Maximal regular boundary value problems in Banach-valued weighted space
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This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banachvalued weighted L p spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties. 1. Introduction and notation Boundary value problems for differential-operator equations have been studied in detail in [4, 15, 22, 35, 40, 42]. The solvability and the spectrum of boundary value problems for elliptic differential-operator equations have also been studied in [5, 6, 12, 14, 16, 18, 29, 30, 31, 32, 33, 34, 37, 41]. A comprehensive introduction to differential-operator equations and historical references may be found in [22, 42]. In these works, Hilbert-valued function spaces have been considered. The main objective of the present paper is to discuss nonlocal boundary value problems for ordinary and partial differential-operator equations (DOE) in Banach-valued weighted L p spaces. In this work, the following is done. (1) The continuity, compactness, and qualitative properties of the embedding operators in the associated Banach-valued weighted function space are considered. (2) An ordinary differential-operator equation Lu =
m
−k (k) ak Am u (x) = f (x), λ
x ∈ (0,b), am = 0
k=0
of arbitrary order on a domain with varying bound is investigated. Copyright © 2005 Hindawi Publishing Corporation Boundary Value Problems 2005:1 (2005) 9–42 DOI: 10.1155/BVP.2005.9
(1.1)
10
Maximal regular BVPs in Banach-valued weighted space (3) An anisotropic partial DOE n k=1
ak Dklk u(x) +
Aα (x)Dα u(x) = f (x),
x = x1 ,x2 ,...,xn
(1.2)
|α:l| 0,
ξ(u,v) ≤ u + v for u = v = 1.
(1.5)
The ξ-convex Banach space E is often called a UMD space and written as E ∈ UMD. It is shown in [9] that a Hilbert operator (H f )(x) = limε→0 | y|>ε f (y)/(x − y)d y is bounded in L p (R,E), p ∈ (1, ∞), for those and only those spaces E which satisfy E ∈ UMD. UMD spaces include, for example, L p , l p spaces, and Lorentz spaces L pq with p, q ∈ (1, ∞). Let C be the set of complex numbers and
Sϕ = λ ∈ C : | arg λ − π | ≤ π − ϕ ∪ {0},
0 < ϕ ≤ π.
(1.6)
A linear operator A is said to be ϕ-positive in a Banach space E with bound M > 0 if D(A) is dense in E and (A − λI)−1
L(E)
−1 ≤ M 1 + |λ|
(1.7)
with λ ∈ Sϕ , ϕ ∈ (0,π], where I is the identity operator in E and L(E) is the space of bounded linear operators acting on E. Sometimes, instead of A + λI we will write A + λ and denote this by Aλ . It is known [38,
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