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

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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 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, veli.sahmurov@okan.edu.tr 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 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. 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 l α ak x Dk k u x Axux Aα x D ux fx, 1.1 |α:l|
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 Lp ; that is, it has l a bounded inverse from Lp to the anisotropic weighted space Wp,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 . γ x be a positive measurable weighted function on the region Ω ⊂ Rn . Let Let γ Lp,γ Ω; E denote the space of all strongly measurable E-valued functions that are deﬁned on Ω with the norm 1/p p 1 ≤ p < ∞. 1.2 f f fx Eγ x dx , Lp,γ Ω;E p,γ For γ x ≡ 1, the space Lp,γ Ω; E will be denoted by Lp Ω; E . The weight γ we will consider satisﬁes an Ap condition; that is, γ ∈ Ap , 1 < p < ∞ if there is a positive constant C such that p −1 1 1 −1/ p−1 ≤ C, 1.3 γ x dx γ x dx |Q | |Q| Q Q for all cubes Q ⊂ Rn . The Banach space E is called a UMD space if the Hilbert operator H f x limε → 0 |x−y|>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 λ; λ ∈ C, arg λ ≤ ϕ ∪ {0}, 0 ≤ ϕ < π. Sϕ 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 −1 −1 ≤M 1 |λ| A λI , 1.5 LE 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 p 1/p p 1 ≤ p < ∞, −∞ < θ < ∞. Aθ u 1.6 u u , E Aθ Let E1 and E2 be two Banach spaces. Now, E1 , E2 θ,p , 0 < θ < 1, 1 ≤ p ≤ ∞ will denote interpolation spaces obtained from {E1 , E2 } by the K method 27, Section 1.3.1 .
Boundary Value Problems 3 A set W ⊂ B E1 , E2 is called R-bounded see 3, 25, 26 if there is a constant C > 0 such that for all T1 , T2 , . . . , Tm ∈ W and u1, u2 , . . . , um ∈ E1 , m ∈ N 1 1 m m dy ≤ C rj y Tj uj rj y uj dy, 1.7 0 0 j1 j1 E2 E1 where {rj } 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 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 F u, C Rn ; B E is called a Fourier multiplier in Lp,γ Rn ; E if the map u → Φu u ∈ S R ; E is well deﬁned and extends to a bounded linear operator in Lp,γ R ; E . The set n n p,γ of all multipliers in Lp,γ Rn ; E will denoted by Mp,γ E . Let ξ1 , ξ2 , . . . , ξn ∈ Rn , ξj / 0 , Vn ξ:ξ 1.8 β1 , β2 , . . . , βn ∈ N n : βk ∈ {0, 1} . Un β 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, ξβ Dβ Ψh ξ : ξ ∈ Rn \ 0, β ∈ U ≤K supR 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/ 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; that is, −1 : ξ ∈ Rn \ 0, β ∈ U ≤ M. ξβ Dβ A x A x supR ξI 1.10 x∈G 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 .
4 Boundary Value Problems For two sequences {aj }∞ and {bj }∞ of positive numbers, the expression aj ∼ bj means 1 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 ⎧ ⎫ ⎨ ⎬ ∞ q A : A ∈ σ∞ E1 , E2 , A < ∞, 1 ≤ q < ∞ . σq E1 , E2 sj 1.12 ⎩ ⎭ j1 Let E0 and E be two Banach spaces and E0 continuously and densely embedded into E and l l1 , l2 , . . . , ln . We let Wp,γ Ω; E0 , E denote the space of all functions u ∈ Lp,γ Ω; E0 possessing l l l l ∂lk u/∂xkk such that Dkk u ∈ Lp,γ Ω; E with the norm generalized derivatives Dkk u n l < ∞. Dkk u u u 1.13 Ω;E0 ,E Lp,γ Ω;E0 l Wp,γ Lp,γ Ω;E k1 i i Let Dk u x γk xk ∂/∂xk u x . Consider the following weighted spaces of func- tions: l l u : u ∈ Lp G; E A , Dk k u ∈ Lp G; E , Wp,γ G; E A , E 1.14 n l Dk k u u u . l Lp G;E A Wp,γ G;E A ,E Lp G;E k1 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 . α α α Theorem A1. Let α D1 1 D2 2 · · · Dnn and suppose that the following con- α1 , α2 , . . . , αn and Dα 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, α1 , α2 , . . . , αn and l l1 , l2 , . . . , ln are n-tuples of nonnegative integer such that 3α n αk κ ≤ 1, 0 ≤ μ ≤ 1 − κ , 1 < p < ∞, 2.1 l 1k k
Boundary Value Problems 5 4 Ω ⊂ Rn is a region such that there exists a bounded linear extension operator from Wp,γ Ω; E A , E to Wp,γ Rn ; E A , E . l 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 l h− 1−μ u ≤ hμ u Dα u . 2.2 Lp,γ Ω;E A1−κ −μ Wp,γ Ω;E A ,E Lp,γ Ω;E l 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 Wp,γ Ω; E A , E ⊂ Lp,γ Ω; E l 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 1 E is an UMD space and A is an operator in σp E , p ∈ 1, ∞ , 2 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, 3 m > 0 is an integer so that the resolvent of A satisﬁes the inequality O | λ| − 1 , R λ, A 2.4 as λ → 0 along any of the arcs μ. Then, the subspace Sp A contains the space E. Let γ γ γ x11 x22 · · · xnn . {x x1 , x2 , . . . , xn : 0 < xk < bk }, 2.5 G γx Let n n β γ ν xkk , xkk . xkk , βk ν γ 2.6 k1 k1 I Wp,β,γ Ω; E A , E , Lp,γ Ω; E denote the embedding operator Wp,β,γ Ω; E A , E → l l Let I Lp,ν Ω; E . From 15, Theorem 2.8 , we have the following.
6 Boundary Value Problems Theorem A4. Let E0 and E be two Banach spaces possessing bases. Suppose that 0 ≤ γk < p − 1, 0 ≤ βk < 1, νk − γk > p βk − 1 , 1 < p < ∞, 2.7 γk − νk n ∼ j −1/k0 , 1, 2, . . . , ∞, κ0 sj I E0 , E k0 > 0, j < 1. p lk − βk k 1 Then, ∼ j −1/ k0 κ0 l sj I Wp,β,γ G; E0 , E , Lp,ν G; E . 2.8 3. Statement of the Problem Consider the BVPs for the degenerate anisotropic DOE n l α ak x Dk k u x Ax λux Aα x D ux fx, 3.1 |α:l|
Boundary Value Problems 7 l l A function u ∈ Wp,γ G; E A , E, Lkj {u ∈ Wp,γ G; E A , E , Lkj 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 Lp -separable if for all f ∈ Lp G; E , there exists a l unique solution u ∈ Wp,γ G; E A , 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 l ≤C f Dk k u Au 3.5 Lp G;E Lp G;E Lp G;E k1 holds. Let Q be a diﬀerential operator generated by problem 3.1 - 3.2 with λ 0; that is, l DQ Wp,γ G; E A , E, Lkj , 3.6 n l α ak x Dk k u Qu Axu Aα x D u. |α:l|
8 Boundary Value Problems where Gk0 τ1 , τ2 , . . . , τk−1 , 0, τk 1 , . . . , τn , Gkb τ1 , τ2 , . . . , τk−1 , bk , τk 1 , . . . , τn , 3.10 ak τ ak x1 τ , x2 τ , . . . , xn τ , Aτ A ak x1 τ , x2 τ , . . . , xn τ , Ak τ Ak ak x1 τ , x2 τ , . . . , xn τ , γτ γ x1 τ , x2 τ , . . . , xn τ . By denoting τ , G, Gk0 , Gkb , ak τ , A τ , Ak y , γk τ again by x, G, Gk0 , Gkb , ak x , A x , Ak x , γk , respectively, we get n l Aα x D α u x ak x Dkk u x Aλ x u x fx, |α:l|
Boundary Value Problems 9 Now, let k l Fkj Yk , Xk 1−γk pmkj /plk ,p , Xk Lp Gk ; E , Yk Wp,γ k Gk ; E A , E , γ γ γ γ l 4.4 −1 lk k x11 , x22 , . . . , xkk−1 , xkk 1 , . . . , xnn , 1 l1 , l2 , . . . , lk−1 , lk 1 , . . . , ln , γ Gkx0 x1 , x2 , . . . , xk−1 , xk0 , xk 1 , . . . , xn . By applying the trace theorem 27, Section 1.8.2 , we have the following. Theorem A5. Let lk and j be integer numbers, 0 ≤ j ≤ lk − 1, θj 1 − γk pj 1 /plk , xk0 ∈ 0, bk . j Then, for any u ∈ G; E0 , E , the transformations u → l Gkx0 are bounded linear from Wp,γ Dk u l Wp,γ G; E0 , E onto Fkj , and the following inequality holds: j ≤C u Dk u Gkx0 . 4.5 l Wp,γ G;E0 ,E Fkj Proof. It is clear that lk l 4.6 Wp,γ G; E0 , E Wp,γk 0, bk ; Yk , Xk . lk Then, by applying the trace theorem 27, Section 1.8.2 to the space Wp,γk 0, bk ; Yk , Xk , 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, ∞ γk n k 1 xk , 0 ≤ γk < 1 − 1/p; and the weight function γ 2 A is an R-positive operator in E for ϕ ∈ 0, π/2 ; 3 ak / 0, and π π arg ωkj − π ≤ − ϕ, arg ωkj ≤ − ϕ, j 1, 2, . . . , dk , j dk 1, . . . , lk 4.7 2 2 for 0 < dk < lk , k 1, 2, . . . , n. Let B denote the operator in Lp G; E generated by BVP 4.1 . In 15, Theorem 5.1 the following result is proved. Theorem A6. Let Condition 1 be satisﬁed. Then, a the problem 4.1 for f ∈ Lp G; E and | arg λ| ≤ ϕ with suﬃciently large |λ| has a unique l solution u that belongs to Wp G; E A , E and the following coercive uniform estimate holds: lk n |λ|1−i/lk Dk u i ≤M f 4.8 Au , Lp G;E Lp G;E Lp G;E k 1i 0 b the operator B is R-positive in Lp 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 l the problem 4.1 has a unique solution u ∈ Wp,γ G; E A , E for all f ∈ Lp G; E and fkj ∈ Fkj . Moreover, the following uniform coercive estimate holds: lk lk n n |λ|1−i/lk Dk u i ≤M f Au fkj . 4.9 Lp G;E Lp G;E Fkj Lp G;E k 1i0 k 1j 1 Consider BVP 3.11 . Let ωk1 x , ωk2 x , . . . , ωklk x be roots of the characteristic equations ak x ωlk 4.10 1 0, k 1, 2, . . . , n. Condition 2. Suppose the following conditions are satisﬁed: 1 ak / 0 and π arg ωkj − π ≤ − ϕ, j 1, 2, . . . , dk , 2 4.11 π arg ωkj ≤ − ϕ, j dk 1, . . . , lk , 2 for π 1, 2, . . . , n, ϕ ∈ 0, 0 < dk < lk , k , 4.12 2 2 E is a Banach space satisfying the multiplier condition with respect to p ∈ 1, ∞ γk γk n k 1 xk bk − xk , 0 ≤ γk < 1 − 1/p. and the weighted function γ −1 mk bk x , where bk are real-valued positive functions. Remark 4.1. Let l 2mk and ak Then, Condition 2 is satisﬁed for ϕ ∈ 0, π/2 . Consider the inhomogenous BVP 3.1 - 3.2 ; that is, L λu f, Lkj u fkj . 4.13 Lemma 4.2. Assume that Condition 2 is satisﬁed and the following hold: 1 A x is a uniformly R-positive operator in E for ϕ ∈ 0, π/2 , and ak x are continuous functions on G, λ ∈ Sϕ , 2 A x A−1 x ∈ C G; B E and A∞ A 1−|α:l|−μ ∈ L∞ G; B E for 0 < μ < 1 − |α : l|. Then, for all λ ∈ S ϕ and for suﬃciently large |λ| the following coercive uniform estimate holds: lk lk n n |λ|1−i/lk Dk u ≤C f i Au fkj , 4.14 Lp,γ G;E Lp,γ G;E Fkj Lp,γ G;E k 1i0 k 1j 1 for the solution of problem 4.13 .
Boundary Value Problems 11 Proof. Let G1, G2 , . . . , GN be regions covering G and let ϕ1 , ϕ2 , . . . , ϕN be a corresponding ∞ supp ϕj ⊂ Gj and N 1 ϕj x partition of unity; that is, ϕj ∈ C0 , σj 1. Now, for j u ∈ Wp,γ G; E A , E and uj x l u x ϕj x , we get n l Φki , ak x Dkk uj x L λ uj A λ x uj x fj x , Lki uj 4.15 k1 where n αk −1 n Cαik Dk k −i ϕj Dk u − α i ϕj Aα x Dα u x , fj f ϕj ak Aα x 4.16 |α:l|
12 Boundary Value Problems From the representation of Fj , Φki and in view of the boundedness of the coeﬃcients, we get n l ≤ fj A x0j − A x uj ak x − a x0j Dkk uj x Fj , Gj ,p,γ Gj ,p,γ Gj ,p,γ Gj ,p,γ k1 Φkj ≤ ϕj Lki u ≤M Bj ϕj Lki u Lki u Lk i u . Fki Fki Fki Fki Fki 4.21 Now, applying Theorem A1 and by using the smoothness of the coeﬃcients of 4.16 , 4.18 and choosing the diameters of σj so small, we see there is an ε > 0 and C ε such that n l ≤ fj Dkk uj x Fj ε A x0j uj ε Gj ,p,γ Gj ,p,γ Gj ,p,γ Gj ,p,γ k1 ≤ f ϕj A α x D α uj x 4.22 M ε uj l Gj ,p,γ Gj ,p,γ Wp,γ Gj ;E A ,E |α:l| 0 and C ε such that ≤ ε uj uj Cε uj l −1 l k k Lp,γk Wp,γk 0,bkj ;Yk ,Xk Wp,γk 0,bkj ;Yk ,Xk 4.24 ≤ ε uj Cε uj , l Wp,γ Gj ;E A ,E Gj ,p,γ where 0, bk ∩ Gj . 0, bkj 4.25 Using the above estimates, we get Φkj ≤ M Lki u ε uj Cε uj . 4.26 Fki l Fki Wp,γ Gj ;E A ,E Gj ,p,γ Consequently, from 4.22 – 4.26 , we have lk n |λ|1−i/lk Dk uj i Auj Gj ,p,γ Gj ,p,γ k 1i0 4.27 lk n ≤C f ε uj Mε uj C fki . 2 Gj ,p,γ Fki Wp,γ Gj ,p,γ k 1i1
Boundary Value Problems 13 Choosing ε < 1 from the above inequality, we obtain lk lk n n |λ|1−i/lk Dk uj ≤C i 4.28 Auj f uj fki . Gj ,p,γ Fki Gj ,p,γ Gj ,p,γ Gj ,p,γ k 1i0 k 1i1 N Then, by using the equality u x uj x and the above estimates, we get 4.14 . j1 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 γk νk n k 1 xk bk − xk , 0 ≤ γk , νk < 1 − 1/p. Consider the problem 3.11 . Reasoning as in the proof of Lemma 4.2, we obtain. Proposition 4.3. Assume Condition 3 hold and suppose that 1 A x is a uniformly R-positive operator in E for ϕ ∈ 0, π/2 , and that ak x are continuous functions on G, λ ∈ Sϕ , 2 A x A−1 x ∈ C G; B E and A∞ A 1−|α:l|−μ ∈ L∞ G; B E for 0 < μ < 1 − |α : l|. Then, for all λ ∈ S ϕ and for suﬃciently large |λ|, the following coercive uniform estimate holds lk n |λ|1−i/lk Dk u ≤C f i 4.29 Au , Lp,γ G;E Lp,γ G;E Lp,γ G;E k 1i0 for the solution of problem 3.11 . Let O denote the operator generated by problem 3.11 for λ 0; that is, l DO Wp,γ G; E A , E, Lkj , 4.30 n l Aα x Dα u. ak x Dkk u Ou Axu |α:l|
14 Boundary Value Problems Proof. By Proposition 4.3 for u ∈ Wp,γ G; E A , E , we have l lk n |λ|1−i/lk Dk u ≤C i 4.32 Au L λu u . p,γ p,γ p,γ p,γ k 1i0 It is clear that 1 1 λ u − Lu ≤ u L L λu Lu . 4.33 p,γ p,γ p,γ p,γ |λ| |λ| l Hence, by using the deﬁnition of Wp,γ G; E A , E and applying Theorem A1, we obtain 1 ≤ u L λu u . 4.34 l p,γ p,γ Wp,γ G;E A ,E |λ| From the above estimate, we have lk n |λ|1−i/lk Dk u ≤C L i 4.35 Au λu p,γ . p,γ p,γ k 1i0 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 Lp,γ G; E ; that is, we have to show that for all f ∈ Lp,γ G; E , there is a unique solution of the problem 3.11 . We consider the smooth functions gj gj x with respect to a partition of unity ϕj ϕj y on the region G that equals one on supp ϕj , where supp gj ⊂ Gj and |gj x | < 1. Let us construct for all j the functions uj that are deﬁned on the regions Ωj G ∩ Gj and satisfying problem 3.11 . The problem 3.11 can be expressed as n l ak x0j Dkk uj x Aλ x0j uj x k1 ⎧ ⎫ ⎨ ⎬ n 4.36 l A x0j − A x uj ak x − ak x0j Dkk uj − A α x D α uj gj f , ⎩ ⎭ |α:l|
Boundary Value Problems 15 −1 By virtue of Theorem A6, the operators Ojλ have inverses Ojλ for | arg λ| ≤ ϕ and −1 for suﬃciently large |λ|. Moreover, the operators Ojλ are bounded from Lp,γ Gj ; E to Wp,γ Gj ; E A , E , and for all f ∈ Lp,γ Gj ; E , we have l lk n |λ|1−i/lk Dk Ojλ f i −1 −1 ≤C f 4.38 AOjλ f . Lp,γ Gj ;E Lp,γ Gj ;E Lp,γ Gj ;E k 1i0 Extending uj to zero outside of supp ϕj in the equalities 4.36 , and using the −1 substitutions uj Ojλ υj , we obtain the operator equations υj Kjλ υj gj f, j 1, 2, . . . , N, 4.39 where Kjλ are bounded linear operators in Lp Gj ; E deﬁned by ⎧ ⎫ ⎨ ⎬ n −1 −1 −1 l A x0j − A x Ojλ ak x − ak x0j Dkk Ojλ − Aα x Dα Ojλ Kjλ gj f . ⎩ ⎭ |α:l| 0 such that −1 A x0j − A x Ojλ υj ≤ ε υj , Lp,γ Gj ;E Lp,γ Gj ;E 4.41 n −1 l ak x − ak x0j ≤ ε υj Dkk Ojλ υj . Lp,γ Gj ;E Lp,γ Gj ;E k1 Moreover, from assumption 2.2 of Theorem 4.4 and Theorem A1 for ε > 0, there is a constant C ε > 0 such that −1 ≤ ε υj Aα x Dα Ojλ υj Cε υj . 4.42 l Wp,γ Gj ;E A ,E Lp,γ Gj ;E Lp,γ Gj ;E |α:l|
16 Boundary Value Problems are solutions of 4.38 . Consider the following linear operator U λ in Lp G; E deﬁned by l DU λ Wp,γ G; E A , E, Lkj , j 1, 2, . . . , lk , k 1, 2, . . . , n, 4.45 N −1 −1 Ojλ I − Kjλ U λf ϕj y Ujλ f gj f. j1 It is clear from the constructions Uj and from the estimate 4.39 that the operators Ujλ are bounded linear from Lp,γ G; E to Wp,γ Gj ; E A , E , and for | arg λ| ≤ ϕ with suﬃciently large l |λ|, we have lk n |λ|1−i/lk Dk Ujλ f ≤C f i 4.46 AUjλ f . p p p k 1i0 Therefore, U λ is a bounded linear operator in Lp,γ G; E . Since the operators Ujλ coincide with the inverse of the operator Oλ in Lp,γ Gj ; E , then acting on Oλ to u N j 1 ϕj Ujλ f gives N N Φλ f Φjλ f, Oλ u ϕj Oλ Ujλ f f 4.47 j1 j1 where Φjλ are bounded linear operators deﬁned by ⎧ ⎫ ⎨ ⎬ lk αk n n Ckν Dk ϕj Dkk −ν Ujλ f Cα,νk Dkk ϕj Dk k −νk Ujλ f l ν α Φjλ f ν ak Aα . 4.48 ⎩k ⎭ |α:l|
Boundary Value Problems 17 Let Q denote the operator generated by BVP 3.1 - 3.2 . From Theorem 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 ∈ Lp G; E , | arg λ| ≤ ϕ and for suﬃciently large |λ| l has a unique solution u ∈ Wp,γ G; E A , E , and the following coercive uniform estimate holds lk n |λ|1−i/lk Dk u i ≤M f 4.51 Au , Lp G;E Lp G;E Lp G;E k 1i0 l b if A−1 ∈ σ∞ E , then the operator O is Fredholm from Wp,γ G; E A , E into Lp G; E . C, l1 Example 4.5. Now, let us consider a special case of 3.1 - 3.2 . Let E 2 and l2 4, 0, 1 × 0, 1 and A q; that is, consider the problem n 2, G 2 4 1 1 Lu a1 Dx u a2 Dy u bDx Dy u a0 u f, m 1j m 1j i m1j ∈ {0, 1}, β1i u i 1, y α1i Dx u 0, y 0, 0, 4.52 i0 i0 m 2j m 2j i 0 ≤ m2j ≤ 3, β2i u i x, 1 α2i Dy u x, 0 0, 0, i0 i0 where ∂i ∂i β2 i i α2 xα1 1 − x y β1 1 − y Dx , Dy , ∂x ∂y 4.53 ak ∈ C G , u u x, y , a1 < 0, a2 > 0. Theorem 4.4 implies that for each f ∈ Lp G , problem 4.52 has a unique solution u ∈ l Wp G satisfying the following coercive estimate: 2 4 ≤C f Dx u Dy u u . 4.54 Lp G Lp G Lp G Lp G Example 4.6. Let lk 2mk and ak bk x −1 mk , where bk are positive continuous function on G, E Cν and A x is a diagonal matrix-function with continuous components dm 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 bk x Dkmk um x 2 mk −1 d m x um x fm x , k1 mkj mkj 4.55 i i αkji Dk um Gk0 0, βkji Dk um Gkb 0, i0 i0 j 1, 2, . . . , mk , m 1, 2, . . . , ν, in the vector-valued space Lp,γ G; Cν . 5. The Spectral Properties of Anisotropic Differential Operators Consider the following degenerated BVP: n l α ak x Dk k u x Axux Aα x D ux fx, |α:l|
Boundary Value Problems 19 Theorem 5.2. Suppose that all the conditions of Theorem 5.1 are satisﬁed with νk 0. Assume that E is a Banach space with a basis and n 1 ∼ j −1/ν , κ 1, 2, . . . , ∞, ν > 0. < 1, sj I E A , E j 5.3 l 1k k Then, a for a suﬃciently large positive d −1 ∼ j −1/ ν κ sj Q d Lp G; E , 5.4 b the system of root functions of the diﬀerential operator Q is complete in Lp G; E . Proof. Let I E0 , E denote the embedding operator from E0 to E. From Result 2, there exists a resolvent operator Q d −1 which is bounded from Lp G; E to Wp,γ G; E A , E . Moreover, l from Theorem A4 and Remark 3.1, we get that the embedding operator l I Wp,γ G; E A , E , Lp G; E 5.5 is compact and ∼ j −1/ ν κ l sj I Wp,γ G; E A , E , Lp G; E . 5.6 It is clear that −1 −1 l Q d Lp G; E Q d Lp G; E , Wp,γ G; E A , E 5.7 l × I Wp,γ G; E A , E , Lp G; E . Hence, from relations 5.6 and 5.7 , we obtain 5.4 . Now, Result 1 implies that the operator Q d is positive in Lp G; E and 1 −1 ∈ σq Lp G; E , 5.8 Q d q> . κ ν Then, from 4.52 and 5.6 , we obtain assertion b .
20 Boundary Value Problems Consider now the operator O in Lp,γ G; E generated by the nondegenerate BVP obtained from 5.1 under the mapping 3.7 ; that is, l DO Wp,γ G; E A , E, Lkj , 5.9 n l Aα x Dα u. ak x Dkk u Ou Axux |α:l|