Linear and nonlinear abstract elliptic equations with VMO coefficients and applications

Veli B Shakhmurov

doi:10.1186/1687-1812-2013-6

Abstract

In this paper, maximal regularity properties for linear and nonlinear elliptic differential-operator equations with VMO (vanishing mean oscillation) coefficients are studied. For linear case, the uniform separability properties for parameter dependent boundary value problems is obtained in L p spaces. Then the existence and uniqueness of a strong solution of the boundary value problem for a nonlinear equation is established. In application, the maximal regularity properties of the anisotropic elliptic equation and the system of equations with VMO coefficients are derived.MSC: 58I10, 58I20, 35Bxx, 35Dxx, 47Hxx, 47Dxx.

Highlights

1 Introduction The goal of the present paper is to study the nonlocal boundary value problems (BVPs) for parameter dependent linear differential-operator equations (DOEs) with discontinuous top-order coefficients, sa(x)u( )
We study maximal regularity properties of anisotropic elliptic equations in mixed Lp spaces and the systems of differential equations with VMO coefficients in the scalar Lp space
Theorem A Suppose the following conditions are satisfied: ( ) E is a Banach space satisfying the multiplier condition with respect to p ∈ (, ∞) and A is an R-positive operator in E; ( )

Summary

Introduction

The goal of the present paper is to study the nonlocal boundary value problems (BVPs) for parameter dependent linear differential-operator equations (DOEs) with discontinuous top-order coefficients, sa(x)u( ). Theorem A Suppose the following conditions are satisfied: ( ) E is a Banach space satisfying the multiplier condition with respect to p ∈ ( , ∞) and A is an R-positive operator in E;. Let Qs denote the operator in Lp( , ; E) generated by the problem ( ) for λ = , i.e., D(Qs) = W ,p , ; E(A), E, Lk , Qsu = sa(x)u( ) + A(x)u. The following coercive uniform estimate holds: Proof First, let us show that the operator Q + λ has a left inverse. The estimate ( ) implies that ( ) has a unique solution and the operator Qs + λ has a bounded inverse in its rank space. It can be derived for the operator function A(x)A– (x ) ∈ VMO(L(E))

Result