Abstract
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 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.
Highlights
Introduction and NotationsIt is well known that many classes of PDEs, pseudo-Des, and integro-DEs can be expressed as differential-operator equations DOEs
We prove that the corresponding differential operator is separable in Lp; that is, it has a bounded inverse from Lp to the anisotropic weighted space Wpl,γ
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
Summary
It is well known that many classes of PDEs, pseudo-Des, and integro-DEs can be expressed as differential-operator equations DOEs. 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. 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 ξβDβ A x A x ξI −1 : ξ ∈ Rn \ 0, β ∈ U ≤ M. We let Wpl ,γ Ω; E0, E denote the space of all functions u ∈ Lp,γ Ω; E0 possessing generalized derivatives Dklk u ∂lk u/∂xklk such that Dklk u ∈ Lp,γ Ω; E with the norm u Wpl ,γ Ω;E0,E u Lp,γ Ω;E0 n k 1 Dklk u Lp,γ Ω;E < ∞. Wpl,γ G; E A , E u : u ∈ Lp G; E A , Dklk u ∈ Lp G; E , u Wpl,γ G;E A ,E u Lp G;E A n k 1 Dklk u Lp G;E
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