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This article is part of the series Proceedings of International Conference on Applied Analysis and Mathematical Modeling 2013.

Open Access Research

Stokes operators with variable coefficients and applications

Veli B Shakhmurov

Author Affiliations

Department of Mechanical Engineering, Okan University, Akfirat, Tuzla, Istanbul, 34959, Turkey

Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan

Boundary Value Problems 2014, 2014:86  doi:10.1186/1687-2770-2014-86

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/86


Received:30 September 2013
Accepted:21 April 2014
Published:2 May 2014

© 2014 Shakhmurov; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

The stationary and instationary Stokes problems with variable coefficients in abstract <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M2">View MathML</a> spaces are considered. The problems contain abstract operators and nonlocal boundary conditions. The well-posedness of these problems is derived.

MSC: 35Q30, 76D05, 34G10, 35J25.

Keywords:
Stokes systems; Navier-Stokes equations; abstract differential equations with variable coefficients; semigroups of operators; boundary value problems; maximal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M2">View MathML</a> regularity

1 Introduction

We consider the initial and boundary value problem (BVP) for the following Stokes type equation with variable coefficients:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M3">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M4">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M5">View MathML</a>

(1.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M6">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M8">View MathML</a> are linear operators in a Banach space E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M10">View MathML</a> are complex numbers, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M11">View MathML</a> are complex-valued functions. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M12">View MathML</a> represents a given, a denotes the initial data and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M13">View MathML</a>

represent the unknown functions. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M15">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M16">View MathML</a> are E-valued functions. This problem is characterized by the presence of abstract operator functions and complex-valued variable coefficients in the principal part. Moreover, boundary conditions are nonlocal, generally. The existence, uniqueness, and coercive estimates of maximal regular solution of problem (1.1)-(1.2) are obtained. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing E and operators A, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M17">View MathML</a> we can obtain maximal regularity properties for numerous class of Stokes type problems with variable coefficients. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M20">View MathML</a> problem (1.1)-(1.2) is reduced to nonlocal Stokes problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M21">View MathML</a>

(1.4)

where ℂ is the set of complex numbers. Note that the existence of weak or strong solutions and regularity properties for the classical Stokes problems were extensively studied, e.g., in [1-10]. There is an extensive literature on the solvability of the initial value problems (IVPs) for the Stokes equation (see, e.g., [1,2,10] and further papers cited there). Solonnikov [8] proved that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23">View MathML</a> the instationary Stokes problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M24">View MathML</a>

(1.5)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M25">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M26">View MathML</a>

Then Giga and Sohr [2] improved the result of Sollonikov for spaces with different exponents in space and time, i.e., they proved that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M27">View MathML</a> there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M25">View MathML</a> of problem (1.5) so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M29">View MathML</a>

(1.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M30">View MathML</a>

Moreover, the estimate obtained was global in time, i.e., the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M31">View MathML</a> is independent of T and f. To derive the global <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M2">View MathML</a>-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M33">View MathML</a> estimates (1.6), Giga and Sohr used abstract parabolic semigroup theory in UMD spaces. The estimate (1.6) allows one to study the existence of solution and regularity properties of the corresponding Navier-Stokes problem (see, e.g., [4]). Consider first at all, the stationary version of problem (1.1)-(1.3), i.e., consider the abstract Stokes problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M34">View MathML</a>

(1.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M35">View MathML</a>

(1.8)

λ is a complex number. By applying the corresponding projection transformation P, (1.7)-(1.8) can be reduced to the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M36">View MathML</a>

(1.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M37">View MathML</a>

(1.10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M38">View MathML</a> denote the operator generated by problem (1.9)-(1.10), i.e., let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M38">View MathML</a> be a Stokes operator in the E-valued solenoidal space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M40">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M41">View MathML</a>

We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M38">View MathML</a> is a positive operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M43">View MathML</a> is a generator of an analytic semigroup in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M40">View MathML</a>. In other words, we consider the instationary Stokes problem (1.1)-(1.3) and prove the well-posedness of this problem. We prove that there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M25">View MathML</a> of problem (1.1)-(1.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M47">View MathML</a>, and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M48">View MathML</a>

(1.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a> is the class of E-valued <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M33">View MathML</a>-spaces and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M51">View MathML</a> is a corresponding interpolation space. The estimate (1.11) allows one to study the existence of solution and regularity properties of the corresponding Navier-Stokes problem. Finally, we give some application of this abstract Stokes problem to anisotropic Stokes equations and systems of equations. Note that the abstract Stokes problem with constant coefficients was studied in [11].

2 Definitions and background

Let E be a Banach space. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M52">View MathML</a> denotes the space of strongly measurable E-valued functions that are defined on the measurable subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M53">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M54">View MathML</a>

The Banach space E is called an UMD space if the Hilbert operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M55">View MathML</a>

is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23">View MathML</a> (see, e.g., [12]). UMD spaces include e.g.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M59">View MathML</a> spaces, and Lorentz spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M61">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M62">View MathML</a>

A linear operator A is said to be ψ-positive in a Banach space E with bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M63">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M64">View MathML</a> is dense on E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M65">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M67">View MathML</a>, where I is the identity operator in E, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M68">View MathML</a> is the space of bounded linear operators in E. It is known [[13], §1.15.1] that there exist fractional powers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M69">View MathML</a> of a positive operator A. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M70">View MathML</a> denote the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M71">View MathML</a> with norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M72">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M74">View MathML</a> be two Banach spaces. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M77">View MathML</a>, will be denoted the interpolation spaces obtained from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M78">View MathML</a> by the K-method [[13], §1.3.2].

Let ℕ denote the set of natural numbers. A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M79">View MathML</a> is called R-bounded (see, e.g., [14]) if there is a positive constant C such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M82">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M83">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M84">View MathML</a> is a sequence of independent symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M85">View MathML</a>-valued random variables on Ω. The smallest C for which the above estimate holds is called a R-bound of the collection Φ and denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M86">View MathML</a>.

A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M87">View MathML</a> is called uniform R-bounded in h, if there is a constant C independent on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M89">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M92">View MathML</a>. It is implied that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M93">View MathML</a>.

The ψ-positive operator A is said to be R-positive in a Banach space E if the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M67">View MathML</a> is R-bounded.

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M96">View MathML</a> is said to be ψ-positive in E uniformly with respect to t with bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M63">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M98">View MathML</a> is independent on t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M99">View MathML</a> is dense in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M100">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M67">View MathML</a>, where M does not depend on t and λ.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M103">View MathML</a> and E be two Banach spaces and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M103">View MathML</a> continuously and densely embedded into E. Let Ω be a domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M105">View MathML</a> and m is a positive integer. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M106">View MathML</a> denotes the space of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M107">View MathML</a> that have generalized derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M108">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M109">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M112">View MathML</a> the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M106">View MathML</a> will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M114">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M115">View MathML</a> the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M116">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M117">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M119">View MathML</a>, denote the E-valued Liouville space of order s such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M120">View MathML</a>. It is known that if E is a UMD space, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M121">View MathML</a> for positive integer m (see, e.g., [[15], §15].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M122">View MathML</a> denote a Liouville-Lions type space, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M123">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M124">View MathML</a> denote the E-valued solenoidal space, i.e., the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M125">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M126">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M127">View MathML</a>

Sometimes we use one and the same symbol C without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say α, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M128">View MathML</a>.

The embedding theorems in vector-valued spaces play a key role in the theory of DOEs. For estimating lower order derivatives we use the following embedding theorems from [16].

Theorem A1Suppose the following conditions are satisfied:

(1) Eis a UMD space andAis anR-positive operator inE;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M129">View MathML</a>andmis a positive integer such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M134">View MathML</a>is a fixed positive number;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M53">View MathML</a>is a region such that there exists a bounded linear extension operator from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M136">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M137">View MathML</a>.

Then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M138">View MathML</a>is continuous and for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M139">View MathML</a>the following uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M140">View MathML</a>

Remark 2.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M53">View MathML</a> is a region satisfying the strong l-horn condition (see [[4], §7]), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M143">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23">View MathML</a> there exists a bounded linear extension operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M145">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M146">View MathML</a>.

Theorem A2Suppose all conditions of Theorem A1are satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M147">View MathML</a>. Moreover, let Ω be a bounded region and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M148">View MathML</a>. Then the embedding

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M149">View MathML</a>

is compact.

Theorem A3Suppose all conditions of Theorem A1satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M147">View MathML</a>. Then the embedding

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M151">View MathML</a>

is continuous and there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M152">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M153">View MathML</a>the uniform estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M154">View MathML</a>

From [[17], Theorem 2.1] we obtain the following.

Theorem A4LetEbe a Banach space, Abe aφ-positive operator inEwith boundM, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M155">View MathML</a>. Letmbe a positive integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M132">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M157">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158">View MathML</a>, an operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M159">View MathML</a>generates a semigroup<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M160">View MathML</a>which is holomorphic for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M161">View MathML</a>. Moreover, there exists a positive constantC (depending only onM, φ, m, αandp) such that for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M164">View MathML</a>

3 The stationary Stokes system with variable coefficients

In this section, we derive the maximal regularity properties of the stationary abstract Stokes problem (1.7)-(1.8).

First at all, we consider the BVP for the variable coefficient differential operator equation (DOE)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M165">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M167">View MathML</a> are linear operators in a Banach space E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M11">View MathML</a> are complex-valued functions, and λ is a complex parameter.

Maximal regularity properties for DOEs studied, e.g., in [1,11,12,14,16-24]. Nonlocal BVPs for PDE were studied in [25].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M170">View MathML</a>, be roots of the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M171">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M173">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M174">View MathML</a>

Condition 3.1 Assume:

(1) E is a UMD space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M166">View MathML</a> is a uniformly R-positive operator in E for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M176">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M180">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M182">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M186">View MathML</a>;

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23">View MathML</a>.

Remark 3.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M192">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M193">View MathML</a> are real-valued positive functions and. Then Condition 3.1 is satisfied.

Remark 3.2 The conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M195">View MathML</a> are given due to the nonlocality of the boundary conditions. For local boundary conditions these assumptions are not required.

From [[19], Theorem 4.1] we have the following.

Theorem 3.1Suppose Condition 3.1 is satisfied. Then problem (3.1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M196">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M197">View MathML</a>and for sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158">View MathML</a>. Moreover, the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M199">View MathML</a>

Consider the differential operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M200">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M201">View MathML</a>generated by problem (1.7)-(1.8), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M202">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M203">View MathML</a>. From Theorem 3.1 we obtain the following.

Result 3.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M204">View MathML</a> there is a resolvent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M205">View MathML</a> and the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M206">View MathML</a>

Let E be a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M207">View MathML</a> denote the class of E-valued system of function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M208">View MathML</a> with norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M209">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M210">View MathML</a> denote the E-valued solenoidal space and A be a positive operator in E. The spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M212">View MathML</a> will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M214">View MathML</a>.

Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M215">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M167">View MathML</a> are linear operators in a Banach space E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M11">View MathML</a> are complex-valued functions, and λ is a complex parameter. From Theorem 3.1 we obtain the following result.

Result 3.2 Suppose Condition 3.1 is satisfied. Then problem (3.2) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M220">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a> and for sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158">View MathML</a>. Moreover, the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M223">View MathML</a>

Consider the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M224">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a> generated by problem (3.2), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M226">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M227">View MathML</a>

From Result 3.2 we obtain the following uniform coercive estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M228">View MathML</a>

(3.3)

Consider the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M229">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M230">View MathML</a> becomes a Banach space with this norm.

It is known that (see, e.g., [3,5]) the vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M231">View MathML</a> has a Helmholtz decomposition. In the following theorem we generalize this result for the E-valued function space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a>.

Theorem 3.2LetEbe an UMD space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M234">View MathML</a>has a Helmholtz decomposition, i.e., there exists a linear bounded projection operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235">View MathML</a>from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a>onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M237">View MathML</a>with null space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M238">View MathML</a>. In particular, all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M234">View MathML</a>have a unique decomposition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M240">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M242">View MathML</a>so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M243">View MathML</a>

(3.4)

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M244">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M245">View MathML</a>.

For proving Theorem 3.2 we need some lemmas.

Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M246">View MathML</a>

(3.5)

Lemma 3.1LetEbe an UMD space, AaR-positive operator inEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M248">View MathML</a>, problem (3.2) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M249">View MathML</a>and the following coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M250">View MathML</a>

(3.6)

Proof Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M251">View MathML</a>

(3.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M252">View MathML</a> is an extension of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M253">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M105">View MathML</a>. Then, by using the Fourier inversion formula, operator-valued multiplier theorems in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M33">View MathML</a> spaces, and by reasoning as in [[17], Theorem 3.2] we see that problem (3.7) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M256">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M257">View MathML</a> and the following coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M258">View MathML</a>

This fact implies that the function u which is a restriction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M259">View MathML</a> on G is a solution of problem (3.5). The estimate (3.6) is obtained from the above estimate.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M260">View MathML</a> be a unit normal to the boundary Γ of the domain G and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M261">View MathML</a> is a normal component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M262">View MathML</a> on Γ, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M263">View MathML</a>

Here and hereafter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M264">View MathML</a> will denoted the conjugate of E, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M265">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M266">View MathML</a>) denotes the duality pairing of functions on G (resp. Γ). □

By reasoning as in [[3], Lemma 2] we get the following.

Lemma 3.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M267">View MathML</a>is dense in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M268">View MathML</a>.

Proposition 3.1There exists a unique bounded linear operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M269">View MathML</a>from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M268">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M271">View MathML</a>onto

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M272">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M273">View MathML</a>

and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M274">View MathML</a>

(3.8)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M275">View MathML</a>

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M276">View MathML</a> consider the linear form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M277">View MathML</a>

(3.9)

By virtue of the trace theorem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M278">View MathML</a>, the interpolation of intersection and dual spaces (see, e.g., [[13], §1.8.2, 1.12.1, 1.11.2]) and by a localization argument we obtain the result that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M279">View MathML</a> is a bounded linear and surjective from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M280">View MathML</a> onto

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M281">View MathML</a>

Hence, we can find for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M282">View MathML</a> an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M283">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M284">View MathML</a>

Therefore, from (3.9) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M285">View MathML</a>

This implies the existence of an element

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M286">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M287">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M288">View MathML</a>

Thus, we have proved the existence of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M269">View MathML</a>. The uniqueness follows from Lemma 3.2. □

Proposition 3.1 implies the following.

Result 3.3 Assume the conditions of Proposition 3.1 are satisfied. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M290">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M291">View MathML</a> is a closed subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M208">View MathML</a>. Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M295">View MathML</a>

(3.10)

Lemma 3.3LetEbe an UMD space, AaR-positive operator inEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a>, problem (3.10) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M298">View MathML</a>and the coercive estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M299">View MathML</a>

(3.11)

Proof Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M300">View MathML</a>

(3.12)

By Lemma 3.1, problem (3.12) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M301">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a> and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M303">View MathML</a>

(3.13)

Consider now the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M304">View MathML</a>

(3.14)

By using Theorem 3.1, Result 3.3, and Proposition 3.1 we conclude that problem (3.14) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a> has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M306">View MathML</a> and the following coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M307">View MathML</a>

(3.15)

Then we conclude that problem (3.10) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M308">View MathML</a> and (3.13), (3.15) imply the estimate (3.11). □

Result 3.4 For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M20">View MathML</a> we obtain from (3.9), (3.11), and (3.12) that the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M310">View MathML</a>

(3.16)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M311">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a> and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M313">View MathML</a>

(3.17)

Result 3.5 For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M20">View MathML</a> we obtain from Lemma 3.1 and from (3.9), (3.10) that the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M315">View MathML</a>

(3.18)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M316">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a> and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M318">View MathML</a>

(3.19)

By (3.9), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M319">View MathML</a>. So, by (3.14) and Proposition 3.1 we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M320">View MathML</a>

(3.20)

From Results 3.4, 3.5, and estimate (3.17) we obtain

Result 3.6 The problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M321">View MathML</a>

(3.21)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M322">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a> and the estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M324">View MathML</a>

(3.22)

Consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M325">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M326">View MathML</a>

where w and υ are solutions of problems (3.18), (3.13), respectively. It is clear that we have the following.

Lemma 3.4LetEbe an UMD space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M328">View MathML</a>is a closed subspace of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a>.

Lemma 3.5LetEbe an UMD space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M233">View MathML</a>. Then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235">View MathML</a>is a bounded linear operator in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M333">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M334">View MathML</a>.

Proof The linearity of the operator P is clear by construction. Moreover, by Result 3.6 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M335">View MathML</a>

(3.23)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M334">View MathML</a> then by Result 3.3 we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M337">View MathML</a>. Moreover, by the estimate (3.20) we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M338">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M339">View MathML</a>. □

Lemma 3.6AssumeEis an UMD space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M340">View MathML</a>. Then the conjugate of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235">View MathML</a>is defined as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M245">View MathML</a>and is bounded linear in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M344">View MathML</a>.

Proof It is known (see, e.g., [13,18]) that the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M345">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M346">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M347">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M348">View MathML</a> we have only to show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M349">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M350">View MathML</a>. But this is deriving by reasoning as in [[3], Lemma 5]. Moreover, by Lemma 3.5 the dual operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M351">View MathML</a> is a bounded linear in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M346">View MathML</a>. □

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M353">View MathML</a>

From Lemmas 3.5, 3.6 we obtain the following.

Result 3.7 Assume E is an UMD space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M340">View MathML</a>. Then any element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a> uniquely can be expressed as a sum of elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M328">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M357">View MathML</a>.

In a similar way as Lemmas 6, 7 of [3] we obtain, respectively, the following.

Lemma 3.7AssumeEis an UMD space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M340">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M359">View MathML</a>

Lemma 3.8AssumeEis an UMD space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M340">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M361">View MathML</a>

Now we are ready to prove Theorem 3.2.

Proof of Theorem 3.2 From Lemmas 3.7, 3.8 we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M362">View MathML</a>. Then, by construction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M364">View MathML</a>. By Lemmas 3.3, 3.5, we obtain the estimate (3.4). Moreover, by Lemma 3.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M357">View MathML</a> is a close subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a>. Then it is known that the dual space of the quotient space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M367">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M368">View MathML</a>. In view of first assertion we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M369">View MathML</a> and by Lemma 3.8 we obtain the second assertion. □

Theorem 3.3Let Condition 3.1 hold. Then problem (1.7)-(1.8) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M370">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M372">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M204">View MathML</a>, and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M374">View MathML</a>

(3.24)

Proof By virtue of Result 3.2, we find that problem (3.2) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M375">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M221">View MathML</a> and for sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M158">View MathML</a>. Moreover, the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M378">View MathML</a>

By applying the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M235">View MathML</a> to problem (1.7)-(1.8) we get the Stokes problem (1.9)-(1.10). It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M380">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M38">View MathML</a> is the Stokes operator and B is a operator generated by problem (3.2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M226">View MathML</a>. Then by Theorem 3.2 we obtain the assertion. □

Result 3.8 From Result 3.2 we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M383">View MathML</a> is a positive operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a> and −O generate a bounded holomorphic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M385">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M386">View MathML</a>.

In a similar way as in [6] we show the following.

Proposition 3.2The following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M387">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M388">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M386">View MathML</a>.

Proof From the estimate (3.3) we see that the operator O is positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a>, i.e., for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M391">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M392">View MathML</a> the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M393">View MathML</a>

where the constant M is independent of λ. Then, by using the Danford integral and operator calculus (see, e.g., in [12]), we obtain the assertion. □

4 Well-posedness of instationary Stokes problems with variable coefficients

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M394">View MathML</a>

In this section, we will show the well-posedness of problem (1.1)-(1.2).

Theorem 4.1Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M395">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M396">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M397">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M398">View MathML</a>, there is a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M25">View MathML</a>of problem (1.1)-(1.2) and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M400">View MathML</a>

(4.1)

Proof Problem (1.1)-(1.2) can be expressed as the following abstract parabolic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M401">View MathML</a>

(4.2)

If we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M402">View MathML</a> then by Proposition 3.2, operator O is positive and generates a bounded holomorphic semigroup in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M49">View MathML</a>. Moreover, by using [[17], Theorem 3.1] we see that the operator O is R-positive in E. Since E is a UMD space, in a similar way as in [[13], Theorem 4.2] we see that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M404">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M405">View MathML</a> there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M406">View MathML</a> of problem (4.2) so that the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M407">View MathML</a>

(4.3)

 □

From the estimates (3.3) and (4.3) we obtain the assertion.

Remark 4.1 There are a lot of positive operators in concrete Banach spaces. Therefore, putting in (1.7)-(1.8) and (1.1)-(1.3) concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of A, by virtue of Theorem 3.3 and Theorem 4.1 we can obtain the maximal regularity properties of different class of stationary and instationary Stokes problems, respectively, which occur in numerous physics and engineering problems.

Let us now show some application of Theorem 3.3 and Theorem 4.1.

5 Application

Consider the stationary Stokes problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M408">View MathML</a>

(5.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M409">View MathML</a>

(5.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M410">View MathML</a>

(5.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M411">View MathML</a> represents a given and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M412">View MathML</a>

are unknown functions;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M413">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M10">View MathML</a> are complex numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M416">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M417">View MathML</a> are complex-valued functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M418">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M419">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M420">View MathML</a>. Now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M421">View MathML</a> will denote the space of all p-summable scalar-valued functions with mixed norm i.e., the space of all measurable functions f defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M422">View MathML</a>, for which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M423">View MathML</a>

Analogously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M424">View MathML</a> denotes the Sobolev space with corresponding mixed norm.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M425">View MathML</a> denotes the class of vector function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M426">View MathML</a>

with norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M427">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M428">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M429">View MathML</a> is the anisotropic Sobolev space with mixed norm.

From Theorem 3.3 we obtain the following.

Theorem 5.1Let the following conditions be satisfied:

(1) Ω is a domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M430">View MathML</a>with sufficiently smooth boundaryΩ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M431">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M432">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M433">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M434">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M435">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M436">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M437">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M438">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M439">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M440">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M441">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M442">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M23">View MathML</a>;

(3) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M444">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M445">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M446">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M447">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M448">View MathML</a>let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M449">View MathML</a>

(4) for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M450">View MathML</a>the local BVPs in local coordinates corresponding to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M451">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M452">View MathML</a>

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M453">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M454">View MathML</a>and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M455">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M456">View MathML</a>;

(5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M457">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M458">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M460">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M462">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M463">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M464">View MathML</a>.

Then, problem (5.1)-(5.3) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M465">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M466">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M467">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M468">View MathML</a>and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M469">View MathML</a>

(5.4)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M470">View MathML</a>. By virtue of [[18], Theorem 4.5.2] <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M471">View MathML</a> is an UMD space. Consider the operator A which is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M472">View MathML</a>

Problem (5.1)-(5.3) can be rewritten in the form of (1.7)-(1.8) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M473">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M474">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M475">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M476">View MathML</a> are functions with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M477">View MathML</a>. In view of [[14], Theorem 8.2] the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M478">View MathML</a>

(5.5)

has a unique solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M479">View MathML</a> and arg <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M480">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M481">View MathML</a>, and the operator A is R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M482">View MathML</a>. Hence, all conditions of Theorem 3.2 are satisfied, i.e., we obtain the assertion.

Consider now the instationary Stokes problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M483">View MathML</a>

(5.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M484">View MathML</a>

(5.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M485">View MathML</a>

(5.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M486">View MathML</a>

(5.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M487">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M488">View MathML</a> are data and solution vector-functions, respectively. □

From Theorem 4.1 and Theorem 5.1 we obtain the following.

Theorem 5.2Assume all conditions of Theorem 5.1 are satisfied<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M489">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M490">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M491">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M492">View MathML</a>, there is a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M493">View MathML</a>of problem (5.6)-(5.9) and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/86/mathml/M494">View MathML</a>

Competing interests

The author declares that they have no competing interests.

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