This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Nonlocal Navier-Stokes problem with a small parameter

Veli B Shakhmurov

Author Affiliations

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

Institute of Mathematics and Mechanics, Azerbaijan National Akademy of Science, Baku, Azerbaijan

Boundary Value Problems 2013, 2013:107  doi:10.1186/1687-2770-2013-107


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/107


Received:5 March 2013
Accepted:12 April 2013
Published:26 April 2013

© 2013 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 cited.

Abstract

Initial nonlocal boundary value problems for a Navier-Stokes equation with a small parameter is considered. The uniform maximal regularity properties of the corresponding stationary Stokes operator, well-posedness of a nonstationary Stokes problem and the existence, uniqueness and uniformly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M1">View MathML</a> estimates for the solution of the Navier-Stokes problem are established.

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

Keywords:
Stokes operators; Navier-Stokes equations; differential equations with small parameters; semigroups of operators; boundary value problems; differential-operator equations; maximal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M2">View MathML</a> regularity

1 Introduction

Consider the following Navier-Stokes problem with a parameter:

(1.1)

(1.2)

(1.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M8">View MathML</a> are complex numbers, ε is a small positive parameter,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M9">View MathML</a>

represent the unknown velocity and pressure, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M10">View MathML</a>

represents a given external force and a denotes the initial velocity. This problem is characterized by nonlocality of boundary conditions and by presence of a small term ε which corresponds to the inverse of Reynolds number Re very large for the Navier-Stokes equations. From both the theoretical and computational points of view, singularly perturbed problems and asymptotic behavior of the Navier-Stokes equations with small viscosity when the boundary is either characteristic or non-characteristic have been well studied; see, e.g., [1-6]. In the present work, we established a uniform time of existence and estimates for solutions of problem (1.1)-(1.3). It is clear that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M11">View MathML</a>, choosing the boundary conditions locally and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M12">View MathML</a>, problem (1.1)-(1.3) is reduced to the classical Navier-Stokes problem

(1.4)

Note that the existence of weak or strong solutions and regularity properties of classical Navier-Stokes problems were extensively studied, e.g., in [1-3,5,7-33]. There is extensive literature on the solvability of the initial value problem for the Navier-Stokes equation ( see, e.g., [25] for further papers cited there ). Hopf [20] proved the existence of a global weak solution of (1.4) using the Faedo-Galerkin approximation and an energy inequality. Another approach to problem (1.4) is to use semigroup theory. Kato and Fujita [18,22,34] and Sobolevskii [27] transformed equation (1.4) into an evolution equation in the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M14">View MathML</a>. They proved the existence of a unique global strong solution for any square-summable initial velocity when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M15">View MathML</a>. On the other hand, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M16">View MathML</a> they proved the existence of a unique local strong solution if the initial velocity has some regularity. Other contributions in this field have also assumed some regularity of the initial velocity corresponding to the Stokes problem; see, for example, Solonnikov [26] and Heywood [21]. Afterward, Giga and Sohr [13] improved this result in two directions. First, they generalized the result of Solonnikov for spaces with different exponents in space and time, and the estimate obtained was global in time. Here, first at all, we consider the nonlocal (boundary value problem) BVP for the following differential operator equation (DOE) with small parameters:

(1.5)

where A is a linear operator in a Banach space E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M8">View MathML</a> are complex numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M20">View MathML</a> are positive and λ is a complex parameter. We show that problem (1.5) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M21">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a> with sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24">View MathML</a>, and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M25">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M26">View MathML</a> independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M28">View MathML</a>, λ and f.

Further, we consider the nonlocal BVP for the stationary Stokes system with small parameters

(1.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M30">View MathML</a>

Then we consider the initial nonlocal BVP for the following nonstationary Stokes equation with small parameters:

(1.7)

Problem (1.7) can be expressed as the abstract parabolic problem with a parameter

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M32">View MathML</a>

(1.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M33">View MathML</a> is a stationary parameter depending on the Stokes operator in a solenoidal space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M34">View MathML</a> defined by

We prove that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M33">View MathML</a> is positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M37">View MathML</a> uniformly with respect to parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M38">View MathML</a> and also is a generator of a holomorphic semigroup. Then, by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M1">View MathML</a>-maximal regularity theorems (see, e.g., [33,35]) for abstract parabolic equations (1.8), we obtain that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M41">View MathML</a>, there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M42">View MathML</a> of problem (1.8) and the following uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M43">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M44">View MathML</a> independent of f and ε. Afterwords, by using the above uniform coercive estimate, we derive local uniform existence and uniform a priori estimates of a solution of problem (1.1)-(1.3).

Modern analysis methods, particularly abstract harmonic analysis, the operator theory, the interpolation of Banach spaces, the theory of semigroups of linear operators, embedding and trace theorems in vector-valued Sobolev-Lions spaces are the main tools implemented to carry out the analysis.

2 Notations, definitions and background

Let E be a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M45">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/2013/1/107/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M46">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M47">View MathML</a>

The Banach space E is called a UMD-space if the Hilbert operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M48">View MathML</a>

is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M50">View MathML</a> (see, e.g., [36]). UMD spaces include, e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M52">View MathML</a> spaces and Lorentz spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M54">View MathML</a>.

Let ℂ be the set of complex numbers and

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/2013/1/107/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M56">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M57">View MathML</a> is dense on E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M58">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M60">View MathML</a>, where I is the identity operator in E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M61">View MathML</a> is the space of bounded linear operators in E. It is known [[30], §1.15.1] that there exist the fractional powers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M62">View MathML</a> of a positive operator A. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M63">View MathML</a> denote the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M64">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M65">View MathML</a>

Let ℕ denote the set of natural numbers. A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M66">View MathML</a> is called R-bounded (see, e.g., [36]) if there is a positive constant C such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M69">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M70">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M71">View MathML</a> is a sequence of independent symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M72">View MathML</a>-valued random variables on Ω. The smallest C for which the above estimate holds is called an R-bound of the collection G and denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M73">View MathML</a>.

A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M74">View MathML</a> is called uniform R-bounded if there is a constant C independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M75">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M78">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M79">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M80">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/2013/1/107/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M60">View MathML</a>, is R-bounded.

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M83">View MathML</a> is said to be ψ-positive in E uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M84">View MathML</a> with bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M56">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M86">View MathML</a> is independent of t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M86">View MathML</a> is dense in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M88">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M90">View MathML</a>, where M does not depend on t and λ.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M91">View MathML</a> and E be two Banach spaces, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M91">View MathML</a> be continuously and densely embedded into E. Let Ω be a measurable set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M93">View MathML</a> and m be a positive integer. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M94">View MathML</a> denote the space consisting of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M95">View MathML</a> that have the generalized derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M96">View MathML</a>, with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M97">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M100">View MathML</a>, the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M94">View MathML</a> will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M102">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/2013/1/107/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M103">View MathML</a>.

3 Boundary value problems for abstract elliptic equations

In this section, we consider problem (1.5). We derive the maximal regularity properties of this problem.

It should be noted that BVPs for DOEs were studied, e.g., in [35-38] and [6,26,27,39-43]. For references, see [35,43]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M105">View MathML</a>. First, we prove the following theorem.

Theorem 3.1Let the following conditions be satisfied:

(1) Eis a UMD space andAis anR-positive operator inEfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M60">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M110">View MathML</a>.

Then problem (1.5) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M21">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a>with sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24">View MathML</a>. Moreover, the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M115">View MathML</a>

(3.1)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M26">View MathML</a>independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M117">View MathML</a>, λandf.

Proof Let us consider the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M118">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M120">View MathML</a> are defined by equalities (3.1)-(3.2). For the investigation (3.4), we consider the following BVP for ordinary DOE:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M121">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M122">View MathML</a> are boundary conditions of type (1.5) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M123">View MathML</a>. By virtue of [[40], Theorem 3.2], we obtain that problem (3.3) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M124">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a>, with sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24">View MathML</a>, and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M128">View MathML</a>

(3.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M129">View MathML</a>, problem (3.2) can be expressed as the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M130">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M131">View MathML</a> is the differential operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M132">View MathML</a> generated by problem (3.3), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M133">View MathML</a>

By virtue of [[35], Theorem 4.5.2], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M134">View MathML</a> provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M136">View MathML</a>. Hence, by virtue of [[40], Theorem 3.2] and [[41], Theorem 3.1], the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M131">View MathML</a> is uniformly R-positive in F. Then, by applying again [[40], Theorem 3.2], we get that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M140">View MathML</a>, problem (3.5) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M141">View MathML</a>, and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M142">View MathML</a>

(3.6)

The estimate (3.4) implies the uniform estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M143">View MathML</a>

(3.7)

By using (3.4) and (3.7), we obtain that problem (3.7) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M144">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a> with sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M140">View MathML</a>, and the coercive uniform estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M148">View MathML</a>

Further, by continuing this process n-times, we obtain the assertion.

From Theorem 3.1 we obtain the following. □

Corollary 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M150">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M107">View MathML</a>and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a>with sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24">View MathML</a>, there is a unique solutionuof problem (1.5) and the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M155">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M156">View MathML</a>independent off, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M20">View MathML</a>andλ.

Proof Let us put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M158">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M159">View MathML</a> in Theorem 3.1. It is known that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M159">View MathML</a> is R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M93">View MathML</a> (see, e.g., [36]). So, the estimate (3.1) implies Corollary 3.1.

Consider the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M162">View MathML</a> generated by problem (1.5), i.e.,

From Theorem 3.1 we obtain the following. □

Result 3.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a>, there is a resolvent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M165">View MathML</a> of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M162">View MathML</a> satisfying the following uniform estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M167">View MathML</a>

It is clear that the solution u of problem (1.5) depends on parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M168">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M169">View MathML</a>. In view of Theorem 3.1, we established estimates for the solution of (1.5) uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M170">View MathML</a>.

4 Regularity properties of solutions for DOEs with parameters

In this section, we show the separability properties of problem (1.5) in Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M171">View MathML</a>. The main result is the following theorem.

Theorem 4.1Let the following conditions be satisfied:

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

(2) mis a positive integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M109">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M174">View MathML</a>

Then problem (3.1)-(3.2) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M175">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a>, with sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24">View MathML</a>, and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M179">View MathML</a>

(4.1)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M180">View MathML</a> independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M117">View MathML</a>, λ and f.

Consider first the following nonlocal BVP for an ordinary DOE with a small parameter:

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M186">View MathML</a> are complex numbers, t is positive, λ is a complex parameter and A is a linear operator in E. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M187">View MathML</a>.

To prove the main result, we need the following result in [[37], Theorem 2.1].

Theorem ALetEbe a UMD space, Abe aψ-positive operator inEwith boundM, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M60">View MathML</a>. Letmbe a positive integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M189">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M190">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M191">View MathML</a>, an operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M192">View MathML</a>generates a semigroup<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M193">View MathML</a>which is holomorphic for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M194">View MathML</a>. Moreover, there exists a positive constantC (depending only onM, ψ, m, αandp) such that for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M195">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M59">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M197">View MathML</a>

In a similar way as in [[43], §1.8.2, Theorem 2], we obtain the following lemma.

Lemma 4.1Letmandjbe integer numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M201">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M202">View MathML</a>, the transformation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M203">View MathML</a>is bounded linear from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M204">View MathML</a>onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M205">View MathML</a>and the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M206">View MathML</a>

Consider at first the homogeneous problem of (4.2)

(4.3)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M208">View MathML</a>

Lemma 4.2LetAbe anR-positive operator in a UMD spaceEand

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M209">View MathML</a>

Then problem (4.3) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M210">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M59">View MathML</a>, and the coercive uniform estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M214">View MathML</a>

(4.4)

Proof In a similar way as in [[40], Theorem 3.1], we obtain the representation of the solution of (4.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M215">View MathML</a>

(4.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M216">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M217">View MathML</a> are uniformly bounded operators. Then, in view of positivity of A, we obtain from (4.5)

(4.6)

(4.7)

By changing the variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M220">View MathML</a> and in view of Theorem A, we obtain

(4.8)

By using the estimate (4.8), by virtue of Theorem A, we get the uniform estimate

(4.9)

Then from (4.6)-(4.9) we obtain (4.4). □

Now we can represent a more general result for nonhomogeneous problem (4.2).

Theorem 4.2Assume that the following conditions are satisfied:

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

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M224">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M50">View MathML</a>.

Then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M228">View MathML</a>is an isomorphism from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M229">View MathML</a>onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M230">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a>with large enough<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24">View MathML</a>. Moreover, the uniform coercive estimate holds

(4.10)

Proof The uniqueness of a solution of problem (4.2) is obtained from Lemma 4.2. Let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M234">View MathML</a>

We will show that problem (4.2) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M235">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M238">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M239">View MathML</a> is the restriction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M240">View MathML</a> of the solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M241">View MathML</a>

(4.11)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M242">View MathML</a> is a solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M243">View MathML</a>

(4.12)

A solution of equation (4.11) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M244">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M245">View MathML</a>. It follows from the above expression that

(4.13)

It is sufficient to show that the operator-functions

are uniform Fourier multipliers in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M248">View MathML</a>. Actually, due to the positivity of A, we have

(4.14)

It is clear to observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M250">View MathML</a>

Due to R-positivity of the operator A, the sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M251">View MathML</a>

are R-bounded. Then, in view of the Kahane contraction principle, from the product properties of the collection of R-bounded operators (see, e.g., [36] Lemma 3.5, Proposition 3.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M252">View MathML</a>

(4.15)

By [[33], Theorem 3.4] it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M254">View MathML</a> are the uniform collection of multipliers in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M248">View MathML</a>. Then in view of (4.13) we obtain that problem (4.11) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M256">View MathML</a> and the uniform coercive estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M257">View MathML</a>

(4.16)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M239">View MathML</a> be the restriction of u on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M259">View MathML</a>. The estimate (4.16) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M260">View MathML</a>. By virtue of Lemma 4.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M261">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M262">View MathML</a>. Thus, by virtue of Lemma 4.2, problem (4.12) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M263">View MathML</a> that belongs to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M264">View MathML</a> and

(4.17)

Moreover, from (4.16) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M266">View MathML</a>

(4.18)

Therefore, by Lemma 4.1 and by estimate (4.17), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M267">View MathML</a>

(4.19)

So, in view of Lemma 4.1 and estimates (4.17)-(4.19), we get

(4.20)

Finally, from (4.18) and (4.20) we obtain (4.10). □

Now, we can prove the main result of this section.

Proof of Theorem 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M269">View MathML</a>. It is clear to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M270">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M272">View MathML</a>.

Let us consider the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M273">View MathML</a>

(4.21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M120">View MathML</a> are defined by equalities (1.5). Problem (4.21) can be expressed as the following BVP for an ordinary DOE:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M275">View MathML</a>

(4.22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M276">View MathML</a> are boundary conditions of type (3.2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M131">View MathML</a> is the operator acting in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M278">View MathML</a> and X defined by

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M278">View MathML</a> and X are UMD spaces, (see, e.g., [[35], Theorem 4.5.2]) by virtue of Theorem 4.2, we obtain that problem (4.22) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M281">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M282">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M23">View MathML</a> with sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M24">View MathML</a>. Moreover, the coercive uniform estimates holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M285">View MathML</a>

(4.23)

From (4.23) we obtain that problem (4.22) has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M286">View MathML</a>

Moreover, the uniform coercive estimates hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M287">View MathML</a>

(4.24)

By applying Theorem 4.2 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M288">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M289">View MathML</a>, we get the following uniform estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M290">View MathML</a>

(4.25)

From estimates (4.24)-(4.25) we conclude the corresponding claim for problem (4.21). Then, by continuing this process n-times, we obtain the assertion. □

5 Nonlocal initial-boundary value problems for the Stokes system with small parameters

In this section, we show the uniform maximal regularity properties of the nonlocal initial value problem for nonstationary Stokes equations (1.6).

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M291">View MathML</a> satisfying equation (1.6) a.e. on G is called the stronger solution of problem (1.6).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M292">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M293">View MathML</a> be the Sobolev space of order s such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M294">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M107">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M296">View MathML</a> denote the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M297">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M298">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M299">View MathML</a>

It is known that ( see, e.g., Fujiwara and Morimoto [17]) a vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M300">View MathML</a> has the Helmholtz decomposition, i.e., all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M300">View MathML</a> can be uniquely decomposed as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M302">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M303">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M304">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M305">View MathML</a> is a projection operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M306">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M307">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M308">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M309">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M310">View MathML</a>

with C independent of u, where B is an open ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M312">View MathML</a> denotes the norm of u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M306">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M314">View MathML</a>.

Then problem (1.6) can be reduced to the following BVP:

(5.1)

Consider the parameter-dependent Stokes operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M316">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M33">View MathML</a> generated by problem (5.1), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M318">View MathML</a>

From Corollary 3.1 we get that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M319">View MathML</a> is positive and also is a generator of a bounded holomorphic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M320">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M321">View MathML</a>.

In a similar way as in [18], we show the following.

Proposition 5.1The following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M322">View MathML</a>

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M168">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M324">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M321">View MathML</a>.

Proof From Result 3.1 we obtain that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M326">View MathML</a> is uniformly positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M327">View MathML</a>, i.e., for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M329">View MathML</a>, the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M330">View MathML</a>

where the constant M is independent of λ and ε. Then, by using the Danford integral and operator calculus as in [18], we obtain the assertion. □

Now consider problem (1.7). The main theorem in this section is the following.

Theorem 5.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M332">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M41">View MathML</a>. Then there is a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M42">View MathML</a>of problem (1.7) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M335">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M336">View MathML</a>. Moreover, the following uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M337">View MathML</a>

(5.2)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M338">View MathML</a>independent offandε.

Proof Problem (1.7) can be expressed as the following abstract parabolic problem with a small parameter:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M339">View MathML</a>

(5.3)

If we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M340">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M159">View MathML</a> in Theorem 3.1, then the Result 3.1 implies that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M342">View MathML</a> is uniformly positive and generates bounded holomorphic semigroup in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M306">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M20">View MathML</a>. Moreover, by using [[41], Theorem 3.1] we get that operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M319">View MathML</a> is R-positive in E. Since E is a UMD space, in a similar way as in [[33], Theorem 4.2], we obtain that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M347">View MathML</a>, there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M348">View MathML</a> of problem (5.3) so that the following uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M349">View MathML</a>

(5.4)

From (5.4) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M350">View MathML</a>, we get the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M351">View MathML</a>

uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M168">View MathML</a>. □

6 Existence and uniqueness for the Navier-Stokes equation with parameters

In this section, we study the Navier-Stokes problem (1.1)-(1.3) in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353">View MathML</a>. Problem (1.1)-(1.3) can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M354">View MathML</a>

(6.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M355">View MathML</a>

We consider equation (6.1) in an integral form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M356">View MathML</a>

(6.2)

To prove the main result, we need the following result which are obtained in a similar way as in [[11], Theorem 2].

Lemma 6.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M357">View MathML</a>, the domain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M358">View MathML</a>is the complex interpolation space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M359">View MathML</a>.

Lemma 6.2For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M110">View MathML</a>, the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M361">View MathML</a>extends uniquely to a uniformly bounded linear operator from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M306">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M319">View MathML</a> is a positive operator, it has fractional powers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M365">View MathML</a>. From Lemma 6.1, it follows that the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M366">View MathML</a> is continuously embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M367">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M368">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M369">View MathML</a> is the vector-valued Bessel space. Then, by using the duality argument and due to uniform positivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M370">View MathML</a>, we obtain the following uniformly in ε estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M371">View MathML</a>

(6.3)

By reasoning as in [12], we obtain the following. □

Lemma 6.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M372">View MathML</a>. Then the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M373">View MathML</a>

uniformly inεwith some constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M374">View MathML</a>provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M375">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M376">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M377">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M378">View MathML</a>

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M379">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M358">View MathML</a> is continuously embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M381">View MathML</a>, and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M382">View MathML</a> is the same as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M383">View MathML</a>, by the Sobolev embedding theorem, we obtain that the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M384">View MathML</a>

is bounded, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M385">View MathML</a>

By the duality argument then, we get that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M386">View MathML</a> is bounded from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M387">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M389">View MathML</a>

Consider first the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M390">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M391">View MathML</a> is bilinear in u, υ, it suffices to prove the estimate on a dense subspace. Therefore, assume that u and υ are smooth. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M392">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M393">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M394">View MathML</a>, using the uniform boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M395">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M387">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353">View MathML</a> and Lemma 6.2 for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M398">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M399">View MathML</a>

By assumption we can take r and η such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M400">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M401">View MathML</a> is continuously embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M367">View MathML</a>, by the Sobolev embedding, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M403">View MathML</a>

i.e., we have the required result for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M390">View MathML</a>. In particular, we get the following uniform estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M405">View MathML</a>

Similarly, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M406">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M407">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M408">View MathML</a>. The above two estimates show that the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M409">View MathML</a> is a uniform bounded operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M410">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M411">View MathML</a> and from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M412">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M353">View MathML</a>. By using Lemma 6.1 and the interpolation of Banach spaces [[30], §1.3.2] for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M414">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M415">View MathML</a>

By using Lemma 6.3 and the iteration argument, by reasoning as in Fujita and Kato [18], we obtain the following. □

Theorem 6.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M332">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M418">View MathML</a>be a real number and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M419">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M420">View MathML</a>

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M421">View MathML</a>, and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M422">View MathML</a>is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M423">View MathML</a>and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M424">View MathML</a>

Then there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M425">View MathML</a>independent ofεand a local solution of (6.2) such that

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M426">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M427">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M428">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M429">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M430">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M431">View MathML</a>for allαwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M432">View MathML</a>uniformly with respect to the parameterε.

Moreover, the solution of (5.2) is unique if

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M433">View MathML</a>;

(5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M434">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M431">View MathML</a>for someβwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M436">View MathML</a>uniformly inε.

Proof We introduce the following iteration scheme:

(6.4)

By estimating the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M438">View MathML</a> in (6.4) and by using Proposition 5.1 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M439">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M440">View MathML</a>

uniformly in ε with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M441">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M442">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M443">View MathML</a> is the beta function. Here we suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M444">View MathML</a>. By induction assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M445">View MathML</a> satisfies the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M446">View MathML</a>

(6.5)

We will estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M447">View MathML</a> by using (6.2). To estimate the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M448">View MathML</a>, we suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M449">View MathML</a>

so that the numbers θ, σ, δ satisfy the assumptions of Lemma 6.3. Using Lemma 6.3 and (6.5), we get the following uniform estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M450">View MathML</a>

Therefore, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M451">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/107/mathml/M452">View MathML</a>

Since we get the uniform estimates with respect to the parameter ε, the remaining part of the proof is the same as in [[12], Theorem 2.3], so this part is omitted. □

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Dedicated to International Conference on the Theory, Methods and Applications of Nonlinear Equations in Kingsville, TX-USA Texas A&M University-Kingsville-2012.

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