Open Access

Linear and nonlinear convolution elliptic equations

Boundary Value Problems20132013:211

DOI: 10.1186/1687-2770-2013-211

Received: 16 May 2013

Accepted: 31 July 2013

Published: 19 September 2013

Abstract

In this paper, the separability properties of elliptic convolution operator equations are investigated. It is obtained that the corresponding convolution-elliptic operator is positive and also is a generator of an analytic semigroup. By using these results, the existence and uniqueness of maximal regular solution of the nonlinear convolution equation is obtained in L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq1_HTML.gif spaces. In application, maximal regularity properties of anisotropic elliptic convolution equations are studied.

MSC:34G10, 45J05, 45K05.

Keywords

positive operators Banach-valued spaces operator-valued multipliers boundary value problems convolution equations nonlinear integro-differential equations

1 Introduction

In recent years, maximal regularity properties for differential operator equations, especially parabolic and elliptic-type, have been studied extensively, e.g., in [113] and the references therein (for comprehensive references, see [13]). Moreover, in [14, 15], on embedding theorems and maximal regular differential operator equations in Banach-valued function spaces have been studied. Also, in [16, 17], on theorems on the multiplicators of Fourier integrals obtained, which were used in studying isotropic as well as anisotropic spaces of differentiable functions of many variables. In addition, multiplicators of Fourier integrals for the spaces of Banach valued functions were studied. On the basis of these results, embedding theorems are proved.

Moreover, convolution-differential equations (CDEs) have been treated, e.g., in [1, 1822] and [23]. Convolution operators in vector valued spaces are studied, e.g., in [2426] and [27]. However, the convolution-differential operator equations (CDOEs) are a relatively less investigated subject (see [13]). The main aim of the present paper is to establish the separability properties of the linear CDOE
| α | l a α D α u + ( A + λ ) u = f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ1_HTML.gif
(1.1)
and the existence and uniqueness of the following nonlinear CDOE
| α | l a α D α u + A u = F ( x , D σ u ) + f ( x ) , | σ | l 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equa_HTML.gif
in E-valued L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq1_HTML.gif spaces, where A = A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq2_HTML.gif is a possible unbounded operator in a Banach space E, and a α = a α ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq3_HTML.gif are complex-valued functions, and λ is a complex parameter. We prove that the problem (1.1) has a unique solution u, and the following coercive uniform estimate holds
| α | l | λ | 1 | α | l a α D α u L p ( R n ; E ) + A u L p ( R n ; E ) + | λ | u L p ( R n ; E ) C f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equb_HTML.gif

for all f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq4_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq5_HTML.gif and λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq6_HTML.gif. The methods are based on operator-valued multiplier theorems, theory of elliptic operators, vector-valued convolution integrals, operator theory and etc. Maximal regularity properties for parabolic CDEs with bounded operator coefficients were investigated in [1].

2 Notations and background

Let L p ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq7_HTML.gif denote the space of all strongly measurable E-valued functions that are defined on the measurable subset Ω R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq8_HTML.gif with the norm
f L p ( Ω ; E ) = ( f ( x ) E p d x ) 1 p , 1 p < , f L ( Ω ; E ) = ess sup x Ω [ f ( x ) E ] , x = ( x 1 , x 2 , , x n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equc_HTML.gif
Let C be the set of complex numbers, and let
S φ = { λ ; | λ C , | arg λ | φ } { 0 } , 0 φ < π . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equd_HTML.gif
A linear operator A = A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq9_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq10_HTML.gif is said to be uniformly positive in a Banach space E if D ( A ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq11_HTML.gif is dense in E, does not depend on x, and there is a positive constant M so that
( A ( x ) + λ I ) 1 B ( E ) M ( 1 + | λ | ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Eque_HTML.gif
for every x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq10_HTML.gif and λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq6_HTML.gif, φ [ 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq12_HTML.gif, where I is an identity operator in E, and B ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq13_HTML.gif is the space of all bounded linear operators in E, equipped with the usual uniform operator topology. Sometimes, instead of A + λ I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq14_HTML.gif, we write A + λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq15_HTML.gif and denote it by A λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq16_HTML.gif. It is known (see [28], §1.14.1) that there exist fractional powers A θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq17_HTML.gifof the positive operator A. Let E ( A θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq18_HTML.gif denote the space D ( A θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq19_HTML.gif with the graphical norm
u E ( A θ ) = ( u p + A θ u p ) 1 p , 1 p < , < θ < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equf_HTML.gif

Let S ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq20_HTML.gif denote Schwartz class, i.e., the space of E-valued rapidly decreasing smooth functions on R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq21_HTML.gif, equipped with its usual topology generated by semi-norms. S ( R n ; C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq22_HTML.gif denoted by just S. Let S ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq23_HTML.gif denote the space of all continuous linear operators L : S E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq24_HTML.gif, equipped with the bounded convergence topology. Recall S ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq20_HTML.gif is norm dense in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq25_HTML.gif when 1 p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq26_HTML.gif.

Let α = ( α 1 , α 2 , , α n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq27_HTML.gif, where α i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq28_HTML.gif are integers. An E-valued generalized function D α f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq29_HTML.gif is called a generalized derivative in the sense of Schwartz distributions of the function f S ( R n , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq30_HTML.gif if the equality
( D α f ) ( φ ) = ( 1 ) | α | f ( D α φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equg_HTML.gif

holds for all φ S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq31_HTML.gif.

Let F denote the Fourier transform. Through this section, the Fourier transformation of a function f will be denoted by f ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq32_HTML.gif. It is known that
F ( D x α f ) = ( i ξ 1 ) α 1 ( i ξ n ) α n f ˆ , D ξ α ( F ( f ) ) = F [ ( i x n ) α 1 ( i x n ) α n f ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equh_HTML.gif

for all f S ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq33_HTML.gif.

Let Ω be a domain in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq21_HTML.gif. C ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq34_HTML.gif and C ( m ) ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq35_HTML.gif will denote the spaces of E-valued bounded uniformly strongly continuous and m-times continuously differentiable functions on Ω, respectively. For E = C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq36_HTML.gif the space C ( m ) ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq35_HTML.gif will be denoted by C ( m ) ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq37_HTML.gif. Suppose E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq38_HTML.gif and E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq39_HTML.gif are two Banach spaces. A function Ψ L ( R n ; B ( E 1 , E 2 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq40_HTML.gif is called a multiplier from L p ( R n ; E 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq41_HTML.gif to L p ( R n ; E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq42_HTML.gif if the map u T u = F 1 Ψ ( ξ ) F u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq43_HTML.gif, u S ( R n ; E 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq44_HTML.gif is well defined and extends to a bounded linear operator
T : L p ( R n ; E 1 ) L p ( R n ; E 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equi_HTML.gif
Let Q denotes a set of some parameters. Let Φ h = { Ψ h M p p ( E 1 , E 2 ) , h Q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq45_HTML.gif be a collection of multipliers in M p p ( E 1 , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq46_HTML.gif. We say that W h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq47_HTML.gif is a collection of uniformly bounded multipliers (UBM) if there exists a positive constant M independent on h Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq48_HTML.gif such that
F 1 Ψ h F u L p ( R n ; E 2 ) M u L p ( R n ; E 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equj_HTML.gif

for all h Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq48_HTML.gif and u S ( R n ; E 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq44_HTML.gif.

A Banach space E is called an UMD-space [29, 30] if the Hilbert operator
( H f ) ( x ) = lim ε 0 { | x y | > ε } f ( y ) x y d y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equk_HTML.gif

is bounded in L p ( R , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq49_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq50_HTML.gif [29]. The UMD spaces include, e.g., L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq1_HTML.gif, l p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq51_HTML.gif spaces and Lorentz spaces L p q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq52_HTML.gif, p , q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq53_HTML.gif.

A set W B ( E 1 , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq54_HTML.gif is called R-bounded (see [5, 6, 12]) if there is a positive constant C such that
0 1 j = 1 m r j ( y ) T j u j E 2 d y C 0 1 j = 1 m r j ( y ) u j E 1 d y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equl_HTML.gif

for all T 1 , T 2 , , T m W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq55_HTML.gif and u 1 , u 2 , , u m E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq56_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq57_HTML.gif, where { r j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq58_HTML.gif is a sequence of independent symmetric { 1 , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq59_HTML.gif-valued random variables on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq60_HTML.gif. The smallest C, for which the above estimate holds, is called an R-bound of the collection W and denoted by R ( W ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq61_HTML.gif.

A set W h B ( E 1 , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq62_HTML.gif, dependent on parameters h Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq48_HTML.gif, is called uniformly R-bounded with respect to h if there is a positive constant C, independent of h Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq48_HTML.gif, such that for all T 1 ( h ) , T 2 ( h ) , , T m ( h ) W h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq63_HTML.gif and u 1 , u 2 , , u m E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq56_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq64_HTML.gif
0 1 j = 1 m r j ( y ) T j ( h ) u j E 2 d y C 0 1 j = 1 m r j ( y ) u j E 1 d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equm_HTML.gif

This implies that sup h Q R ( W h ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq65_HTML.gif.

Definition 2.1 A Banach space E is said to be a space, satisfying the multiplier condition, if for any Ψ C ( n ) ( R n { 0 } ; B ( E ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq66_HTML.gif the R-boundedness of the set
{ | ξ | | β | D ξ β Ψ ( ξ ) : ξ R n 0 , β = ( β 1 , β 2 , , β n ) , β k { 0 , 1 } } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equn_HTML.gif

implies that Ψ is a Fourier multiplier, i.e., Ψ M p p ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq67_HTML.gif for any p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq68_HTML.gif.

The uniform R-boundedness of the set
{ | ξ | | β | D β Ψ h ( ξ ) : ξ R n { 0 } , β { 0 , 1 } } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equo_HTML.gif
i.e.,
sup h Q R ( { | ξ | | β | D β Ψ h ( ξ ) : ξ R n 0 , β k { 0 , 1 } } ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equp_HTML.gif

implies that Ψ h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq69_HTML.gif is a uniformly bounded collection of Fourier multipliers (UBM) in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq25_HTML.gif.

Remark 2.2 Note that if E is UMD space, then by virtue of [5, 7, 12, 25], it satisfies the multiplier condition. The UMD spaces satisfy the uniform multiplier condition (see Proposition 2.4).

Definition 2.3 A positive operator A is said to be a uniformly R-positive in a Banach space E if there exists φ [ 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq12_HTML.gif such that the set
L A = { ξ ( A + ξ ) 1 : ξ S φ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equq_HTML.gif

is uniformly R-bounded.

Note that every norm bounded set in Hilbert spaces is R-bounded. Therefore, all sectorial operators in Hilbert spaces are R-positive.

Let h R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq70_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq57_HTML.gif and e k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq71_HTML.gif, k = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq72_HTML.gif be standard unit vectors of R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq21_HTML.gif,
Δ k ( h ) f ( x ) = f ( x + h e k ) f ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equr_HTML.gif
and let A = A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq2_HTML.gif, x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq73_HTML.gif be a closed linear operator in E with domain D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq74_HTML.gif independent of x. The Fourier transformation of A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq75_HTML.gif is a linear operator with the same domain D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq74_HTML.gif defined as
A ˆ u ( φ ) = A u ( φ ˆ ) for  u S ( R n ; E ( A ) ) , φ S ( R n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equs_HTML.gif
(For details see [[2], p.7].) Let A = A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq2_HTML.gif be a closed linear operator in E with domain D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq74_HTML.gif independent of x. Then, it is differentiable if there is the limit
( A x k ) u = lim h 0 Δ k ( h ) A ( x ) u h , k = 1 , 2 , , n , u D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equt_HTML.gif

in the sense of E-norm.

Let A = A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq2_HTML.gif, x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq73_HTML.gif be closed linear operator in E with domain D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq74_HTML.gif independent of x and u S ( R n , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq76_HTML.gif. We can define the convolution A u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq77_HTML.gif in the distribution sense by
A u ( x ) = R n A ( x y ) u ( y ) d y = R n A ( y ) u ( x y ) d y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equu_HTML.gif

(see [2]).

Let E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq78_HTML.gif and E be two Banach spaces, where E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq78_HTML.gif is continuously and densely embedded into E. Let l be a integer number. W p l ( R n ; E 0 , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq79_HTML.gif denote the space of all functions from S ( R n ; E 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq80_HTML.gif such that u L p ( R n ; E 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq81_HTML.gif and the generalized derivatives D k l u L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq82_HTML.gif with the following norm
u W p l ( R n ; E 0 , E ) = u L p ( R n ; E 0 ) + k = 1 n D k l u L p ( R n ; E ) < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equv_HTML.gif
It is clearly seen that
W p l ( R n ; E 0 , E ) = W p l ( R n ; E ) L p ( R n ; E 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equw_HTML.gif

A function u W p l ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq83_HTML.gif satisfying the equation (1.1) a.e. on R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq21_HTML.gif, is called a solution of equation (1.1).

The elliptic CDOE (1.1) is said to be separable in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq84_HTML.gif if for f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq85_HTML.gif the equation (1.1) has a unique solution u, and the following coercive estimate holds
| α | l a α D α u L p ( R n ; E ) + A u L p ( R n ; E ) C f L p ( R n ; E ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equx_HTML.gif

where the constant C do not depend on f.

In a similar way as Theorem A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq86_HTML.gif in [31], Theorem A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq86_HTML.gif and by reasoning as Theorem 3.7 in [7], we obtain the following.

Proposition 2.4 Let E be UMD space, Ψ h C n ( R n { 0 } ; B ( E ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq87_HTML.gif and suppose there is a positive constant K such that
sup h Q R ( { | ξ | | β | D β Ψ h ( ξ ) : ξ R n { 0 } , β k { 0 , 1 } } ) K . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equy_HTML.gif

Then Ψ h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq69_HTML.gif is UBM in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq25_HTML.gif for p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq5_HTML.gif.

Proof Really, some steps of proof trivially work for the parameter dependent case (see [7]). Other steps can be easily shown by setting
ϕ h = { | ξ | | β | D β Ψ h ( ξ ) : ξ R n { 0 } , β k { 0 , 1 } } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equz_HTML.gif
instead of
{ | ξ | | β | D β Ψ ( ξ ) : ξ R n { 0 } , β k { 0 , 1 } } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equaa_HTML.gif

and by using uniformly R-boundedness of set ϕ h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq88_HTML.gif. However, parameter depended analog of Proposition 3.4 in [7] is not straightforward. Let M h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq89_HTML.gif and M h , N L 1 loc ( R n , B ( E ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq90_HTML.gif be Fourier multipliers in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq84_HTML.gif. Let M h , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq91_HTML.gif converge to M h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq89_HTML.gif in L 1 loc ( R n , B ( E ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq92_HTML.gif, and let T h , N = F 1 M h , N F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq93_HTML.gif be uniformly bounded with respect to h and N. Then by reasoning as Proposition 3.4 in [7], we obtain that the operator function T h = F 1 M h F = lim N F 1 M h , N F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq94_HTML.gif is uniformly bounded with respect to h. Hence, by using steps above, in a similar way as Theorem 3.7 in [7], we obtain the assertion.

Let E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq38_HTML.gif and E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq39_HTML.gif be two Banach spaces. Suppose that T B ( E 1 , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq95_HTML.gif and 1 p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq26_HTML.gif. Then T ˜ B ( L p ( R n ; E 1 ) , L p ( R n ; E 2 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq96_HTML.gif will denote operator ( T ˜ f ) ( x ) = T ( f ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq97_HTML.gif for f L p ( R n ; E 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq98_HTML.gif and x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq73_HTML.gif. □

In a similar way as Proposition 2.11 in [12], we have

Proposition 2.5 Let 1 p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq26_HTML.gif. If W B ( E 1 , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq54_HTML.gif is R-bounded, then the collection W ˜ = { T ˜ : T W } B ( L p ( R n ; E 1 ) , L p ( R n ; E 2 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq99_HTML.gif is also R-bounded.

From [11], we obtain the following.

Theorem 2.6 Let the following conditions be satisfied
  1. 1.

    E is a Banach space satisfying the uniform multiplier condition, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq50_HTML.gif and 0 < h h 0 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq100_HTML.gif are certain parameters;

     
  2. 2.

    l is a positive integer, and α = ( α 1 , α 2 , , α n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq101_HTML.gif are n-tuples of nonnegative integer numbers such that ϰ = | α | l < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq102_HTML.gif, 0 μ < 1 ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq103_HTML.gif;

     
  3. 3.

    A is an R-positive operator in E with 0 φ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq104_HTML.gif.

     
Then the embedding D α W p l ( R n ; E ( A ) , E ) L p ( R n ; E ( A 1 ϰ μ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq105_HTML.gif is continuous, and there exists a positive constant C μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq106_HTML.gif such that
D α u L p ( R n ; E ( A 1 ϰ μ ) ) C μ [ h μ u W p l ( R n ; E ( A ) , E ) + h ( 1 μ ) u L p ( R n ; E ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equab_HTML.gif
Theorem 2.7 Let the following conditions be satisfied
  1. 1.

    E is a Banach space satisfying the uniform multiplier condition, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq50_HTML.gif and 0 < h h 0 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq100_HTML.gif are certain parameters;

     
  2. 2.

    l is a positive integer, and α = ( α 1 , α 2 , , α n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq101_HTML.gif are n-tuples of nonnegative integer numbers such that ϰ = p | α | + n p l < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq107_HTML.gif, 0 μ < 1 ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq103_HTML.gif;

     
  3. 3.

    A is an R-positive operator in E with 0 φ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq104_HTML.gif.

     
Then the embedding D α W p l ( R n ; E ( A ) , E ) C ( R n ; E ( A 1 ϰ μ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq108_HTML.gif is continuous, and there exists a positive constant C μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq109_HTML.gif such that
D α u C ( R n ; E ( A 1 ϰ μ ) ) C μ [ h μ u W p l ( R n ; E ( A ) , E ) + h ( 1 μ ) u L p ( R n ; E ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equac_HTML.gif

for all u W p l ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq110_HTML.gif.

3 Elliptic CDOE

Condition 3.1 Assume that a α L ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq111_HTML.gif and the following hold
L ( ξ ) = | α | l a α ( ξ ) ( i ξ ) α S φ 1 , | L ( ξ ) | C k = 1 n | a k | | ξ k | l , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equad_HTML.gif

where φ 1 [ 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq112_HTML.gif, ξ = ( ξ 1 , ξ 2 , , ξ n ) R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq113_HTML.gif.

In the following, we denote the operator functions by σ i ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq114_HTML.gif for i = 0 , 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq115_HTML.gif.

Lemma 3.2 Assume Condition 3.1 holds, and A ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq116_HTML.gif is a uniformly φ-positive operator in E with 0 φ < π φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq117_HTML.gif. Then, the following operator functions
σ 0 ( ξ , λ ) = λ D ( ξ , λ ) , σ 1 ( ξ , λ ) = A ( ξ ) D ( ξ , λ ) , σ 2 ( ξ , λ ) = | α | l | λ | 1 | α | l a α ( ξ ) ( i ξ ) α D ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equae_HTML.gif

are uniformly bounded, where D ( ξ , λ ) = [ A ( ξ ) + L ( ξ ) + λ ] 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq118_HTML.gif.

Proof By virtue of Lemma 2.3 in [4] for L ( ξ ) S φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq119_HTML.gif, λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq6_HTML.gif and φ 1 + φ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq120_HTML.gif there is a positive constant C such that
| λ + L ( ξ ) | C ( | λ | + | L ( ξ ) | ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ2_HTML.gif
(3.1)
Since L ( ξ ) S φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq121_HTML.gif, in view of (3.1) and resolvent properties of positive operators, we get that A ( ξ ) + L ( ξ ) + λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq122_HTML.gif is invertible and
σ 0 ( ξ , λ ) B ( E ) M | λ | [ 1 + | λ | + | L ( ξ ) | ] 1 M 0 , σ 1 ( ξ , λ ) B ( E ) = I ( λ + L ( ξ ) ) D ( ξ , λ ) B ( E ) 1 + M | λ + L ( ξ ) | ( 1 + | λ + L ( ξ ) | ) 1 M 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equaf_HTML.gif
Next, let us consider σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq123_HTML.gif. It is clearly seen that
σ 2 ( ξ , λ ) B ( E ) C | α | l | λ | [ | ξ | | λ | 1 l ] | α | D ( ξ , λ ) B ( E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ3_HTML.gif
(3.2)
Since A is uniformly φ-positive and L ( ξ ) S φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq121_HTML.gif, then setting y k = ( | λ | 1 l | ξ k | ) α k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq124_HTML.gif in the following well-known inequality
y 1 α 1 y 2 α 2 y n α n C ( 1 + k = 1 n y k l ) , y k 0 , | α | l , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ4_HTML.gif
(3.3)
we obtain
σ 2 ( ξ , λ ) B ( E ) C | α | l | λ | [ 1 + k = 1 n | ξ k | l | λ | 1 ] [ 1 + | λ + L ( ξ ) | ] 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equag_HTML.gif
Taking into account the Condition 3.1 and (3.1)-(3.3), we get
σ 2 ( ξ , λ ) B ( E ) C ( | λ | + k = 1 n | ξ k | l ) [ 1 + | λ | + | L ( ξ ) | ] 1 C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equah_HTML.gif

 □

Lemma 3.3 Assume Condition 3.1 holds, and a α C ( n ) ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq125_HTML.gif. Let A ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq116_HTML.gif be a uniformly φ-positive operator in a Banach space E with 0 φ < π φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq126_HTML.gif, [ D β A ( ξ ) ] A 1 ( ξ ) C ( R n ; B ( E ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq127_HTML.gif and let
| ξ | β | D β a α ( ξ ) | C 1 , β k { 0 , 1 } , ξ R n { 0 } , 0 | β | n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ5_HTML.gif
(3.4)
| ξ | β [ D β A ( ξ ) ] A 1 ( ξ ) B ( E ) C 2 , β k { 0 , 1 } , ξ R n { 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ6_HTML.gif
(3.5)

Then, operator functions | ξ | β D β σ i ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq128_HTML.gif are uniformly bounded.

Proof Let us first prove that ξ k σ 1 ξ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq129_HTML.gif is uniformly bounded. Really,
ξ k σ 1 ξ k B ( E ) I 1 B ( E ) + I 2 B ( E ) + I 3 B ( E ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equai_HTML.gif
where
I 1 = [ ξ k A ( ξ ) ξ k ] D ( ξ , λ ) , I 2 = A ( ξ ) [ ξ k A ( ξ ) ξ k ] [ D ( ξ , λ ) ] 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equaj_HTML.gif
and
I 3 = A ( ξ ) [ ξ k L ( ξ ) ξ k ] D 2 ( ξ , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equak_HTML.gif
By using (3.1) and (3.5), we get
I 1 B ( E ) [ ξ k A ( ξ ) ξ k ] A 1 ( ξ ) B ( E ) σ 1 B ( E ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equal_HTML.gif
Due to positivity of A, by using (3.1) and (3.5), we obtain
I 2 B ( E ) [ ξ k A ( ξ ) ξ k ] A 1 ( ξ ) B ( E ) σ 1 B ( E ) 2 C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equam_HTML.gif
Since, A ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq116_HTML.gif is uniformly φ-positive, by using (3.1), (3.3) and (3.4) for λ S ( φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq130_HTML.gif and φ 1 + φ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq131_HTML.gif, we get
I 3 B ( E ) | ξ k L ξ k | D ( ξ , λ ) B ( E ) σ 1 ( ξ , λ ) B ( E ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equan_HTML.gif
In a similar way, the uniform boundedness of σ 0 ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq132_HTML.gif is proved. Next, we shall prove ξ k σ 2 ξ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq133_HTML.gif is uniformly bounded. Similarly,
ξ k σ 2 ξ k B ( E ) J 1 B ( E ) + J 2 B ( E ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equao_HTML.gif
where
J 1 = | α | l | λ | 1 | α | l ( ξ k a α ξ k ) [ ( i ξ ) α + a α ( ξ ) i α k ( i ξ ) α ] D ( ξ , λ ) , J 2 = | α | l | λ | 1 | α | l a α ( ξ ) ( i ξ ) α [ ξ k a α ξ k + a α ( ξ ) ( i ξ ) α + ξ k A ( ξ ) ξ k ] [ D ( ξ , λ ) ] 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equap_HTML.gif
Let us first show that J 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq134_HTML.gif is uniformly bounded. It is clear that
J 1 B ( E ) | α | l | ξ k a α ξ k | ξ α | λ | 1 | α | l D ( ξ , λ ) B ( E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equaq_HTML.gif
Due to positivity of A, by virtue of (3.1) and (3.3)-(3.5), we obtain J 1 B ( E ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq135_HTML.gif. In a similar way, we have J 2 B ( E ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq136_HTML.gif. Hence, operator functions ξ k σ i ξ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq137_HTML.gif, i = 0 , 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq115_HTML.gif are uniformly bounded. From the representations of σ i ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq138_HTML.gif, it easy to see that operator functions | ξ | β D β σ i ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq139_HTML.gif contain similar terms as I k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq140_HTML.gif, namely, the functions | ξ | β D β σ i ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq141_HTML.gif will be represented as combinations of principal terms
ξ σ [ D ξ γ A ( ξ ) + D ξ γ a α ( ξ ) ] [ D ( ξ , λ ) ] | β | , | α | l | λ | 1 | α | l ξ σ D ξ γ [ A ( ξ ) + a α ( ξ ) ] [ D ( ξ , λ ) ] | β | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ7_HTML.gif
(3.6)
where | σ | + | γ | | β | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq142_HTML.gif. Therefore, by using similar arguments as above and in view of (3.6), one can easily check that
| ξ | β D β σ i ( ξ , λ ) C , i = 0 , 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equar_HTML.gif

 □

Lemma 3.4 Let all conditions of the Lemma 3.2 hold. Suppose that E is a Banach space satisfying the uniform multiplier condition, and A ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq143_HTML.gif is a uniformly R positive operator in E. Then, the following sets
S 0 ( ξ , λ ) = { | ξ | β D ξ β σ 0 ( ξ , λ ) ; ξ R n { 0 } } , S 1 ( ξ , λ ) = { | ξ | β D ξ β σ 1 ( ξ , λ ) ; ξ R n { 0 } } , S 2 ( ξ , λ ) = { | ξ | β D ξ β σ 2 ( ξ , λ ) ; ξ R n { 0 } } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equas_HTML.gif

are uniformly R-bounded for β k { 0 , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq144_HTML.gif and 0 | β | n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq145_HTML.gif.

Proof Due to R-positivity of A we obtain that the set
B 1 ( ξ , λ ) = { [ λ + L ( ξ ) ] D ( ξ , λ ) ; ξ R n { 0 } } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equat_HTML.gif
is R bounded. Since
I σ ( ξ , λ ) = A D ( ξ , λ ) , σ ( ξ , λ ) = [ λ + L ( ξ ) ] D ( ξ , λ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equau_HTML.gif
the set B 2 ( ξ , λ ) = { A D ( ξ , λ ) ; ξ R n { 0 } } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq146_HTML.gif is R -bounded. Moreover, in view of Condition 3.1 and (3.1), there is a positive constant M such that
| λ | | λ + L ( ξ ) | 1 M . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equav_HTML.gif
Then, by virtue of Kahane’s contraction principle, Lemma 3.5 in [5], we obtain that the set B 3 ( ξ , λ ) = { λ D ( ξ , λ ) ; ξ R n { 0 } } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq147_HTML.gif is uniformly R-bounded. Then by Lemma 3.2, we obtain the uniform R-boundedness of sets B k ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq148_HTML.gif, i.e,
sup λ R { B k ( ξ , λ ) } M k , k = 1 , 2 , 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ8_HTML.gif
(3.7)
Moreover, due to boundedness of a α ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq149_HTML.gif, in view of Condition 3.1 and by virtue of (3.1) and (3.3), we obtain
| α l | λ | 1 | α | l a α ( ξ ) ( i ξ ) α | C 1 ( 1 + | λ | + | L ( ξ ) | ) C ( 1 + | λ + L ( ξ ) | ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ9_HTML.gif
(3.8)
In view of representation (3.6) and estimate (3.8), we need to show uniform R-boundedness of the following sets
{ ξ σ [ D ξ γ A ( ξ ) + D ξ γ a α ( ξ ) ] [ D ( ξ , λ ) ] | β | ; ξ R n { 0 } } , { | α | l | λ | 1 | α | l ξ σ [ D ξ γ A ( ξ ) + D ξ γ a α ( ξ ) ] [ D ( ξ , λ ) ] | β | ; ξ R n { 0 } } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equaw_HTML.gif
for | σ | + | γ | | β | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq142_HTML.gif. By virtue of Kahane’s contraction principle, additional and product properties of R-bounded operators, see, e.g., Lemma 3.5, Proposition 3.4 in [5], and in view of (3.7), it is sufficient to prove uniform R-boundedness of the following set
B ( ξ , λ ) = { Q ( ξ , λ ) ; ξ R n { 0 } } , Q ( ξ , λ ) = | α | l | λ | 1 | α | l a α ( ξ ) ξ α D ( ξ , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equax_HTML.gif
Since
Q ( ξ , λ ) = | α | l | λ | 1 | α | l a α ( ξ ) ξ α [ λ + L ( ξ ) ] 1 σ ( ξ , λ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equay_HTML.gif
thanks to R-boundedness of B 2 ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq150_HTML.gif, we have
0 1 j = 1 m r j ( y ) σ ( η j , λ ) u j E d y C 0 1 j = 1 m r j ( y ) u j E d y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ10_HTML.gif
(3.9)
for all ξ 1 , ξ 2 , , ξ m R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq151_HTML.gif, η j = ( ξ j 1 , ξ j 2 , , ξ j n ) R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq152_HTML.gif, u 1 , u 2 , , u m E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq153_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq57_HTML.gif, where { r j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq58_HTML.gif is a sequence of independent symmetric { 1 , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq59_HTML.gif-valued random variables on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq154_HTML.gif. Thus, in view of Kahane’s contraction principle, additional and product properties of R-bounded operators and (3.9), we obtain
0 1 j = 1 m r j ( y ) Q ( η j , λ ) u j E d y C 0 1 j = 1 m σ ( η j , λ ) r j ( y ) u j E d y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ11_HTML.gif
(3.10)
C 0 1 j = 1 m r j ( y ) u j E d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ12_HTML.gif
(3.11)
The estimate (3.10) implies R-boundedness of the set B ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq155_HTML.gif. Moreover, from Lemma 3.2, we get
sup λ R { Q ( ξ , λ ) : ξ R n { 0 } } C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equaz_HTML.gif

i.e., we obtain the assertion. □

The following result is the corollary of Lemma 3.4 and Proposition 2.4.

Result 3.5 Suppose that all conditions of Lemma 3.3 are satisfied, E is UMD space, and A ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq156_HTML.gif is a uniformly R-positive operator in E. Then the sets S i ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq157_HTML.gif, i = 0 , 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq115_HTML.gif are uniformly R-bounded.

Now, we are ready to present our main results. We find sufficient conditions that guarantee separability of problem (1.1).

Condition 3.6 Suppose that the following are satisfied
  1. 1.

    For φ 1 [ 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq112_HTML.gif and ξ R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq158_HTML.gif, L ( ξ ) = | α | l a ˆ α ( ξ ) ( i ξ ) α S φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq159_HTML.gif, | L ( ξ ) | C k = 1 n | a ˆ k ξ k | l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq160_HTML.gif;

     
  2. 2.

    a ˆ α C ( n ) ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq161_HTML.gif and | ξ | β | D β a ˆ α ( ξ ) | C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq162_HTML.gif, β k { 0 , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq163_HTML.gif, 0 | β | n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq145_HTML.gif;

     
  3. 3.
    For 0 | β | n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq145_HTML.gif and ξ R n { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq164_HTML.gif,
    [ D β A ˆ ( ξ ) ] A ˆ 1 ( ξ ) C ( R n ; B ( E ) ) , | ξ | β [ D β A ˆ ( ξ ) ] A ˆ 1 ( ξ ) B ( E ) C 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equba_HTML.gif
     
Theorem 3.7 Suppose that Condition 3.6 holds, and E is a Banach space satisfying the uniform multiplier condition. Let A ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq165_HTML.gif be a uniformly R-positive in E with 0 φ < π φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq117_HTML.gif. Then, problem (1.1) has a unique solution u, and the following coercive uniform estimate holds
| α | l | λ | 1 | α | l a α D α u L p ( R n ; E ) + A u L p ( R n ; E ) + | λ | u L p ( R n ; E ) C f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ13_HTML.gif
(3.12)

for all f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq4_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq166_HTML.gif and λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq6_HTML.gif.

Proof By applying the Fourier transform to equation (1.1), we get
u ˆ ( ξ ) = D ( ξ , λ ) f ˆ ( ξ ) , D ( ξ , λ ) = [ A ˆ ( ξ ) + L ( ξ ) + λ ] 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbb_HTML.gif
Hence, the solution of equation (1.1) can be represented as u ( x ) = F 1 D ( ξ , λ ) f ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq167_HTML.gif. Then there are positive constants C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq168_HTML.gif and C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq169_HTML.gif, so that
C 1 | λ | u L p ( R n ; E ) F 1 [ σ 0 ( ξ , λ ) f ˆ ] L p ( R n ; E ) C 2 | λ | u L p ( R n ; E ) , C 1 A u L p ( R n ; E ) F 1 [ σ 1 ( ξ , λ ) f ˆ ] L p ( R n ; E ) C 2 A u L p ( R n ; E ) , C 1 | α | l | λ | 1 | α | l a α D α u L p ( R n ; E ) F 1 [ σ 2 ( ξ , λ ) f ˆ ] L p ( R n ; E ) C 2 | α | l | λ | 1 | α | l a α D α u L p ( R n ; E ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ14_HTML.gif
(3.13)
where σ i ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq138_HTML.gif are operator functions defined in Lemma 3.3. Therefore, it is sufficient to show that the operator-functions σ i ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq138_HTML.gif are UBM in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq25_HTML.gif. However, these follow from Lemma 3.4. Thus, from (3.13), we obtain
| λ | u L p ( R n ; E ) C 0 f L p ( R n ; E ) , A u L p ( R n ; E ) C 1 f L p ( R n ; E ) , | α | l | λ | 1 | α | l a α D α u L p ( R n ; E ) C 2 f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbc_HTML.gif

for all f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq4_HTML.gif. Hence, we get assertion.

Let O be an operator in X = L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq170_HTML.gif that is generated by the problem (1.1) for λ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq171_HTML.gif, i.e.,
D ( O ) W p l ( R n ; E ( A ) , E ) , O u = | α | l a α D α + A u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbd_HTML.gif

 □

Result 3.8 Theorem 2.6 implies that the operator O is separable in X, i.e., for all f X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq172_HTML.gif, all terms of equation (1.1) also are from X, and for solution u of equation (1.1), there are positive constants C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq168_HTML.gif and C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq169_HTML.gif so that
C 1 O u X | α | l a α D α u X + A u X C 2 O u X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Eqube_HTML.gif
Condition 3.9 Let D ( A ) = D ( A ˆ ) = D ( A ˆ ( ξ 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq173_HTML.gif for ξ 0 R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq174_HTML.gif. Moreover, there are positive constants C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq168_HTML.gif and C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq169_HTML.gif so that for u D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq175_HTML.gif, x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq73_HTML.gif
C 1 A ˆ ( ξ 0 ) u A ( x ) u C 2 A ˆ ( ξ 0 ) u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbf_HTML.gif

Remark 3.10 Condition 3.9 is checked for the regular elliptic operators with smooth coefficients on sufficiently smooth domains Ω R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq176_HTML.gif considered in the Banach space E = L p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq177_HTML.gif, p 1 ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq178_HTML.gif (see Theorem 5.1).

Theorem 3.11 Assume that all conditions of Theorem 3.7 and Condition 3.9 are satisfied. Let E be a Banach space satisfying the uniform multiplier condition. Then, problem (1.1) has a unique solution u W p l ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq179_HTML.gif, and the following coercive uniform estimate holds
| α | l | λ | 1 | α | l D α u L p ( R n ; E ) + A u L p ( R n ; E ) M f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbg_HTML.gif

for all f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq4_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq180_HTML.gif and λ S ( φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq181_HTML.gif.

Proof By applying the Fourier transform to equation (1.1), we obtain D ( ξ , λ ) u ˆ ( ξ ) = f ˆ ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq182_HTML.gif, where
D ( ξ , λ ) = [ A ˆ ( ξ ) + L ( ξ ) + λ ] 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbh_HTML.gif
So, we obtain that the solution of equation (1.1) can be represented as u ( x ) = F 1 D ( ξ , λ ) f ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq167_HTML.gif. Moreover, by Condition 3.9, we have
A F 1 D ( ξ , λ ) f ˆ L p ( R n ; E ) M A ˆ ( ξ 0 ) F 1 D ( ξ , λ ) f ˆ L p ( R n ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbi_HTML.gif

Hence, by using estimates (3.12), it is sufficient to show that the operator functions | α | l | λ | 1 | α | l ξ α D ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq183_HTML.gif and A ˆ ( ξ 0 ) D ( ξ , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq184_HTML.gif are UBM in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq185_HTML.gif. Really, in view of Condition 3.9, and uniformly R-positivity of A ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq165_HTML.gif, these are proved by reasoning as in Lemma 3.4. □

Condition 3.12 There are positive constants C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq168_HTML.gif and C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq169_HTML.gif such that
C 1 k = 1 n | a k ξ k | l | L ( ξ ) | C 2 k = 1 n | a k ξ k | l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbj_HTML.gif
for ξ R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq158_HTML.gif and
C 1 A ( x 0 ) u A ( x ) u C 2 A ( x 0 ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbk_HTML.gif

in cases, where D ( A ) = D ( A ˆ ) = D ( A ( x 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq186_HTML.gif, A ˆ ( ξ ) A 1 ( x 0 ) L ( R n ; B ( E ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq187_HTML.gif for ξ , x , x 0 R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq188_HTML.gif and u D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq189_HTML.gif.

Theorem 3.13 Let all conditions of Theorem 3.11 and Condition 3.12 hold. Then for u W p l ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq190_HTML.gif, there are positive constants M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq191_HTML.gif and M 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq192_HTML.gif, so that
M 1 u W p l ( R n ; E ( A ) , E ) | α | l a α D α u X + A u X M 2 u W p l ( R n ; E ( A ) , E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbl_HTML.gif
Proof The left part of the inequality above is derived from Theorem 3.11. So, it remains to prove the right side of the estimate. Really, from Condition 3.12 for u W p l ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq193_HTML.gif we have
A u X M F 1 A ˆ u ˆ X C F 1 A ˆ A 1 ( x 0 ) A ( x 0 ) u ˆ X C F 1 A ( x 0 ) u ˆ X C A u X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbm_HTML.gif
Hence, applying the Fourier transform to equation (1.1), and by reasoning as Theorem 3.11, it is sufficient to prove that the function
| α | l a ˆ α ξ α [ k = 1 n ξ k l k ] 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbn_HTML.gif

is a multiplier in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq194_HTML.gif. In fact, by using Condition 3.12 and the proof of Lemma 3.2, we get desired result. □

Result 3.14 Theorem 3.13 implies that for all u W p l ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq195_HTML.gif, there are positive constants C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq168_HTML.gif and C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq169_HTML.gif, so that
C 1 u W p l ( R n ; E ( A ) , E ) O u L p ( R n ; E ) C 2 u W p l ( R n ; E ( A ) , E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbo_HTML.gif

From Theorem 3.7, we have the following.

Result 3.15 Assume all conditions of Theorem 3.7 hold. Then, for all λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq6_HTML.gif, the resolvent of operator O exists, and the following sharp estimate holds
| α | l | λ | 1 | α | l a α D α ( O + λ ) 1 B ( X ) + A ( O + λ ) 1 B ( X ) + λ ( O + λ ) 1 B ( X ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbp_HTML.gif

Result 3.16 Theorem 3.7 particularly implies that the operator O + a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq196_HTML.gif for a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq197_HTML.gif is positive in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq194_HTML.gif, i.e., if A ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq198_HTML.gif is uniformly R-positive for φ ( π 2 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq199_HTML.gif, then (see, e.g., [28], § 1.14.5) the operator O + a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq200_HTML.gif is a generator of an analytic semigroup in L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq201_HTML.gif.

From Theorems 3.7, 3.11, 3.13 and Proposition 2.4, we obtain the following.

Result 3.17 Let conditions of Theorems 3.7, 3.11, 3.13 hold for Banach spaces E UMD https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq202_HTML.gif, respectively. Then assertions of Theorems 3.7, 3.11, 3.13 are valid.

4 The quasilinear CDOE

Consider the equations
| α | = l a α D α u + ( A D σ u ) u = F ( x , D σ u ) + f ( x ) , x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ15_HTML.gif
(4.1)
in E-valued L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq1_HTML.gif spaces, where A = A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq2_HTML.gif is a possible unbounded operator in Banach space E, a α = a α ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq203_HTML.gif are complex-valued functions, and D σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq204_HTML.gif denote all differential operators that | σ | l 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq205_HTML.gif. Let
X = L p ( R n ; E ) , Y = W p l ( R n ; E ( A ) , E ) , E j = ( E ( A ) , E ) ϰ σ , p , ϰ σ = p | σ | + 1 p l , E 0 = | σ | < l 1 E ϰ σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbq_HTML.gif
Remark 4.1 By using Theorem 2.7, we obtain that the embedding D ϰ σ Y E ϰ σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq206_HTML.gif is continuous, and by trace theorem [32] (or [19]) for w Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq207_HTML.gif, W = { w ϰ σ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq208_HTML.gif, w ϰ σ = D σ w ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq209_HTML.gif, | σ | < l 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq210_HTML.gif,
| σ | < l 1 D σ w C ( ( R n ) , E ϰ σ ) = | σ | < l 1 sup x R n D j w ( x ) E ϰ σ w Y , E r = { υ E 0 , υ E 0 r } , 0 < r r 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbr_HTML.gif
Let A ( x , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq211_HTML.gif denote by A 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq212_HTML.gif. Consider the linear CDOE
| α | = l a α D α w + A 0 w = Q ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ16_HTML.gif
(4.2)
From Theorem 3.7, we conclude that problem (4.2) has a unique solution w W p l ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq213_HTML.gif, and the coercive uniform estimate holds
| α | l D α w L p ( R n ; E ) + A 0 w L p ( R n ; E ) M f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ17_HTML.gif
(4.3)

for all Q L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq214_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq5_HTML.gif.

Condition 4.2 Assume that all conditions of Theorem 3.11 are satisfied for A = A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq215_HTML.gif and a α L 1 < 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq216_HTML.gif. Suppose that
  1. 1.
    The function: υ A ( x , υ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq217_HTML.gif is a Lipschitz function from E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq78_HTML.gif to B ( E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq218_HTML.gif, i.e.,
    A ( x , u ) A ( x , υ ) B ( E ( A ) , E ) L u υ E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbs_HTML.gif
     
for all x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq73_HTML.gif;
  1. 2.
    F : R n × E 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq219_HTML.gif is a measurable function for each u, u ¯ E r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq220_HTML.gif, u = { υ ϰ σ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq221_HTML.gif, u ¯ = { u ¯ ϰ σ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq222_HTML.gif, u ϰ σ , u ¯ ϰ σ E ϰ σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq223_HTML.gif, and F ( x , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq224_HTML.gif is continuous with respect to x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq225_HTML.gif, F ( x , 0 ) X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq226_HTML.gif. Moreover, there exists g i ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq227_HTML.gif such that
    F ( x , u ) E g 1 ( x ) u E 0 , F ( x , u ) F ( x , u ¯ ) E g 2 ( x ) u u ¯ E 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbt_HTML.gif
     

for all x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq73_HTML.gif, u , υ E r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq228_HTML.gif, g i L p ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq229_HTML.gif and g i L p ( R n ) M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq230_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq231_HTML.gif.

Theorem 4.3 Let Condition 4.2 hold. Then, there exist a radius 0 < r r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq232_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq233_HTML.gif such that for each f L p ( R n , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq234_HTML.gif with f L p ( R n E ) δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq235_HTML.gif there exists a unique u W p l ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq236_HTML.gif with u W p l ( R n ; E ( A ) , E ) r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq237_HTML.gif satisfying equation (3.13).

Proof We want to to solve problem (4.1) locally by means of maximal regularity of the linear problem (4.2) via the contraction mapping theorem. For this purpose, let w be a solution of the linear BVP (4.2). Consider the following ball
B r = { υ Y , υ Y r } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbu_HTML.gif

Let f L p ( R n ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq238_HTML.gif such that f L p ( R n ) δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq239_HTML.gif. Let υ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq240_HTML.gif, υ Y r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq241_HTML.gif.

Define a map G on B r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq242_HTML.gif by
G υ = u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ18_HTML.gif
(4.4)
where u is a solution of problem (4.1). We want to show that Q ( B r ) B r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq243_HTML.gif, and that L is a contraction operator in Y. Consider the function
Q ( x ) = ( ( A 0 A ) D σ υ ) υ + F ( x , D σ υ ) + f ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbv_HTML.gif
We claim that Q X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq244_HTML.gif, moreover, δ and g i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq245_HTML.gif can be chosen such that M Q X δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq246_HTML.gif. In fact, since by Theorem 2.7, υ C ( R n ; E ϰ σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq247_HTML.gif, and one has
A ( x , u ) A 0 ( x ) C ( R n ; B ( E ( A 0 ) , E ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbw_HTML.gif
Thus, Q is measurable and
Q E L υ C ( R n ; E ϰ σ ) υ E ( A 0 ) + g 1 ( x ) υ C ( R n ; E ϰ σ ) + f X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbx_HTML.gif
Now, by Remark 4.1, υ C ( R n ; E ϰ σ ) υ Y r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq248_HTML.gif, by choosing M L r + M h 1 L p < 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq249_HTML.gif and δ = r ( 1 2 M 1 L r h 1 L p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq250_HTML.gif, it follows that
M Q Y M [ L r υ L p ( R n E ) + r h 1 L p + δ ] M [ L r 2 + r h 1 L p + δ ] < 1 2 r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equby_HTML.gif
Moreover, by Theorem 3.11 and by embedding Theorem 2.6, we get
| α | = l a α D α υ L p ( R n E ) < 1 2 r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equbz_HTML.gif
Thus, G maps the set B r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq242_HTML.gif to B r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq242_HTML.gif. Let us show that G is a strict contraction. Let
u 1 = G υ 1 , u 2 = G υ 2 , υ 1 , υ 2 B r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equca_HTML.gif
It is clearly seen that u 1 u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq251_HTML.gif is a solution of the linear problem (4.2) for
Q = ( ( A 0 A ) D σ υ ) υ + F ( x , D σ υ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcb_HTML.gif
Then, by using estimate (4.3) and reasoning as above, we get
u 1 u 2 Y M Q X M { L r υ 1 υ 2 X + L υ 1 υ 2 Y υ 1 L p ( R n ; E ( A 0 ) ) h 2 L p υ 1 υ 2 Y } M ( 2 L r + h 2 L p ) υ 1 υ 2 Y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcc_HTML.gif

Choose h 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq252_HTML.gif, so that h 2 L p < 1 M 2 L r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq253_HTML.gif, we obtain that G is a strict contraction. Then by virtue of contraction mapping principle, we obtain that problem (4.1) has a unique solution u W p l ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq190_HTML.gif. □

5 Boundary value problems for integro-differential equations

In this section, by applying Theorem 3.7, the BVP for the anisotropic type convolution equations is studied. The maximal regularity of this problem in mixed L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq254_HTML.gif norms is derived. In this direction, we can mention, e.g., the works [2, 18, 21] and [33].

Let Ω ˜ = R n × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq255_HTML.gif, where Ω R μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq256_HTML.gif is an open connected set with a compact C 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq257_HTML.gif-boundary Ω. Consider the BVP for integro-differential equation
( L + λ ) u = | α | l a α D α u + | α | 2 m ( b α η α D y α + λ ) u = f ( x , y ) , x R n , y Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ19_HTML.gif
(5.1)
B j u = | β | m j b j β ( y ) D y β u ( x , y ) = 0 , y Ω , j = 1 , 2 , , m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ20_HTML.gif
(5.2)
where
D j = i y j , y = ( y 1 , , y μ ) , b α = b α ( x ) , η α = η α ( y ) , a α = a α ( x ) , α = ( α 1 , α 2 , , α n ) , a α = a α ( x ) , u = u ( x , y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcd_HTML.gif
In general, l 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq258_HTML.gif, so equation (4.4) is anisotropic. For l = 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq259_HTML.gif, we get isotropic equation. If Ω ˜ = R n × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq260_HTML.gif, p = ( p 1 , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq261_HTML.gif, L p ( Ω ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq262_HTML.gif will denote the space of all p-summable scalar-valued functions with a mixed norm (see, e.g., [34]), i.e., the space of all measurable functions f defined on Ω ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq263_HTML.gif, for which
f L p ( Ω ˜ ) = ( R n ( Ω | f ( x , y ) | p 1 d x ) p p 1 d y ) 1 p < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equce_HTML.gif

Analogously, W p l ( Ω ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq264_HTML.gif denotes the Sobolev space with a corresponding mixed norm [34]. Let Q denote the operator, generated by problem (4.4) and (5.1). In this section, we present the following result.

Theorem 5.1 Let the following conditions be satisfied
  1. 1.

    η α C ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq265_HTML.gif for each | α | = 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq266_HTML.gif and η α L ( Ω ) + L r k ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq267_HTML.gif for each | α | = k < 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq268_HTML.gif with r k p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq269_HTML.gif, p 1 ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq270_HTML.gif and 2 m k > l r k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq271_HTML.gif, ν α L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq272_HTML.gif;

     
  2. 2.

    b j β C 2 m m j ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq273_HTML.gif for each j , β , m j < 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq274_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq50_HTML.gif, λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq6_HTML.gif, φ [ 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq275_HTML.gif;

     
  3. 3.

    For y Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq276_HTML.gif, ξ R μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq277_HTML.gif, σ S φ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq278_HTML.gif, φ 0 ( 0 , π 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq279_HTML.gif, | ξ | + | σ | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq280_HTML.gif let σ + | α | = 2 m η α ( y ) ξ α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq281_HTML.gif;

     
  4. 4.
    For each y 0 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq282_HTML.gif local BVP in local coordinates corresponding to y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq283_HTML.gif
    σ + | α | = 2 m η α ( y 0 ) D α ϑ ( y ) = 0 , B j 0 ϑ = | β | = m j b j β ( y 0 ) D β ϑ ( y ) = h j , j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcf_HTML.gif
     
has a unique solution ϑ C 0 ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq284_HTML.gif for all h = ( h 1 , h 2 , , h m ) R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq285_HTML.gif and for ξ R μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq286_HTML.gif with | ξ | + | λ | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq287_HTML.gif;
  1. 5.
    The (1) part of Condition 3.6 is satisfied, a ˆ α , b ˆ α C ( n ) ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq288_HTML.gif, and there are positive constants C i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq289_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq290_HTML.gif, so that
    | ξ | β | D β a ˆ α ( ξ ) | C 1 , | ξ | β | D β b ˆ α ( ξ ) | C 2 | b ˆ α ( ξ ) | , ξ R n { 0 } , β k { 0 , 1 } , 0 | β | n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcg_HTML.gif
     
Then, for f W p l ( Ω ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq291_HTML.gif and λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq6_HTML.gif problems (4.4) and (5.1) have a unique solution u W p l ( Ω ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq292_HTML.gif, and the following coercive uniform estimate holds
| α | l | λ | 1 | α | l a α D α u L p ( Ω ˜ ) + | λ | u L p ( Ω ˜ ) + | α | 2 m b α η α D α u L p ( Ω ˜ ) C f L p ( Ω ˜ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equch_HTML.gif
Proof Let E = L p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq177_HTML.gif. It is known [29] that L p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq293_HTML.gif is UMD space for p 1 ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq178_HTML.gif. Consider the operator A in L p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq293_HTML.gif, defined by
D ( A ) = W p 1 2 m ( Ω ; B j u = 0 ) , A ( x ) u = | α | 2 m b α ( x ) η α ( y ) D α u ( y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ21_HTML.gif
(5.3)
Therefore, problems (4.4) and (5.1) can be rewritten in the form of (1.1), where u ( x ) = u ( x , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq294_HTML.gif, f ( x ) = f ( x , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq295_HTML.gif are functions with values in E = L p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq177_HTML.gif. It is easy to see that A ˆ ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq296_HTML.gif and D β A ˆ ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq297_HTML.gif are operators in L p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq293_HTML.gif defined by
D ( A ˆ ) = D ( D β A ˆ ) = W p 1 2 m ( Ω ; B j u = 0 ) , A ˆ ( ξ ) u = | α | 2 m b ˆ α ( ξ ) η α ( y ) D α u ( y ) , D ξ β A ˆ ( ξ ) u = | α | 2 m D ξ β b ˆ α ( ξ ) η α ( y ) D α u ( y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ22_HTML.gif
(5.4)
In view of conditions and by [[5], Theorem 8.2] operators A ˆ ( ξ ) + μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq298_HTML.gif and D β A ˆ ( ξ ) + μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq299_HTML.gif for sufficiently large μ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq300_HTML.gif, are uniformly R-positive in L p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq293_HTML.gif. Moreover, by (3.3), the problems
μ u ( y ) + | α | 2 m b ˆ α ( ξ ) η α ( y ) D α u ( y ) = f ( y ) , B j u = | β | m j b j β ( y ) D β u ( y ) = 0 , j = 1 , 2 , , m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ23_HTML.gif
(5.5)
μ u ( y ) + α 2 m D β b ˆ α ( ξ ) η α ( y ) D α u ( y ) = f ( y ) , B j u = | β | m j b j β ( y ) D β u ( y ) = 0 , j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ24_HTML.gif
(5.6)
for f L p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq301_HTML.gif and for sufficiently large μ, have unique solutions that belong to W p 1 l ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq302_HTML.gif, and the coercive estimates hold
u W p 1 l ( Ω ) C ( A ˆ + μ ) u L p 1 ( Ω ) , u W p 1 2 m ( Ω ) C ( D β A ˆ + μ ) u L p 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equci_HTML.gif
for solutions of problems (5.4) and (5.5). Then in view of (5) condition and by virtue of embedding theorems [34], we obtain
( A ˆ + μ ) u L p 1 ( Ω ) C u W p 1 2 m ( Ω ) C ( A ˆ + μ ) u L p 1 ( Ω ) , ( D β A ˆ + μ ) u L p 1 ( Ω ) C u W p 1 2 m ( Ω ) C ( D β A ˆ + μ ) u L p 1 ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ25_HTML.gif
(5.7)
Moreover by using (5) condition for u W p 1 2 m ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq303_HTML.gif we have
| ξ | β ( D ξ β A ˆ + μ ) u L p 1 ( Ω ) C ( A ˆ + μ ) u L p 1 ( Ω ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcj_HTML.gif

i.e., all conditions of Theorem 3.7 hold, and we obtain the assertion. □

6 Infinite system of IDEs

Consider the following infinity system of a convolution equation
| α | l a α D α u m + j = 1 ( d j + λ ) u j ( x ) = f m ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equ26_HTML.gif
(6.1)

for x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq73_HTML.gif and m = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq304_HTML.gif .

Condition 6.1 There are positive constants C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq168_HTML.gif and C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq169_HTML.gif, so that for { d j ( x ) } 1 l q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq305_HTML.gif for all x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq73_HTML.gif and some x 0 R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq306_HTML.gif,
C 1 | d j ( x 0 ) | | d j ( x ) | C 2 | d j ( x 0 ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equck_HTML.gif
Suppose that a ˆ α , d ˆ m C ( n ) ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq307_HTML.gif, and there are positive constants M i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq308_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq290_HTML.gif, so that
| ξ | β | D β a ˆ α ( ξ ) | M 1 , | ξ | β | D β d ˆ m ( ξ ) | M 2 | d ˆ m ( ξ ) | , ξ R n { 0 } , β k { 0 , 1 } , 0 | β | n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcl_HTML.gif
Let
D ( x ) = { d m ( x ) } , d m > 0 , u = { u m } , D u = { d m u m } , l q ( D ) = { u l q , u l q ( D ) = ( m = 1 | d m ( x 0 ) u m | q ) 1 q < } , 1 < q < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcm_HTML.gif

Let Q be a differential operator in L p ( R n ; l q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq309_HTML.gif, generated by problem (5.7) and B = B ( L p ( R n ; l q ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq310_HTML.gif. Applying Theorem 3.7, we have the following.

Theorem 6.2 Suppose that (1) part of Condition 3.6 and Condition 6.1 are satisfied. Then
  1. 1.
    For all f ( x ) = { f m ( x ) } 1 L p ( R n ; l q ( D ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq311_HTML.gif, for λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq6_HTML.gif, φ [ 0 , π ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq275_HTML.gif the equation (6.1) has a unique solution u = { u m ( x ) } 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq312_HTML.gif that belongs to W p l ( R n ; l q ( D ) , l q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq313_HTML.gif, and the coercive uniform estimate holds
    | α | l | λ | 1 | α | l a α D α u L p ( R n ; l q ) + D u L p ( R n ; l q ) + | λ | u L p ( R n ; l q ) C f L p ( R n ; l q ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcn_HTML.gif
     
  2. 2.
    For λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq6_HTML.gif, there exists a resolvent ( Q + λ ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq314_HTML.gif of operator Q and
    | α | l | λ | 1 | α | l a α [ D α ( Q + λ ) 1 ] B + D ( Q + λ ) 1 B + λ ( Q + λ ) 1 B C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equco_HTML.gif
     
Proof Really, let E = l q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq315_HTML.gif and A = [ d m ( x ) δ j m ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq316_HTML.gif, m , j = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq317_HTML.gif . Then
A ˆ ( ξ ) = [ d ˆ m ( ξ ) δ j m ] , D β A ˆ ( ξ ) = [ D β d ˆ m ( ξ ) δ j m ] , m , j = 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_Equcp_HTML.gif

It is easy to see that A ˆ ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq296_HTML.gif is uniformly R-positive in l q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-211/MediaObjects/13661_2013_Article_459_IEq318_HTML.gif, and all conditions of Theorem 3.7 are hold. Therefore, by virtue of Theorem 3.7 and Result 4.1, we obtain the assertions. □

Remark 6.3 There are a lot of positive operators in concrete Banach spaces. Therefore, putting concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of operator A in (1.1) and (4.1), we can obtain the maximal regularity of different class of convolution equations, Cauchy problems for parabolic CDEs or it’s systems, by virtue of Theorem 3.7 and Theorem 3.11, respectively.

Declarations

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this paper.

Authors’ Affiliations

(1)
Department of Mechanical Engineering, Okan University
(2)
Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences
(3)
Department of Mathematics, Dumlupınar University

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