Maximal regular boundary value problems in Banach-valued function spaces and applications

Veli B Shakhmurov

doi:10.1155/ijmms/2006/92134

Abstract

The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non-selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach-valuedLp-spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied.

Highlights

Background materialsEmbedding theorems of vector-valued Sobolev spaces played important role in the present investigation
A Banach space E is said to be the space satisfying a multiplier condition with respect to p ∈ (1, ∞) when for Ψ ∈ C(n)(Rn; B(E)), if the sets
This section concentrates on anisotropic Banach-valued Sobolev spaces Wpl (Ω; E0, E) associated with Banach spaces E0, E

Summary

Maximal regular problems

By Lp,q(Ω) and Wpl ,q(Ω), we will denote a scalar-valued (p, q)-integrable function space and Sobolev space with mixed norms, respectively [8]. The operator A(t) is said to be positive in a Banach space E uniformly with respect to t, if D(A(t)) is independent of t, D(A(t)) is dense in E, and. Let S(Rn;E) denote a Schwarz class, that is, the space of all E-valued rapidly decreasing smooth functions φ on Rn. The function Ψ ∈ C(Rn; L(E1, E2)) is called a multiplier from Lp(Rn; E1) to Lq(Rn; E2) if the map u → Ku = F−1Ψ(ξ)Fu, u ∈ S(Rn; E1), is well defined and extends to a bounded linear operator. Let. Φh is a uniformly bounded multiplier with respect to h if there exists a constant C > 0, independent on h ∈ B(h), such that. A Banach space E is said to be the space satisfying a multiplier condition with respect to p ∈ (1, ∞) when for Ψ ∈ C(n)(Rn; B(E)), if the sets. By σ∞(E) will be denoted a space of all compact operators in E

Background materials

Statement of problems

Ordinary DOE with constant coefficients

Result