The Uniformly Parabolic Equations of Higher Order with Discontinuous Data in Generalized Morrey Spaces and Elliptic Equations in Unbounded Domains

Tair Gadjiev,Konul Suleymanova

doi:10.5772/intechopen.96781

Abstract

We study the regularity of the solutions of the Cauchy-Dirichlet problem for linear uniformly parabolic equations of higher order with vanishing mean oscillation (VMO) coefficients. We prove continuity in generalized parabolic Morrey spaces Mp,φ of sublinear operators generated by the parabolic Calderon-Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. We obtain strong solution belongs to the generalized Sobolev-Morrey space Wp,φm,1∘Q. Also we consider elliptic equation in unbounded domains.

Highlights

We consider the higher order linear Cauchy-Dirichlet problem in Q 1⁄4 Ω Â ð0, TÞ, being a cylinder in nþ1, Ω ⊂ Rn be a bounded domain 0 < T < ∞ut À X aαβðx, tÞDαβuðx, tÞ 1⁄4 f ðx, tÞ, a:e: in Q (1) ∣α∣ ≤ m,∣β∣ ≤ m uðx, tÞ 1⁄4 0 on ∂pQ, (2)where ∂pQ 1⁄4 ð∂Ω Â 1⁄20, TÞ ∪ ðΩ Â ft 1⁄4 0gÞ stands for the parabolic boundary ofQ and Dαβ 1⁄4 ∂xα11,∂⋯∣α∣, ∂xαnn ⋯ ∂yβ11,∂⋯∣β∣, ∂yβnn, ∣α∣ 1⁄4 Pnk1⁄41αk, β 1⁄4 Pnk1⁄41βk
The generalized Morrey space is defined to be the set of all f ∈ Lp,locÀ nþ1Á such that
: Let φ : nþ1 Â Rþ ! Rþ be a measurable function and p ∈ 1⁄21, ∞Þ: The generalized parabolic Morrey space Mp,φÀ nþ1Á consists of all f ∈ Lp,locÀ nþ1Á such that

Summary

Introduction

We consider the higher order linear Cauchy-Dirichlet problem in Q 1⁄4 Ω Â ð0, TÞ, being a cylinder in nþ, Ω ⊂ Rn be a bounded domain 0 < T < ∞. In [3] proved boundedness of maximal and Calderon-Zygmund operators in Mp,φ imposing suitable integral and doubling conditions on φ. These results allow to study the regularity of the solutions of various linear elliptic and parabolic value problems in Mp,φ (see [4–6]). We study the boundedness of the sublinear operators, generated by Calderon-Zygmund operators in generalized Morrey spaces and the regularity of the solutions of higher order uniformly elliptic boundary value problem in local generalized Morrey spaces where domain is bounded.

Some notation and definition

Non-singular integrals in generalized Morrey spaces

Singular and non-singular integrals in generalized Morrey spaces

Proof of the first main result

The higher order elliptic equations in unbounded domains