inE. SometimesA λIwill be written asA λand denoted byAλ. It is known27, Section 1.15.1that there exists fractional powersA^θ of the sectorial operatorA. LetEA^θ denote the spaceDA^θwith graphical norm

u _EAθ

u ^p A^θu^p1/p

, 1≤p <∞, −∞< θ <∞. 1.6 LetE1andE2be two Banach spaces. Now,E1, E2_θ,p, 0< θ <1, 1≤p≤ ∞will denote interpolation spaces obtained from{E1, E₂}by theKmethod27, Section 1.3.1.

A setW ⊂ BE1, E2is calledR-boundedsee3,25,26if there is a constantC >0 such that for allT₁, T₂, . . . , T_m∈Wandu_1,u₂, . . . , u_m∈E₁,m∈N

₁

0

m j1

r_j y

T_ju_j E2

dy≤C ₁

0

m j1

r_j y

u_j E1

dy, 1.7

where{rj}is a sequence of independent symmetric{−1,1}-valued random variables on0,1.

The smallest C for which the above estimate holds is called an R-bound of the collectionWand is denoted byRW.

Let SRⁿ;E denote the Schwartz class, that is, the space of all E-valued rapidly decreasing smooth functions on Rⁿ. Let F be the Fourier transformation. A functionΨ ∈ CRⁿ;BEis called a Fourier multiplier inL_p,γRⁿ;Eif the mapu → Φu F⁻¹ΨξFu, u∈SRⁿ;Eis well defined and extends to a bounded linear operator inL_p,γRⁿ;E. The set of all multipliers inLp,γRⁿ;Ewill denoted byM^p,γ_p,γE.

Let

V_n

ξ:ξ ξ₁, ξ₂, . . . , ξ_n∈Rⁿ, ξ_j/0 , U_n

β

β₁, β₂, . . . , β_n

∈Nⁿ:β_k∈ {0,1}

. 1.8

Definition 1.1. A Banach spaceEis said to be a space satisfying a multiplier condition if, for anyΨ∈CⁿRⁿ;BE, theR-boundedness of the set{ξ^βD^β_ξΨξ:ξ∈Rⁿ\0, β∈U_n}implies thatΨis a Fourier multiplier inL_p,γRⁿ;E, that is,Ψ∈M^p,γ_p,γEfor anyp∈1,∞.

LetΨh ∈ M^p,γ_p,γEbe a multiplier function dependent on the parameterh ∈ Q. The uniformR-boundedness of the set{ξ^βD^βΨhξ:ξ∈Rⁿ\0, β∈U}; that is,

sup

h∈QR

ξ^βD^βΨhξ:ξ∈Rⁿ\0, β∈U

≤K 1.9

implies thatΨhis a uniform collection of Fourier multipliers.

Definition 1.2. Theϕ-positive operatorAis said to beR-positive in a Banach spaceEif there existsϕ∈0, πsuch that the set{AA ξI⁻¹:ξ∈Sϕ}isR-bounded.

A linear operator Ax is said to be ϕ-positive in E uniformly in x if DAx is independent ofx,DAxis dense inEand Ax λI⁻¹ ≤M/1 |λ|for anyλ ∈S_ϕ, ϕ∈0, π.

Theϕ-positive operator Ax,x ∈ Gis said to be uniformlyR-positive in a Banach spaceEif there existsϕ ∈0, πsuch that the set{AxAx ξI⁻¹ :ξ ∈S_ϕ}is uniformly R-bounded; that is,

sup

x∈GR ξ^βD^β

AxAx ξI⁻¹

:ξ∈Rⁿ\0, β∈U

≤M. 1.10

Letσ_∞E1, E2denote the space of all compact operators fromE1toE2. ForE1 E2E, it is denoted byσ_∞E.

For two sequences{aj}^∞₁ and{bj}^∞₁ of positive numbers, the expressionaj∼bjmeans that there exist positive numbersC₁andC₂such that

C₁a_j≤b_j≤C₂a_j. 1.11

Letσ_∞E1, E₂denote the space of all compact operators fromE₁toE₂. ForE₁ E₂E, it is denoted byσ_∞E.

Now,sjAdenotes the approximation numbers of operatorAsee, e.g.,27, Section 1.16.1. Let

σqE1, E2

⎧⎨

⎩A:A∈σ_∞E1, E2,^∞

j1

s^q_jA<∞,1≤q <∞

⎫⎬

⎭. 1.12

LetE₀andEbe two Banach spaces andE₀continuously and densely embedded into Eandl l1, l2, . . . , ln.

We let W_p,γ^l Ω;E₀, E denote the space of all functions u ∈ L_p,γΩ;E₀ possessing generalized derivativesD^l_k^ku∂^lku/∂x^l_k^ksuch thatD^l_k^ku∈L_p,γΩ;Ewith the norm

u _Wp,γ^l _Ω;E0,E u _Lp,γΩ;E0 ⁿ

k1

D^l_k^ku

Lp,γΩ;E<∞. 1.13

LetD_kⁱux γkxk∂/∂xkⁱux. Consider the following weighted spaces of func- tions:

W_p,γ^lG;EA, E

u:u∈LpG;EA, D^l_k^ku∈LpG;E,

u _W^l

p,γG;EA,E u _LpG;EA ⁿ

k1

D_k^lku

LpG;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 from24.

Theorem A1. Letα α1, α₂, . . . , α_nandD^αD^α₁¹D^α₂²· · ·D^α_nⁿand suppose that the following con- ditions are satisfied:

1Eis a Banach space satisfying the multiplier condition with respect topandγ, 2Ais anR-positive operator inE,

3α α1, α2, . . . , αnandl l1, l2, . . . , lnaren-tuples of nonnegative integer such that κⁿ

k1

α_k

lk ≤1, 0≤μ≤1−κ, 1< p <∞, 2.1

4 Ω ⊂ Rⁿ is a region such that there exists a bounded linear extension operator from W_p,γ^l Ω;EA, EtoW_p,γ^l Rⁿ;EA, E.

Then, the embeddingD^αW_p,γ^l Ω;EA, E ⊂ Lp,γΩ;EA^1−κ−μ is continuous. Moreover, for all positive numberh <∞andu∈W_p,γ^l Ω;EA, E, the following estimate holds

D^αu _Lp,γΩ;EA^1−κ−μ ≤h^μ u _Wp,γ^l _Ω;EA,E h^−1−μ u _Lp,γΩ;E. 2.2

Theorem A2. Suppose that all conditions of Theorem A1 are satisfied. Moreover, letγ ∈A_p,Ωbe a bounded region andA⁻¹∈σ_∞E. Then, the embedding

W_p,γ^l Ω;EA, E⊂L_p,γΩ;E 2.3

is compact.

Let SpAdenote the closure of the linear span of the root vectors of the linear operatorA.

From18, Theorem 3.4.1, we have the following.

Theorem A3. Assume that

1Eis an UMD space andAis an operator inσ_pE,p∈1,∞,

2μ₁, μ₂, . . . , μ_s are non overlapping, differentiable arcs in the complex plane starting at the origin. Suppose that each of thesregions into which the planes are divided by these arcs is contained in an angular sector of opening less thenπ/p,

3m >0 is an integer so that the resolvent ofAsatisfies the inequality Rλ, A O

|λ|⁻¹

, 2.4

asλ → 0 along any of the arcsμ.

Then, the subspace SpAcontains the spaceE.

Let

G{x x₁, x₂, . . . , x_n: 0< x_k< b_k}, γx x^γ₁¹x₂^γ2· · ·x_n^γn. 2.5 Let

β_kx^β_k^k, νⁿ

k1

x^ν_k^k, γⁿ

k1

x^γ_k^k. 2.6

Let I IW_p,β,γ^l Ω;EA, E, Lp,γΩ;E denote the embedding operator W_p,β,γ^l Ω;EA, E → Lp,νΩ;E.

From15, Theorem 2.8, we have the following.

Theorem A4. LetE0andEbe 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_jIE0, E∼j^−1/k0, k₀ >0, j 1,2, . . . ,∞, κ0ⁿ

k1

γk−νk

p lk−βk

<1. 2.7

Then,

s_j I

W_p,β,γ^l G;E₀, E, Lp,νG;E

∼j^{−1/k0 κ0}. 2.8

3. Statement of the Problem

Consider the BVPs for the degenerate anisotropic DOE n

k1

akxD_k^lkux Ax λux

|α:l|<1

AαxD^αux fx, 3.1

mkj

i0

αkjiDⁱ_k uGk0 0, j1,2, . . . , dk,

mkj

i0

β_kjiDⁱ_k uGkb 0, j1,2, . . . , l_k−d_k, d_k∈0, lk,

3.2

where

α α₁, α₂, . . . , α_n, l l₁, l₂, . . . , l_n, |α:l|ⁿ

k1

α_k l_k, G{x x1, x2, . . . , xn,0< xk< bk,}, α α1, α2, . . . , αn, D^α D^α₁ ¹D₂^α2· · ·D^α_nⁿ, Dⁱ_k ux

x^γ_k^kbk−xk^νk ∂

∂x_k i

ux,

0≤γ_k, ν_k<1− 1

p, k1,2, . . . , n, G_k0 x₁, x₂, . . . , x_k−1,0, x_{k 1}, . . . , x_n, G_kb x₁, x₂, . . . , x_k−1, b_k, x_{k 1}, . . . , x_n, 0≤m_kj≤l_k−1,

xk x1, x₂, . . . , x_k−1, x_{k 1}, . . . , x_n, G_k

j /k

0, b_j

, j, k1,2, . . . , n,

3.3

αjk,βjk,λare complex numbers,akare complex-valued functions onG,Ax, andAαxare linear operators inE. Moreover,γ_kandν_kare such that

_xk

0

x_k^−γkbk−x_k^−νkdx_k<∞, x_k∈0, bk, k 1,2, . . . , n. 3.4

A functionu∈W_p,γ^lG;EA, E, Lkj {u∈W_p,γ^lG;EA, E, Lkju0}and satisfying 3.1a.e. onGis said to be solution of the problem3.1-3.2.

We say the problem3.1-3.2 isLp-separable if for allf ∈ LpG;E, there exists a unique solutionu ∈ W_p,γ^lG;EA, Eof the problem 3.1-3.2and a positive constant C depending onlyG, p, γ, l, E, Asuch that the coercive estimate

n k1

D^l_k^ku

LpG;E Au _LpG;E≤Cf

LpG;E 3.5

holds.

LetQbe a differential operator generated by problem3.1-3.2withλ0; that is, DQ W_p,γ^l

G;EA, E, Lkj

,

Quⁿ

k1

akxD^l_k^ku Axu

|α:l|<1

AαxD^αu. 3.6

We say the problem3.1-3.2is Fredholm inLpG;Eif dimKerQdimKerQ^∗<∞, whereQ^∗is a conjugate ofQ.

Remark 3.1. Under the substitutions

τ_{k xk}

0

x^−γ_k^kbk−x_k^−νkdx_k, k1,2, . . . , n, 3.7

the spaces LpG;E and W_p,γ^lG;EA, E are mapped isomorphically onto the weighted spacesL_{p, γ}G;EandW_p, ^l_γG;EA, E, where

Gⁿ

k1

0,b_k , b_k

_bk

0

x_k^−γkbk−x_k^−νkdx_k. 3.8

Moreover, under the substitution3.7the problem3.1-3.2reduces to the nondegenerate BVP

n k1

a_kτD_k^lkuτ

Aτ λ

uτ

|α:l|<1

A_ατD^αuτ fτ,

mkj

i0

α_kjiD_kⁱu G_k0

0, j1,2, . . . , d_k, xk∈G_k,

mkj

i0

βkjiDⁱ_ku Gkb

0, xk∈Gk, j1,2, . . . , lk−dk, dk∈0, lk,

3.9

where

Gk0 τ1, τ2, . . . , τ_k−1,0, τ_{k 1}, . . . , τn, Gkb

τ1, τ2, . . . , τ_k−1,bk, τ_{k 1}, . . . , τn

, a_kτ a_kx1τ, x2τ, . . . , xnτ, Aτ Aakx1τ, x2τ, . . . , xnτ, Akτ Akakx1τ, x2τ, . . . , xnτ, γτ γx1τ, x2τ, . . . , xnτ.

3.10

By denotingτ,G,Gk0,Gkb,akτ,Aτ,Aky,γkτagain byx,G,Gk0,Gkb,akx,Ax, A_kx,γ_k, respectively, we get

n k1

akxD^l_k^kux Aλxux

|α:l|<1

AαxD^αux fx,

mkj

i0

αkjiDⁱ_kuGkb 0, j1,2, . . . , lk−dk, xk∈Gk,

mkj

i0

β_kjiDⁱ_kuGkb 0, xk∈G_k, j1,2, . . . , l_k−d_k, d_k∈0, lk.

3.11

4. BVPs for Partial DOE

Let us first consider the BVP for the anisotropic type DOE with constant coefficients

L λuⁿ

k1

a_kD^l_k^kux A λux fx,

Lkjufkj, j 1,2, . . . , dk, Lkjufkj, j1,2, . . . , lk−dk,

4.1

where

Dⁱ_k ux

x^γ_k^k ∂

∂xk

i

ux, 4.2

Lkj are boundary conditions defined by 3.2, ak are complex numbers, λ is a complex parameter, andAis a linear operator in a Banach spaceE. Letω_k1, ω_k2, . . . , ω_klk be the roots of the characteristic equations

akω^lk 10, k1,2, . . . , n. 4.3

Now, let

Fkj Yk, Xk_{1−γk pmkj/plk,p}, XkLpGk;E, YkW_p,γ^lk_kGk;EA, E, l^k l1, l2, . . . , l_k−1, l_{k 1}, . . . , ln, γ^k

x^γ₁¹, x^γ₂², . . . , x^γ_k−1^k−1, x^γ_{k 1}^{k 1}, . . . , x_n^ln , Gkx0 x1, x2, . . . , x_k−1, xk0, x_{k 1}, . . . , xn.

4.4

By applying the trace theorem27, Section 1.8.2, we have the following.

Theorem A5. Letlkandjbe integer numbers, 0≤j ≤lk−1,θj 1−γk pj 1/plk,xk0∈0, bk. Then, for anyu ∈ W_p,γ^l G;E₀, E, the transformationsu → D_k^juGkx0are bounded linear from W_p,γ^l G;E₀, EontoF_kj, and the following inequality holds:

D^j_kuGkx0

Fkj

≤C u _Wp,γ^l _G;E0,E. 4.5

Proof. It is clear that

W_p,γ^l G;E₀, E W_p,γ^lk_k0, bk;Y_k, X_k. 4.6

Then, by applying the trace theorem27, Section 1.8.2to the spaceW_p,γ^lk_k0, bk;Yk, Xk, we obtain the assertion.

Condition 1. Assume that the following conditions are satisfied:

1Eis a Banach space satisfying the multiplier condition with respect top ∈ 1,∞ and the weight functionγ !_n

k1x^γ_k^k, 0≤γ_k<1−1/p;

2Ais anR-positive operator inEforϕ∈0, π/2;

3a_k/0, and argω_kj−π≤ π

2 −ϕ, j1,2, . . . , d_k, argω_kj≤ π

2 −ϕ, jd_k 1, . . . , l_k 4.7 for 0< d_k< l_k,k1,2, . . . , n.

LetBdenote the operator inL_pG;Egenerated by BVP4.1. In15, Theorem 5.1the following result is proved.

Theorem A6. Let Condition1be satisfied. Then,

athe problem4.1forf ∈L_pG;Eand|argλ| ≤ϕwith sufficiently large|λ|has a unique solutionuthat belongs toW_p^lG;EA, Eand the following coercive uniform estimate holds:

n k1 lk

i0

|λ|^1−i/lkDⁱ_k u

LpG;E Au _LpG;E≤Mf

LpG;E, 4.8

bthe operatorBisR-positive inL_pG;E.

From Theorems A5 and A6 we have.

Theorem A7. Suppose that Condition1is satisfied. Then, for sufficiently large|λ|with|argλ| ≤ϕ the problem4.1has a unique solutionu ∈W_p,γ^lG;EA, Efor allf ∈ LpG;Eandfkj ∈ Fkj. Moreover, the following uniform coercive estimate holds:

n k1

lk

i0

|λ|^1−i/lkDⁱ_k u

LpG;E Au _LpG;E≤Mf

LpG;E

n k1

lk

j1

f_kj

Fkj. 4.9

Consider BVP3.11. Letωk1x, ωk2x, . . . , ωklkxbe roots of the characteristic equations a_kxω^lk 10, k1,2, . . . , n. 4.10 Condition 2. Suppose the following conditions are satisfied:

1ak/0 and

argωkj−π≤ π

2 −ϕ, j1,2, . . . , dk, argωkj≤ π

2 −ϕ, jdk 1, . . . , lk,

4.11

for

0< dk< lk, k1,2, . . . , n, ϕ∈ 0,π

2

, 4.12

2Eis a Banach space satisfying the multiplier condition with respect top ∈ 1,∞ and the weighted functionγ!_n

k1x^γ_k^kbk−xk^γk, 0≤γk<1−1/p.

Remark 4.1. Letl 2m_k anda_k −1^mkb_kx, whereb_k are real-valued positive functions.

Then, Condition2is satisfied forϕ∈0, π/2.

Consider the inhomogenous BVP3.1-3.2; that is,

L λuf, Lkjufkj. 4.13 Lemma 4.2. Assume that Condition2is satisfied and the following hold:

1Axis a uniformlyR-positive operator inEforϕ ∈0, π/2, anda_kxare continuous functions onG,λ∈Sϕ,

2AxA⁻¹x∈CG;BEandA_∞A^{1−|α:l|−μ}∈L_∞G;BEfor 0< μ <1− |α:l|.

Then, for allλ ∈ Sϕand for sufficiently large|λ|the following coercive uniform estimate holds:

n k1

lk

i0

|λ|^1−i/lkDⁱ_ku

Lp,γG;E Au _Lp,γG;E≤Cf

Lp,γG;E

n k1

lk

j1

fkj

Fkj, 4.14

for the solution of problem4.13.

Proof. Let G1,G2, . . . , GN be regions covering G and let ϕ1, ϕ2, . . . , ϕN be a corresponding partition of unity; that is, ϕ_j ∈ C^∞₀, σ_j suppϕ_j ⊂ G_j and "_N

j1ϕ_jx 1. Now, for u∈W_p,γ^l G;EA, Eandujx uxϕjx, we get

L λujⁿ

k1

a_kxD^l_k^ku_jx A_λxujx f_jx, L_kiu_j Φki, 4.15

where

f_j fϕ_j n k1

a_k

|α:l|<1

A_αxⁿ

k1 αk−1

i0

C_αikD^α_k^k−iϕ_jDⁱ_ku−

|α:l|<1

ϕ_jA_αxD^αux, ΦkiϕjLkiu Bki

ϕj

L_kiu,

4.16

here, L_ki and Bki are boundary operators which orders less than mki − 1. Freezing the coefficients of4.15, we have

n k1

a_k x_0j

D^l_k^ku_jx A_λ x_0j

u_jx F_j, L_kiu_j Φki, i1,2, . . . , l_k, k 1,2, . . . , n,

4.17

where

Fjfj

#A x0j

−Ax$ uj

n k1

#akx−a x0j

$D^l_k^kujx. 4.18

It is clear thatγx∼!_n

k1x^γ_k^kon neighborhoods ofG_j∩G_k0and

γx∼ⁿ

k1

bk−x_k^νk, 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 problems4.17are generated locally only on parts of the boundary. Then, by Theorem A7 problem4.17has a unique solutionujand for|argλ| ≤ϕ the following coercive estimate holds:

n k1

lk

i0

|λ|^1−i/lkDⁱ_ku_j

Gj,p,γ Au_j

Gj,p,γ≤C

%F_j

Gj,p,γ

n k1

lk

i1

Φkj

Fki

&

. 4.20

From the representation ofFj,Φkiand in view of the boundedness of the coefficients, we get F_j

Gj,p,γ≤f_j

Gj,p,γ #A x_0j

−Ax$ u_j

Gj,p,γ

n k1

#

a_kx−a x_0j$

D_k^lku_jx

Gj,p,γ, Φkj

Fki≤ϕ_jL_kiu

Fki B_j ϕ_j

L_kiu

Fki ≤M

Lkiu _Fki L_kiu

Fki

.

4.21 Now, applying Theorem A1 and by using the smoothness of the coefficients of4.16,4.18 and choosing the diameters ofσ_jso small, we see there is anε >0 andCεsuch that

Fj

Gj,p,γ ≤fj

Gj,p,γ εA x0j

uj

Gj,p,γ ε n k1

D_k^lkujx

Gj,p,γ

≤fϕj

Gj,p,γ M

|α:l|<1

AαxD^αujx

Gj,p,γ εuj

Wp,γ^l Gj;EA,E

≤fϕ_j

Gj,p,γ εu_j

W_p,γ^l Gj;EA,E Cεu_j

Gj,p,γ.

4.22

Then, using Theorem A5 and using the smoothness of the coefficients of4.16,4.18, we get Φki Fkj ≤M

Lkiu _Fki L_kiu

Fki

≤M

Lkiu _Fki u_j

Wp,γ^lk−10,bkj;Yk,Xk

. 4.23

Now, using Theorem A1, we get that there is anε >0 andCεsuch that uj

W_p,γk^lk−10,bkj;Yk,Xk≤εuj

W_p,γk^lk 0,bkj;Yk,Xk Cεuj

L_p,γk

≤εuj

Wp,γ^l Gj;EA,E Cεuj

Gj,p,γ,

4.24

where

0, bkj

0, bk∩Gj. 4.25

Using the above estimates, we get Φkj

Fki≤M Lkiu _Fki εu_j

Wp,γ^l Gj;EA,E Cεu_j

Gj,p,γ. 4.26

Consequently, from4.22–4.26, we have n

k1 lk

i0

|λ|^1−i/lkDⁱ_kuj

Gj,p,γ

Auj

Gj,p,γ

≤Cf

Gj,p,γ εu_j

Wp,γ² Mεu_j

Gj,p,γ C n k1

lk

i1

f_ki

Fki.

4.27

Choosingε <1 from the above inequality, we obtain n

k1 lk

i0

|λ|^1−i/lkD_kⁱuj

Gj,p,γ

Auj

Gj,p,γ ≤C

%f

Gj,p,γ uj

Gj,p,γ

n k1

lk

i1

fki

Fki

&

. 4.28

Then, by using the equalityux "_N

j1ujxand the above estimates, we get4.14.

Condition 3. Suppose that part1.1of Condition1is satisfied and thatEis a Banach space satisfying the multiplier condition with respect top∈1,∞and the weighted functionγ

!_n

k1x^γ_k^kbk−xk^νk, 0≤γk,νk<1−1/p.

Consider the problem3.11. Reasoning as in the proof ofLemma 4.2, we obtain.

Proposition 4.3. Assume Condition3hold and suppose that

1Ax is a uniformly R-positive operator in E for ϕ ∈ 0, π/2, and that a_kx are continuous functions onG,λ∈Sϕ,

2AxA⁻¹x∈CG;BEandA_∞A^{1−|α:l|−μ}∈L_∞G;BEfor 0< μ <1− |α:l|.

Then, for allλ ∈ Sϕand for sufficiently large|λ|, the following coercive uniform estimate holds

n k1

lk

i0

|λ|^1−i/lkDⁱ_ku

Lp,γG;E Au _Lp,γG;E ≤Cf

Lp,γG;E, 4.29

for the solution of problem3.11.

LetOdenote the operator generated by problem3.11forλ0; that is, DO W_p,γ^l

G;EA, E, Lkj

,

Ouⁿ

k1

akxD_k^lku Axu

|α:l|<1

AαxD^αu. 4.30

Theorem 4.4. Assume that Condition3is satisfied and that the following hold:

1Axis a uniformlyR-positive operator inE, andakxare continuous functions onG, 2AxA⁻¹x∈CG;BE, andAαA^{1−|α:l|−μ}∈L_∞G;BEfor 0< μ <1− |α:l|.

Then, problem3.11has a unique solutionu ∈ W_p,γ^l G;EA, Eforf ∈ Lp,γG;Eand λ∈Sϕwith large enough|λ|. Moreover, the following coercive uniform estimate holds:

n k1

lk

i0

|λ|^1−i/lkDⁱ_ku

Lp,γG;E Au _Lp,γG;E ≤Cf

Lp,γG;E. 4.31

Proof. ByProposition 4.3foru∈W_p,γ^l G;EA, E, we have n

k1 lk

i0

|λ|^1−i/lkDⁱ_ku

p,γ Au _p,γ ≤C

L λu _p,γ u _p,γ

. 4.32

It is clear that u _p,γ 1

|λ| L λu−Lu _p,γ ≤ 1

|λ|

L λu _p,γ Lu _p,γ

. 4.33

Hence, by using the definition ofW_p,γ^l G;EA, Eand applying Theorem A1, we obtain

u _p,γ ≤ 1

|λ|

L λu _p,γ u _Wp,γ^l _G;EA,E

. 4.34

From the above estimate, we have n

k1 lk

i0

|λ|^1−i/lkDⁱ_ku

p,γ Au _p,γ ≤C L λu _p,γ. 4.35

The estimate4.35implies that problem3.11has a unique solution and that the operator O λhas a bounded inverse in its rank space. We need to show that this rank space coincides with the spaceL_p,γG;E; that is, we have to show that for allf ∈L_p,γG;E, there is a unique solution of the problem3.11. We consider the smooth functionsg_j g_jxwith respect to a partition of unityϕj ϕjyon the regionGthat equals one on suppϕj, where suppgj ⊂Gj

and |gjx| < 1. Let us construct for allj the functions uj that are defined on the regions ΩjG∩G_jand satisfying problem3.11. The problem3.11can be expressed as

n k1

a_k x_0j

D_k^lku_jx A_λ x_0j

u_jx

g_j

⎧⎨

⎩f # A

x_0j

−Ax$ u_j

n k1

#a_kx−a_k x_0j$

D_k^lku_j−

|α:l|<1

A_αxD^αu_j

⎫⎬

⎭, Lkiuj0, j 1,2, . . . , N.

4.36

Consider operatorsOjλinLp,γGj;Ethat are generated by the BVPs4.17; that is, D

O_jλ

W_p,γ^l

G_j;EA, E, Lki

, i1,2, . . . , l_k, k1,2, . . . , n,

O_jλuⁿ

k1

a_k x_0j

D^l_k^ku_jx A_λ x_0j

u_jx, j1, . . . , N. 4.37