Degenerate Differential Operators with Parameters

The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valued Lp− spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.

Boundary value problems (BVPs) for differential-operator equations (DOEs) in H-valued (Hilbert space-valued) function spaces have been studied extensively by many researchers (see [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein).BVPs for DOE on E-valued (Banach space valued) function spaces are studied in [1,[14][15][16][17].The main aim of the present paper is to discuss the BVPs for regular degenerate DOE with the parameter on E-valued function spaces.The maximal regularity and Fredholmness of these problems in Banach-valued L p -spaces are established.In applications, the nonlocal BVPs for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.
A Banach space E is called the UMD-space (see, e.g., [19,20]) if the Hilbert operator is bounded in the space L p (R,E), p ∈ (1,∞).UMD spaces include, for example, L p , l p spaces and Lorentz spaces L pq , p, q ∈ (1,∞).Let C be a set of complex numbers and A linear operator A is said to be positive in a Banach space E, with bound M if D(A) is dense on E and with ξ ∈ S ϕ ,ϕ ∈ (0,π], where M is a positive constant and I is an identity operator in E, where L(E) is the space of bounded linear operators acting in E. Sometimes instead of A + ξI will be written A + ξ and denoted by A ξ .It is known [33,Section 1.15.1]there exist fractional powers A θ of the positive operator A. By the definition of the positive operator A for all ξ ∈ S(ϕ), ξ(A − ξI) −1  B(E) ≤ M. (1.5) The operator A(t) is said to be positive in the Banach space E uniformly with respect to t if D(A(t)) is independent of t, D(A(t)) is dense in E, and for all λ ∈ S(ϕ), ϕ ∈ (0,π].Let E(A θ ) denote the space D(A θ ) with graphical norm defined as (1.7) Let E 1 and E 2 be two Banach spaces.By (E 1 ,E 2 ) θ,p , 0 < θ < 1, 1 ≤ p ≤ ∞, will be denoted an interpolation space for {E 1 ,E 2 } by the K-method [21,Section 1.3.1].
We denote by D(R n ;E) the space of E-valued C ∞ -functions with compact support, equipped with the usual inductive limit topology and S = S(R n ;E) denotes the E-valued Schwartz space of rapidly decreasing, smooth functions.For E = C we simply write D(R n ) and S(R n ), respectively.D (R n ;E) = L(D(R n ),E) denote the space of E-valued distributions and S (E) = S (R n ;E) is a space of linear continued mapping from S(R n ) into Veli B. Shakhmurov 3 E. Let E 1 and E 2 be two Banach spaces.The Fourier transform for u ∈ S (R n ;E) is defined by (1.8) Let γ such that S(R n ;E 1 ) is dense in L p,γ (R n ;E 1 ) (see, e.g., Lemma 2.1).A function Ψ ∈ C(R n ;L(E 1 ,E 2 )) is called a Fourier multiplier from L p,γ (R n ;E 1 ) to L q,γ (R n ;E 2 ) if the map u → Φu = F −1 Ψ(ξ)Fu, u ∈ S(R n ;E 1 ) is well defined and extends to a bounded linear operator We denote the set of all multipliers from depending on h ∈ H is called a uniformly collection of multipliers with respect to h if there exists a positive constant C independent on h ∈ H such that for all h ∈ H and u ∈ S(R n ;E 1 ).The exposition of the theory of L p -multipliers of the Fourier transformation, and some related references, can be found in [33, Sections 2.2.1-2.2.4].In vector-valued function spaces, Fourier multipliers have been studied in [14,22,23,[25][26][27]29].
A set K ⊂ B(E 1 ,E 2 ) is called R-bounded (see, e.g., [14,22,28]) if there is a positive constant C such that for all T 1 ,T 2 ,...,T m ∈ K and u 1 ,u 2 ,...,u m ∈ E 1 , m ∈ N, where {r j } is a sequence of independent symmetric [−1,1]-valued random variables on [0,1] and N denotes the set of natural numbers.The smallest such constant C is called the R-bound of K and is denoted by R(K).

A family of sets
where the constant C is independent on parameter h (i.e., sup h∈H R( . We say that W h is a uniform collection of multipliers if there exists a constant M > 0 independent on h ∈ H such that for all h ∈ H and u ∈ S(R n ;E 1 ). Let (1.14) Definition 1.1.The Banach space E is said to be a space satisfying a multiplier condition with respect to p ∈ (1,∞) and weight function γ, when for every p,γ (E).A Banach space E is said to be a space satisfying a uniform multiplier condition, when for then Ψ h is a uniform collection of multipliers in M p p (E) for p ∈ (1,∞).A Banach space E has a property (α) (see, e.g., [22,29]) if there exists a constant α such that for all N ∈ N, x i, j ∈ E, α i j ∈ {0, 1}, i, j = 1,2,...,N, and all choices of independent, symmetric, {−1, 1}-valued random variables ε 1 ,ε 2 ,...,ε N , ε 1 ,ε 2 ,...,ε N on probability spaces Ω,Ω .For example, the spaces L p (Ω), 1 ≤ p < ∞, have the property (α).
Remark 1.2.The result [21] implies that the space l p , p ∈ (1,∞), satisfies multiplier condition with respect to p and the weight functions Moreover, the UMD spaces with (α) properties satisfy the multiplier condition with respect to p ∈ (1,∞) and the weighted function It is well known (see [25,26]) that any Hilbert space satisfies the multiplier condition.There are, however, Banach spaces which are not Hilbert spaces but satisfy the multiplier condition, for example, UMD spaces (see [14,17,22,27]).

Definition 1.3. A positive operator
A is said to be R-positive in the Banach space E if there exists ϕ ∈ (0,π] such that the set Veli B. Shakhmurov 5 Note that in a Hilbert space every norm bounded set is R-bounded.Therefore, in a Hilbert space, all positive operators are R-positive.If A is a generator of a contraction semigroup on L q ,1 ≤ q ≤ ∞ [30], A has bounded imaginary powers with (−A it ) B(E) ≤ Ce ν|t| , ν < π/2, [31] or if A is generator of a semigroup with Gaussian bound [23] in E ∈ UMD, then those operators are R-positive.
σ ∞ (E) will denote the space of compact operators in E. Let E 0 and E be two Banach spaces and E 0 is continuously and densely embedded into E. Let Ω be a domain on R n and l = (l 1 ,l 2 ,...,l n ).W l p,γ (Ω;E 0 ,E) denotes a space that consists of functions u ∈ L p,γ (Ω;E 0 ) such that it has the generalized derivatives For E 0 = E the space W l p,γ (Ω;E 0 ,E) will be denoted by W l p,γ (Ω;E).The weight γ is said to satisfy an A p condition, that is, , where γ k ∈ A p and there exist constants C 1 , C 2 such that (1.23)

Background materials
Embedding theorems for vector-valued Sobolev spaces played important role in the present investigation.Embedding theorems in Hilbert-valued function spaces have been studied, for example, in [11][12][13]32].This section is concentrated on weighted anisotropic Banach-valued Sobolev spaces W l p,γ (Ω;E 0 ,E) associated with Banach spaces E 0 , E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of E 0 and E. In particular, the most regular class of interpolation spaces E α between E 0 , E, depending on α and order of spaces are found that mixed derivatives D α are bounded and compact from W l p,γ (Ω;E 0 ,E) to L p,γ (Ω;E α ).These results generalize and improve the results [11][12][13]32].Multiplier theorems in the operator-valued L p spaces are important tools in the theory of embedding of function spaces and in BVPs.Since our consideration take place in weighted case with parameterized estimates, so we have to generalize multiplier theorems [22] for the case of L p,γ and for multipliers depending on parameters.Lets first show the following needed lemma.Lemma 2.1.Let E be a Banach space, 1 ≤ p < ∞, and γ a positive measurable function on an open subset Ω of R n , essentially bounded on compact subsets of Ω.Then the space D(Ω;E) is dense in L p,γ (Ω;E). (2.1) By the dominated convergence theorem lim n→∞ u − u n Lp,γ(Ω;E) = 0, hence a compactly supported function can be approximated with bounded compactly supported functions, that is, with compactly supported function belonging to L p (Ω;E).From the standard proof of the denseness theorem in case of spaces without weight, it follows that if u is a compactly supported function belonging to L p (Ω;E), then there exists a compact subset K ⊂ Ω, with suppu ⊆ K, and a sequence of functions we have From [15] we have the proof.

Veli B. Shakhmurov 7
Theorem 2.3.Suppose the following conditions are satisfied: (1) E is a Banach space that satisfies the multiplier condition with respect to p and weighted function γ(x) and A is an R-positive operator in E for ϕ with 0 < ϕ ≤ π; (3) Ω ∈ R n is a region such that there exists a bounded linear extension operator from W l p,γ (Ω;E(A),E) to W l p,γ (R n ;E(A),E).Then the following embedding: is continuous and there exists a positive constant C μ such that for all u ∈ W l p,γ (Ω;E(A),E), and h with 0 < h ≤ h 0 < ∞.Proof.It is sufficient to prove the estimate (2.6).At first we prove the estimate (2.6) for Ω = R n .Really, it is easy to see that Thus the inequality (2.6) for Ω = R n will be proved if the estimate is provided for a suitable positive constant C μ .Let By virtue of (2.8) it is easy to see that inequality (2.9) will follow immediately if we can prove that the operator-function p,γ (E) depend on parameters t and h.To see this, it is sufficing to show that the sets are R-bounded in E and the R-bounds do not depend on t and h.In fact, by using a similar technique as in [14, Lemma 3.1] we have uniformly with respect to t and h.Due to R-positivity of operator A and by estimate (2.12) we obtain that the sets are R bounded uniformly with respect to t and h.Moreover, for u 1 ,u 2 ,...,u m ∈ E, m ∈ N, and where {r j } is a sequence of independent symmetric {−1, 1}-valued random variables on [0,1].By virtue of Kahane's contraction principle [14, Lemma 3.5] we obtain from the above equality Then by the above estimate, in view of (2.12), and by product properties of the collection of R-bounded operators (see, e.g., [14,Proposition 3.4]) we get that the set {Ψ t,h (ξ) : is R-bounded uniformly with respect to t and h.In a similar way, by using Kahane's contraction principle and by product and additional properties of the collection of R-bounded operators [14, Proposition 3.4], we obtain that the sets are R-bounded uniformly with respect to t and h.Then we obtain that operator-function Ψ t,h (ξ) is a uniform collection of multipliers in M q,γ p,γ (E).Therefore, we obtain the estimate (2.12).Then by using an extension operator in W l p,γ (Ω;E(A),E), we obtain from (2.9) estimate (2.6).
Theorem 2.4.Suppose all conditions of Theorem 2.3 are satisfied; Ω is a bounded region on R n satisfy the l-horn conditions and A −1 ∈ σ ∞ (E).Let the weighted function γ satisfy Condition 1.4.Then for 0 < μ ≤ 1 − κ, an embedding , the following multiplicative inequality is obtained: ( By virtue of [16, Theorem 2], the embedding is compact.Then from the above estimate we obtain assertion of Theorem 2.4.By a similar manner as Theorem 2.3, we have the following.
Lemma 2.8.Let the following conditions be satisfied: (1) , a m = 0, and the roots ω j are d-separated; (2) A is a closed operator in a Banach space E with a dense domain D(A) and

.31)
Then for a function u (x) to be a solution of (2.29), which belongs to the space W m p,ν (0,b;E(A m ),E), it is necessary and sufficient that where (2.33)
Theorem 4.1.Let A be a positive operator in a Banach space E for ϕ ∈ (0,π], 0 for sufficiently large |λ| and t, has a unique solution u belongs to W [2]   p,ν (0,1;E(A),E), and the coercive uniform estimate |λ| u Lp(0,1;E) + tu [2]  Lp(0,1; holds with respect to parameters t and λ. Proof.Under the substitution (3.6), the problem (4.1) reduces to a nondegenerate problem L 0 (λ,t)u = −tu (2) (y) + (A + λ)u(y) = 0, (4.3) in the weighted space L p,γ (0,b;E).Dividing both sides of (4.3) to t > 0, we obtain a boundary value problem ) Since A is the positive operator in E and 0 < t < t 0 < ∞, then A/t is positive uniformly with respect to t, and for all λ ∈ S ϕ , we have By virtue of condition (1) together with estimate (4.7) and by virtue of [4, Lemma 5.4.2/6],there is a holomorphic semigroup e −x(t −1 Aλ) 1/2 for x > 0, which is strongly continuous for x ≥ 0. Then by virtue of Lemma 2.8 an arbitrary solution of (4.5), for | arg λ| ≤ π − ϕ, belonging to W 2 p,γ (0,b;E(A),E) has the form where Now taking into account the boundary conditions (4.6), we obtain algebraic linear equations with respect to g 1 , g 2 : (4.10) The system (4.10) can be expressed as the following matrix-operator equation: where Let Q(λ,t) denote a determinant-operator of the matrix-operator D(λ,t).It is clear that Using the properties of positive operators and holomorphic semigroups (see [4, Lemma 5.4.2/6]) it is clear to see that for | arg λ| ≤ π − ϕ, |λ| → ∞ and 0 < t ≤ t 0 , The above estimate implies Due to the positivity of operator A in E and by (4.15) we obtain that operator By virtue of estimate (4.15) it is clear that the operator Q −1 (λ,t) is bounded uniformly with respect to the parameter λ, that is, Veli B. Shakhmurov 15 Consequently, the system (4.10) has a unique solution for | arg λ| ≤ π − ϕ, sufficiently large |λ|, and the solution can be expressed in the form By virtue of the properties of the golomorphic semigroups [33, Section 1.13.1], in view of uniformly boundedness of Q 0 , and by changing of variable, we obtain from (4.20) a uniformly estimate, with respect to t and λ, By the properties of resolvent of positive operator A, we have Then by virtue of estimate (4.22) and Remark 3.1 we obtain the estimate (4.2).
It is shown by taking into account R-positivity of the operator A and by using the equivalent definition of the interpolation spaces [33, Section 1.14.5].