Báo cáo hóa học: " Research Article Degenerate Anisotropic Differential Operators and Applications" pot

Maximalregularity properties for higher-order degenerate anisotropic DOEs with constant coefficientsand nondegenerate equations with variable coefficients were studied in 15,16.. Note, the p

Volume 2011, Article ID 268032, 27 pages

doi:10.1155/2011/268032

Research Article

Degenerate Anisotropic Differential Operators

and Applications

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

Received 2 December 2010; Accepted 18 January 2011

Academic Editor: Gary Lieberman

Commons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

variable coefficients are studied Several conditions for the separability and Fredholmness in

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 havebeen studied extensively in the literaturesee 1 14 and the references therein Maximalregularity properties for higher-order degenerate anisotropic DOEs with constant coefficientsand 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 forhigher-order degenerate DOEs; that is,

where D i k u x  γ k x k ∂/∂x ki u x, 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, , n reduces 1.1 to elliptic form

Note, the principal part of the corresponding differential operator is adjoint Nevertheless, the sharp uniform coercive estimate for the resolvent, Fredholmness,discreetness of the spectrum, and completeness of root elements of this operator areestablished.

nonself-We prove that the corresponding differential operator is separable in Lp; that is, it has

a bounded inverse from L p to the anisotropic weighted space W p,γ l This fact allows us toderive some significant spectral properties of the differential operator For the exposition ofdifferential equations with bounded or unbounded operator coefficients in Banach-valuedfunction spaces, we refer the reader to8,15–25

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

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

for all cubes Q ⊂ R n

The Banach space E is called a UMD space if the Hilbert operator Hfx 

limε→ 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, ∞.

LetC 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 > 0 if

D A is dense on E and

denote the space DA θ with graphical norm

u E A θ u p A θ up1/p

, 1≤ p < ∞, −∞ < θ < ∞. 1.6

Let E1and E2be two Banach spaces Now,E1, E2θ,p , 0 < θ < 1, 1 ≤ p ≤ ∞ will denote

interpolation spaces obtained from{E1, E2} by the K method 27, Section 1.3.1

A set W ⊂ BE1, E2 is called R-bounded see 3,25,26 if there is a constant C > 0 such that for all T1, T2, , T m ∈ W and u 1, u2, , u m ∈ E1, m ∈ N

10

Let SR n ; E denote the Schwartz class, that is, the space of all E-valued rapidly decreasing smooth functions on R n Let F be the Fourier transformation A function Ψ ∈

C R n ; BE is called a Fourier multiplier in L p,γ R n ; E if the map u → Φu  F−1ΨξFu,

u ∈ SR n ; E is well defined and extends to a bounded linear operator in L p,γ R n ; E The set

of all multipliers in L p,γ R n ; E will denoted by M p,γ

Definition 1.1 A Banach space E is said to be a space satisfying a multiplier condition if, for

anyΨ ∈ C n R n ; BE, the R-boundedness of the set {ξ β D β ξ Ψξ : ξ ∈ R n \ 0, β ∈ U n} impliesthatΨ is a Fourier multiplier in L p,γ R n ; E, that is, Ψ ∈ M p,γ

p,γ E for any p ∈ 1, ∞.

implies thatΨhis 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 Ax λI−1 ≤ M/1 |λ| for any λ ∈ S ϕ,

For two sequences{a j}∞1 and{b j}∞1 of positive numbers, the expression a j ∼ b jmeans

that there exist positive numbers C1and C2such that

con-ditions are satisfied:

1 E is a Banach space satisfying the multiplier condition with respect to p and γ,

4 Ω ⊂ R n is a region such that there exists a bounded linear extension operator from

p,γ Ω; EA, E to W l

p,γ R n ; EA, E.

p,γ Ω; EA, E ⊂ L p,γ Ω; EA1−κ−μ is continuous Moreover,

p,γ Ω; EA, E, the following estimate holds

D α u L p,γ Ω;EA1−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

is compact.

Let Sp A denote the closure of the linear span of the root vectors of the linear operator A.

From18, Theorem 3.4.1, we have the following

Theorem A3 Assume that

1 E is an UMD space and A is an operator in σ p E, p ∈ 1, ∞,

2 μ1, μ2, , 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

as λ → 0 along any of the arcs μ.

Then, the subspace Sp A contains the space E.

Theorem A4 Let E0and E be two Banach spaces possessing bases Suppose that

3 Statement of the Problem

Consider the BVPs for the degenerate anisotropic DOE

x −γ k

k b k − x k−ν k dx k < ∞, x k ∈ 0, b k , k  1, 2, , n. 3.4

A function u ∈ W p,γ l G; EA, E, L kj   {u ∈ W p,γ l 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 problem3.1-3.2 is L p -separable if for all f ∈ L p G; E, there exists a unique solution u ∈ W p,γ l 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

4 BVPs for Partial DOE

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

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

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 .

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

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

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 problem4.1 has a unique solution u ∈ W p,γ l G; EA, E for all f ∈ L p G; E and f kj ∈ F kj Moreover, the following uniform coercive estimate holds:

Then, Condition2is satisfied for ϕ ∈ 0, π/2.

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

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−1x ∈ CG; BE and A∞A 1−|α:l|−μ ∈ L∞G; BE for 0 < μ < 1 − |α : l|.

Proof Let G 1, G2, , G N be regions covering G and let ϕ1, ϕ2, , ϕ N be a corresponding

partition of unity; that is, ϕ j ∈ C∞

on neighborhoods of G j ∩ G kb and γ x ∼ C j on other parts of the domains G j , where C j

are positive constants Hence, the problems4.17 are generated locally only on parts of theboundary Then, by Theorem A7 problem4.17 has a unique solution u jand for| arg λ| ≤ ϕ

the following coercive estimate holds:

From the representation of F j,Φkiand in view of the boundedness of the coefficients, we get

and choosing the diameters of σ j so small, we see there is an ε > 0 and Cε such that

≤fϕ j

G j ,p,γ εu j

W l p,γ G j ;EA,E Cεu j

G j ,p,γ 4.26Consequently, from4.22–4.26, we have

Choosing ε < 1 from the above inequality, we obtain

Then, by using the equality ux "N

j1u j x and the above estimates, we get 4.14

satisfying the multiplier condition with respect to p ∈ 1, ∞ and the weighted function γ 

!n

k1x γ k

k b k − x kν k, 0≤ γ k , ν k < 1 − 1/p.

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

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

2 AxA−1x ∈ CG; BE and A∞A 1−|α:l|−μ ∈ L∞G; BE for 0 < μ < 1 − |α : l|.

for the solution of problem3.11.

Let O denote the operator generated by problem3.11 for λ  0; that is,

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−1x ∈ CG; BE, and A α A 1−|α:l|−μ ∈ L∞G; BE for 0 < μ < 1 − |α : l|.

p,γ G; EA, E for f ∈ L p,γ G; E and

Proof ByProposition 4.3for u ∈ W l

p,γ G; EA, E, we have

Hence, by using the definition of W l

p,γ G; EA, E and applying Theorem A1, we obtain

The estimate4.35 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 problem3.11 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

Ωj  G ∩ G jand satisfying problem3.11 The problem 3.11 can be expressed as

By virtue of Theorem A6, the operators O jλ have inverses O jλ−1 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 p,γ l G j ; EA, E, and for all f ∈ L p,γ G j ; E, we have

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−1g j f Moreover,

are solutions of4.38 Consider the following linear operator U λ in L p G; E defined by

It is clear from the constructions U jand from the estimate4.39 that the operators U jλare

bounded linear from L p,γ G; E to W l

p,γ G j ; EA, E, and for | arg λ| ≤ ϕ with sufficiently large

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 

sufficiently large |λ|, there is an ε ∈ 0, 1 such that Φjλ < ε Therefore, there exists a

bounded linear invertible operator I "N

j1Φjλ−1; that is, we infer for all f ∈ L p,γ G; E

that the BVP3.11 has a unique solution

type sharp estimate:

Let Q denote the operator generated by BVP 3.1-3.2 From Theorem 4.4 and

Remark 3.1, we get the following

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 p,γ l G; EA, E, and the following coercive uniform

n  2, G  0, 1 × 0, 1 and A  q; that is, consider the problem

Theorem 4.4 implies that for each f ∈ L p G, problem 4.52 has a unique solution u ∈

W p l G satisfying the following coercive estimate:

on G, E Cν and Ax is a diagonal matrix-function with continuous components d m x > 0.

Then, we obtain the separability of the following BVPs for the system of anisotropicPDEs with varying coefficients:

in the vector-valued space L p,γ G; C ν

5 The Spectral Properties of Anisotropic Differential Operators

Consider the following degenerated BVP:

Consider the operator Q generated by problem5.1

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 p,γ l G; EA, E into L p G; E.

inverseO λ−1from L p G; E to W p l G; EA, E; that is, the operator Q λ is Fredholm from W p l 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 p,γ l G; EA, E into

L p G; E.

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

b the system of root functions of the differential operator Q is complete in L p G; E.

resolvent operatorQ d−1which is bounded from L p G; E to W p,γ l G; EA, E Moreover,

from Theorem A4 andRemark 3.1, we get that the embedding operator

Hence, from relations5.6 and 5.7, we obtain 5.4 Now, Result1implies that the

operator Q d is positive in L p G; E and

Consider now the operator O in L p,γ G; E generated by the nondegenerate BVP

obtained from5.1 under the mapping 3.7; that is,

FromTheorem 5.2andRemark 3.1, we get the following

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 equationsare studied Maximal regularity properties for PDEs have been studied, for example, in3for smooth domains and in28 for nonsmooth domains

y ∈∂Ω

 0, x ∈ G, j  1, 2, , m,

6.1

where D j  − i∂/∂y j , α kji , β kji are complex number, y  y1, , y μ  ∈ Ω ⊂ R μand

Let Ω  G × Ω, p  p1, p  Now, Lp Ω will denote the space of all p-summable

scalar-valued functions with mixed normsee, e.g., 29, Section 1, page 6, that is, the space of all

measurable functions f defined on Ω, for which

p  Ω denotes the Sobolev space with corresponding mixed norm

Let ω kj  ω kj x, j  1, 2, , l k , k  1, 2, , n denote the roots of the equations

Let Q denote the operator generated by BVP6.1 Let

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 ≥ p1,

From Theorems A5 and A6... space E.

Theorem A4 Let E0and E be two Banach spaces possessing... and a positive constant C depending only G, p, γ, l, E, A such that the coercive estimate

4