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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 Shakhmurov3 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA Department of Mathematics, National University of Ireland, Galway, Ireland Department 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 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 Lp 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 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 l ak x Dk k u x k i A x u x Aα x D |α:l|ε f y / x−y dy is bounded in Lp R, E , p ∈ 1, ∞ see, e.g., 26 UMD spaces include, for example, 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}, ≤ ϕ < π 1.4 A linear operator A is said to be ϕ-positive in a Banach space E with bound M > if D A is dense on E and A λI −1 LE ≤M |λ| −1 , 1.5 for all λ ∈ Sϕ , ϕ ∈ 0, π , I is an identity operator in E, and B E is the space of bounded linear operators in E Sometimes A λI will be written as A λ and denoted by Aλ It is known 27, Section 1.15.1 that there exists fractional powers Aθ of the sectorial operator A Let E Aθ denote the space D Aθ with graphical norm u E Aθ u p Aθ u p 1/p , ≤ p < ∞, −∞ < θ < ∞ 1.6 Let E1 and E2 be two Banach spaces Now, E1 , E2 θ,p , < θ < 1, ≤ p ≤ ∞ will denote interpolation spaces obtained from {E1 , E2 } by the K method 27, Section 1.3.1 Boundary Value Problems A set W ⊂ B E1 , E2 is called R-bounded see 3, 25, 26 if there is a constant C > such that for all T1 , T2 , , Tm ∈ W and u1, u2 , , um ∈ E1 , m ∈ N m j E2 m dy ≤ C rj y Tj uj j rj y uj dy, 1.7 E1 where {rj } is a sequence of independent symmetric {−1, 1}-valued random variables on 0, The smallest C for which the above estimate holds is called an R-bound of the collection W and is denoted by R W Let S Rn ; 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 function Ψ ∈ F −1 Ψ ξ Fu, C Rn ; B E is called a Fourier multiplier in Lp,γ Rn ; E if the map u → Φu n n u ∈ S R ; E is well defined and extends to a bounded linear operator in Lp,γ R ; E The set p,γ of all multipliers in Lp,γ Rn ; E will denoted by Mp,γ E Let Vn Un ξ:ξ β ξ1 , ξ2 , , ξn ∈ Rn , ξj / , 1.8 β1 , β2 , , βn ∈ N n : βk ∈ {0, 1} Definition 1.1 A Banach space E is said to be a space satisfying a multiplier condition if, for β any Ψ ∈ C n Rn ; B E , the R-boundedness of the set {ξβ Dξ Ψ ξ : ξ ∈ Rn \ 0, β ∈ Un } implies p,γ that Ψ is a Fourier multiplier in Lp,γ Rn ; E , that is, Ψ ∈ Mp,γ E for any p ∈ 1, ∞ p,γ Let Ψh ∈ Mp,γ E be a multiplier function dependent on the parameter h ∈ Q The uniform R-boundedness of the set {ξβ Dβ Ψh ξ : ξ ∈ Rn \ 0, β ∈ U}; that is, supR ξβ Dβ Ψh ξ : ξ ∈ Rn \ 0, β ∈ U ≤K 1.9 h∈Q implies that Ψh 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 {A A ξI −1 : ξ ∈ Sϕ } is R-bounded A linear operator A x is said to be ϕ-positive in E uniformly in x if D A x is independent of x, D A x is dense in E and A x λI −1 ≤ M/ |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; that is, supR ξβ Dβ A x A x ξI −1 : ξ ∈ Rn \ 0, β ∈ U ≤ M 1.10 x∈G Let σ∞ E1 , E2 denote the space of all compact operators from E1 to E2 For E1 denoted by σ∞ E E2 E, it is Boundary Value Problems For two sequences {aj }∞ and {bj }∞ of positive numbers, the expression aj ∼ bj means 1 that there exist positive numbers C1 and C2 such that C1 aj ≤ bj ≤ C2 aj 1.11 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 Now, sj A denotes the approximation numbers of operator A see, e.g., 27, Section 1.16.1 Let ⎧ ⎨ ∞ ⎩ j A : A ∈ σ∞ E1 , E2 , σq E1 , E2 q sj ⎫ ⎬ A < ∞, ≤ q < ∞ ⎭ 1.12 Let E0 and E be two Banach spaces and E0 continuously and densely embedded into E and l l1 , l2 , , ln l We let Wp,γ Ω; E0 , E denote the space of all functions u ∈ Lp,γ Ω; E0 possessing l generalized derivatives Dkk u l l ∂lk u/∂xkk such that Dkk u ∈ Lp,γ Ω; E with the norm n u l Wp,γ u Ω;E0 ,E Lp,γ Ω;E0 l Dkk u k i tions: Let Dk u x γk xk ∂/∂xk l i Lp,γ Ω;E < ∞ 1.13 u x Consider the following weighted spaces of func- l u : u ∈ Lp G; E A , Dk k u ∈ Lp G; E , Wp,γ G; E A , E n u u l Wp,γ G;E A ,E 1.14 l Dk k u Lp G;E A k Lp G;E 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 ditions are satisfied: α1 , α2 , , αn and Dα α α α D1 D2 · · · Dnn and suppose that the following con- E is a Banach space satisfying the multiplier condition with respect to p and γ, A is an R-positive operator in E, α α1 , α2 , , αn and l κ n k l1 , l2 , , ln are n-tuples of nonnegative integer such that αk ≤ 1, l k ≤ μ ≤ − κ, < p < ∞, 2.1 Boundary Value Problems Ω ⊂ Rn is a region such that there exists a bounded linear extension operator from l l Wp,γ Ω; E A , E to Wp,γ Rn ; E A , E l Then, the embedding Dα Wp,γ Ω; E A , E ⊂ Lp,γ Ω; E A1−κ−μ is continuous Moreover, l for all positive number h < ∞ and u ∈ Wp,γ Ω; E A , E , the following estimate holds Dα u Lp,γ Ω;E A1−κ−μ ≤ hμ u h− 1−μ u l Wp,γ Ω;E A ,E Lp,γ Ω;E 2.2 Theorem A2 Suppose that all conditions of Theorem A1 are satisfied Moreover, let γ ∈ Ap , Ω be a bounded region and A−1 ∈ σ∞ E Then, the embedding l Wp,γ Ω; E A , E ⊂ Lp,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 , we have the following Theorem A3 Assume that E is an UMD space and A is an operator in σp E , p ∈ 1, ∞ , μ1 , μ2 , , μs 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, m > is an integer so that the resolvent of A satisfies the inequality O |λ|−1 , R λ, A 2.4 as λ → along any of the arcs μ Then, the subspace Sp A contains the space E Let G {x x1 , x2 , , xn : < xk < bk }, γ γ γ x11 x22 · · · xnn γ x 2.5 Let βk β xkk , n ν ν xkk , k l Let I I Wp,β,γ Ω; E A , E , Lp,γ Ω; E Lp,ν Ω; E n γ xkk γ l denote the embedding operator Wp,β,γ Ω; E A , E From 15, Theorem 2.8 , we have the following 2.6 k → Boundary Value Problems Theorem A4 Let E0 and E be two Banach spaces possessing bases Suppose that ≤ γk < p − 1, ≤ βk < 1, ∼ j −1/k0 , sj I E0 , E νk − γk > p βk − , 1, 2, , ∞, k0 > 0, j < p < ∞, n κ0 k γk − νk < p lk − βk 2.7 Then, ∼ j −1/ k0 l sj I Wp,β,γ G; E0 , E , Lp,ν G; E κ0 2.8 Statement of the Problem Consider the BVPs for the degenerate anisotropic DOE n l ak x Dk k u x A x λ u x Aα x D mkj i mkj i αkji Dk u Gk0 0, u x f x , 3.1 1, 2, , dk , j 3.2 i 1, 2, , lk − dk , dk ∈ 0, lk , βkji Dk u Gkb 0, j α1 , α2 , , αn , i α |α:l|