Linear and nonlinear abstract elliptic equations with VMO coefficients and applications

R E S E A R C H

Open Access

Linear and nonlinear abstract elliptic

equations with

VMO

coefficients and

applications

Veli B Shakhmurov

*_{Correspondence:}

[email protected] Department of Mechanical Engineering, Okan University, Akfirat, Tuzla, Istanbul 34959, Turkey

Abstract

In this paper, maximal regularity properties for linear and nonlinear elliptic

differential-operator equations withVMO(vanishing mean oscillation) coefficients are studied. For linear case, the uniform separability properties for parameter dependent boundary value problems is obtained inLp_{spaces. Then the existence and uniqueness} of a strong solution of the boundary value problem for a nonlinear equation

is established. In application, the maximal regularity properties of the anisotropic elliptic equation and the system of equations withVMOcoefficients are derived. MSC: 58I10; 58I20; 35Bxx; 35Dxx; 47Hxx; 47Dxx

Keywords: differential equations withVMOcoefficients; boundary value problems; differential-operator equations; maximalLp_{regularity; abstract function spaces;} nonlinear elliptic equations

1 Introduction

The goal of the present paper is to study the nonlocal boundary value problems (BVPs) for parameter dependent linear differential-operator equations (DOEs) with discontinuous top-order coefficients,

sa(x)u()(x) +A(x)u(x) +s_A(x)u()_{(x) +A(x)u(x) +λu(x) =f(x), ()} and the nonlinear equation

a(x)u()(x) +Bx,u,u()u(x) =Fx,u,u(),

where a is a complex valued function,s is a positive, λ is a complex parameter, and A=A(x),Ai=Ai(x) are linear andBis a nonlinear operator in a Banach spaceE. Here,

the principal coefficientsaandAmay be discontinuous. More precisely, we assume that aandA(·)A–_(x

) belong to the operator-valued Sarason classVMO. The Sarason class VMOwas first defined in []. In the recent years, there has been considerable interest in elliptic and parabolic equations withVMOcoefficients. This is mainly due to the fact that theVMOclass contains as a subspaceC(¯) that ensures the extension ofLp-theory of

operators with continuous coefficients to discontinuous coefficients (see,e.g., [–]). On the other hand, the Sobolev spacesW,n_{() andW}σ,σn_{(),  <σ< , are also contained in}

VMO. Global regularity of the Dirichlet problem for elliptic equations withVMO

cients has been studied in [–]. We refer to the survey [], where an excellent presenta-tion and relapresenta-tions with another similar results can be found concerning the regularizing properties of these operators in the framework of Sobolev spaces.

It is known that many classes of PDEs (partial differential equations), pseudo DEs and integro DEs can be expressed in the form of DOEs. Many researchers (see,e.g., [–]) investigated similar spaces of functions and classes of PDEs under a single DOE. Moreover, the maximal regularity properties of DOEs with continuous coefficients were studied,e.g., in [, , ].

Here, the equation with top-orderVMO-operator coefficients is considered in abstract spaces. We shall prove the uniform separability of the problem (),i.e., we show that, for eachf ∈Lp_{(, ;E), there exists a unique strong solutionuof the problem () and a positive}

constantCdepending only onp,γ,EandAsuch that 

i=

si|_λ|–i_u(i)

Lp_(,;E)+AuLp_(,;E)≤Cf_Lp_(,;E).

Note that the principal part of a corresponding differential operator is nonselfadjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent and Fredholmness are established. Then the existence and uniqueness of the above nonlinear problem is derived. In application, we study maximal regularity properties of anisotropic elliptic equations in mixedLp _{spaces and the systems (finite or infinite) of differential equations withVMO}

coefficients in the scalarLp_space.

Since () involves unbounded operators, it is not easy to get representation for the Green function and estimate of solutions. Therefore, we use the modern harmonic analysis el-ements,e.g., the Hilbert operators and the commutator estimates inE-valuedLp_spaces,

embedding theorems of Sobolev-Lions spaces and some semigroups estimates to over-come these difficulties. Moreover, we also use our previous results on equations with con-tinuous leading coefficients and the perturbation theory of linear operators to obtain our main assertions.

2 Notations and background

Throughout the paper, we setE to be a Banach space and⊂Rn_.Lp_{(;E) denotes the}

space of all strongly measurableE-valued functions that are defined onwith the norm fp=fLp_(;E)=

f(x)p_Edx 

p

, ≤p<∞.

BMO(E) (bounded mean oscillation) (see [, ]) is the space of allE-valued local inte-grable functions with the norm

sup

B

B

f(x) –fB_Edx=f∗,E<∞,

whereBranges in the class of the balls inRn_andf

Bis the average_|B| _Bf(x)dx.

Forf ∈BMO(E) andr> , we set

sup ρ≤r

B

f(x) –fB_Edx=η(r),

We will say that a functionf∈BMO(E) is inVMO(E) iflim_r→+η(r) = . We will call the

η(r) aVMOmodulus off.

Note that, ifE=C, whereCis the set of complex numbers, thenBMO(E) andVMO(E) coincide with the John-Nirenberg classBMOand Sarason classVMO, respectively.

The Banach spaceEis called aUMD(unconditional martingale difference)-space if the Hilbert operator

(Hf)(x) =lim ε→

|x–y|>ε f(y) x–ydy

is bounded inLp(R,E),p∈(,∞) (see,e.g., [, ]).UMDspaces include,e.g.,Lp,lpspaces

and Lorentz spacesLpq,p,q∈(,∞).

Let

Sϕ=

λ∈C,|argλ| ≤ϕ∪ {}, ≤ϕ<π.

A linear operatorAis said to be aϕ-positive (or positive) in a Banach spaceEwith the boundM>  ifD(A) is dense onEand

(A+λI)–_L(E)≤M +|λ|–

forλ∈Sϕ,ϕ∈(,π],Iis an identity operator inEandL(E) is the space of bounded linear

operators inE. Sometimes instead ofA+λI, it will be written asA+λand denoted byAλ.

It is known [, §..] that there exist fractional powersAθ _{of the positive operatorA.}

LetE(Aθ_{) denote the spaceD(A}θ_{) with the graphical norm}

uE(Aθ₎=

up_+Aθ up



p_{, }≤_p<∞_{, –}∞_<θ<∞_.

Let E andE be two Banach spaces. A setW ⊂L(E,E) is called R-bounded (see [, ]) if there is a positive constant C such that for all T,T, . . . ,Tm ∈ W and

u,u, . . . ,um∈E,m∈N





m

j=

rj(y)Tjuj

E dy≤C





m

j= rj(y)uj

E dy,

where {rj} is a sequence of independent symmetric{–, }-valued random variables on

[, ].

Let S(Rn_{;E) denote the Schwarz class,i.e., the space of all E-valued rapidly}

decreas-ing smooth functions onRn_{. LetF denote the Fourier transformation. A function ∈}

L∞(Rn_;B(E

,E)) is called a Fourier multiplier fromLp(Rn;E) to Lp(Rn;E) if the map u→u=F–(ξ)Fu,u∈S(Rn;E) is well defined and extends to a bounded linear

op-erator

:Lp

Rn;E

→Lp

Rn;E

.

The set of all multipliers fromLp(Rn;E) toLp(Rn;E) will be denoted byMpp(E,E). For

Let

Un=

β= (β,β, . . . ,βn)∈Nn:βk∈ {, }

.

Definition  A Banach spaceE is said to be a space satisfying the multiplier condition if, for any∈C(n)_(Rn_{;L(E)), theR-boundedness of the set{ξ}β_Dβ

ξ(ξ) :ξ∈Rn\,β∈Un}

implies thatis a Fourier multiplier inLp(Rn;E),i.e.,∈Mpp(E) for anyp∈(,∞). Definition  Theϕ-positive operatorAis said to be anR-positive in a Banach spaceEif there existsϕ∈[,π) such that the setLA={A(A+λ)–:λ∈Sϕ}isR-bounded.

A linear operatorA(x) is said to be positive inEuniformly inxifD(A(x)) is independent ofx,D(A(x)) is dense inEand

A(x) +λ–≤M +|λ|–

for allλ∈S(ϕ),ϕ∈[,π).

Letσ_∞(E,E) denote the space of all compact operators fromEtoE. ForE=E=E, it is denoted byσ∞(E).

AssumeEandEare two Banach spaces andEis continuously and densely embedded intoE. Letmbe a natural number.Wm,p_{(;E,E) (the so-called Sobolev-Lions type space)}

denotes a space of all functionsu∈Lp_{(;E) possessing the generalized derivativesD}m ku=

∂mu

∂xm_k such thatD m

ku∈Lp(;E) endowed with the norm

uWm,p_(;E

,E)=uLp(;E)+

n

k=

Dm_ku_Lp_(;E)<∞.

ForE=E, the spaceWm,p(;E,E) will be denoted byWm,p(;E). It is clear that

Wm,p(;E,E) =Wm,p(;E)∩Lp(;E).

Let s be a positive parameter. We define inWm,p_(;E

,E) the following parameter-dependent norm:

u_Wm,p

s (;E,E)=uLp(;E)+

n

k=

sDm_ku_Lp_(;E).

The spaceBs

p,p(;E,E) is defined as a trace space forWm,p(;E,E) as in a scalar case

(see,e.g., [, §..]).

Function u ∈ W,p_{(, ;E(A),E,L}

k) = {u ∈ W,p(, ;E(A),E),Lku = } satisfying the

equation () a.e. on (, ) is said to be a solution of the problem () on (, ). From [] we have

Theorem A Suppose the following conditions are satisfied:

() α= (α,α, . . . ,αn)arentuples of nonnegative integer numbers such thatκ=|_mα|≤

and <μ≤ –κ;

() ∈Rn_{is a region such that there exists a bounded linear extension operator from}

Wm,p(;E(A),E)toWm,p(Rn;E(A),E). Then the embedding

Dα_Wm,p_;E(A),E⊂Lp_;EA–κ–μ

is continuous and there exists a positive constant Cμsuch that

s|mα|_Dα_u

Lp_(;E(A–κ–μ₎₎≤Cμ

hμ_u

Wsm,p(;E(A),E)+h

–(–μ)_u

Lp_(;E)

for all u∈Wm,p_{(;E(A),E)and <h≤h}

<∞.

Theorem A Suppose all the conditions of Theorem A are satisfied. Assume is a bounded region in Rn_{and A}–_∈σ

∞(E).Then,for <μ≤ –κ,the embedding DαWm,p;E(A),E⊂Lp;EA–κ–μ

is compact.

In a similar way as in [, Theorem .], we have the following result.

Lemma A Let E be a Banach space and f ∈VMO(E).The following conditions are equiv-alent:

() f∈VMO(E);

() f is in theBMOclosure of the set of uniformly continuous functions which belong to VMO;

() lim_y→f(x–y) –f(x)∗,E= .

For f ∈Lp_{(;E),p∈(,∞),a∈L}∞_(Rn_{),consider the commutator operator}

H[a,f](x) =a(x)Hf(x) –H(af)(x) =lim ε→

|x–y|>ε,

[a(x) –a(y)] x–y f(y)dy.

From [, Theorem ] and [, Corollary .], we have

Theorem A Let E be a UMD space and a∈VMO∩L∞(Rn).Then H[a,f]is a bounded operator in Lp_{(R;E),p∈(,∞).}

From Theorem Aand the property () of Lemma A we obtain, respectively, the fol-lowing.

Theorem A_ Assume all the conditions of TheoremAare satisfied.Also,let a∈VMO∩ L∞(Rn)and let ηbe the VMO modulus of a.Then,for any ε> ,there exists a positive numberδ=δ(ε,η)such that

Theorem A Let E be a UMD space and A(·)uniformly R-positive in E.Moreover, let A(·)A–(x)∈L∞(R;L(E))∩BMO(L(E)),x∈R.Then the commutator operator

H[A,f](x) =A(x)A–(x)Hf(x) –H

A(x)A–(x)f

(x)

=lim ε→

|x–y|>ε,

[A(x)A–_{(x) –A(y)A}–_(x)] x–y f(y)dy

is bounded in Lp(R;E),p∈(,∞).

Theorem A Assume all the conditions of TheoremAare satisfied andηis a VMO mod-ulus of A(·)A–_(x).

Then,for anyε> ,there exists a positive numberδ=δ(ε,η)such that H[A,f]_Lp₍

r;E)≤MεfLp(r;E), r∈(,δ).

Consider the nonlocal BVP for a parameter dependent DOE with constant coefficients

(L+λ)u=sau()(x) + (A+λ)u(x) =f(x), x∈(, ) ,

Lku= mk

i= sμi_α

kiu(i)() +βkiu(i)()

=fk, k= , , ()

where mk∈ {, },a,αki,βkiare complex numbers,s is a positive parameter,λis a complex

spectral parameter,μi=_i +__p,θk=mk_ +__p,Aλ=A+λand A is a linear operator in E. Letω,ωbe roots of the equation aω+  = and let

αk=αkmk, βk=βkmk, μ=

(–ω)m_α

 βωm (–ω)m_α

 βωm .

It is known that if the operator A isϕ-positive in E,then operatorsωkA

 

λ,k= , generate analytic semigroups

Uλs(x) =eωs– A

 

λx_{, Uλs(x) =e}–ωs– A  

λ(–x) _forλ∈_S(ϕ).

Let

Ek=

E(A),E_θ

k,p.

From [, Theorem ] and [, Theorem .], we obtain

Theorem A Assume the following conditions are satisfied:

() Eis a Banach space satisfying the multiplier condition with respect top∈(,∞); () Ais anR-positive operator inEfor≤ϕ<πandμ= ;

Then

()for f ∈Lp(, ;E),fk∈Ek,λ∈S(ϕ)and for sufficiently large|λ|,the problem()has a

unique solution u∈W,p(, ;E(A),E).Moreover,the coercive uniform estimate holds



i=

si|_λ|–i_u(i)

Lp_(,;E)+AuLp_(,;E)≤C

fLp_(,;E)+



k= fkEk

.

()For fk= ,the solution is represented as

u(x) = 



Gλs(x,y)f(y)dy, ()

Gλs(x,y) =



i,j=

m

k=

Bij(λ)

s–Aλ

–(+mi–k–)_U

jλs(x)Uλs( –y)

+ 

i,j=

m

k=

Dij(λ)A

–(+mi–k–)

λ Ujλs(x)Uλs(y)

+Uλs(x–y),

where Bij(λ)and Dij(λ)are uniformly bounded operators in E and

Uλs(x–y) =

⎧ ⎨ ⎩

a–_s_A–

λ Uλs(x–y), x≥y,

–a–_s_A–  

λ Uλs(x–y), x≤y.

Consider the BVP for a DOE with variable coefficients

sa(x)u()(x) +A(x) +λu(x) =f(x), x∈(, ), Lku=

mk

i= sθk_α

kiu(i)() +βkiu(i)()

= , k= , , ()

wherea=a(x) is a complex valued function,sis a positive parameter,mk∈ {, },αki,βki

are complex numbers,λis a spectral parameter,θk=m_k +__p andA(x) is a linear operator

inE.

Letω=ω(x),ω=ω(x) be roots of the equationa(x)ω+  =  and let

αk=αkmk, βk=βkmk, μ(x) =

(–ω)m_α

 βωm (–ω)mα βωm .

In the next theorem, let us consider the case that principal coefficients are continuous. The well-posedness of this problem occurs in the studying of equations withVMO coef-ficients. From [, Theorem ] and [, Theorem .], we get

Theorem A Suppose the following conditions are satisfied:

() Eis a Banach space satisfying the multiplier condition with respect top∈(,∞); () a∈C[, ],–a∈S(ϕ),a() =a(),andμ(x)= for a.e.x∈[, ];

() Reωk(x)=  andωλk ∈S(ϕ)forλ∈S(ϕ),k= , .a.e.x∈[, ];

() A(x)is a uniformlyR-positive operator inEand

Then,for all f ∈Lp_{(, ;E),λ∈S(ϕ)and for sufficiently large|λ|,there is a unique solution}

u∈W,p_{(, ;E(A),E)of the problem().Moreover,the following coercive uniform estimate}

holds:



i=

si|_λ|–i_u(i)

Lp(,;E)+AuLp(,;E)≤CfLp(,;E).

3 DOEs with VMO coefficients

Consider the principal part of the problem ()

(L+λ)u=sa(x)u()(x) +A(x) +λu(x) =f(x), x∈(, ), Lku=

mk

i= sμi_α

kiu(i)() +βkiu(i)()

= , k= , . ()

Condition  Assume the following conditions are satisfied: () Eis aUMDspace,p∈(,∞);

() a∈VMO∩L∞(R),ηis aVMOmodulus ofa,–a∈S(ϕ),μ(x)= ; () Reωk(x)= andωλk ∈S(ϕ)forλ∈S(ϕ),≤ϕ<π,k= , . a.e.x∈[, ];

() A(x)is a uniformlyR-positive operator inEand

A(·)A–(x)∈L∞, ;L(E)∩VMOL(E), x∈(, );

() a() =a(),A() =A()andηis aVMOmodulus ofA(·)A–(x). First, we obtain an integral representation formula for solutions.

Lemma  Let Conditionhold and f ∈Lp_{(, ;E).Then,for all solutions u of the problem}

()belonging to W,p_{(, ;E(A),E),we have}

u(i)(x) = 



iλs(x,x–y)

a(x) –a(y)u()(y)

+A(x) –A(y)u(y) +f(y)dy+f(x), ()

A(x)u(x) = 



_λs(x,x–y)a(x) –a(y)u()(y) +A(x) –A(y)u(y) +f(y)dy+f(x),

where

_iλs(x,x–y) = 

i,j=

m

k=

B_ij(λ)s–Aλ

–_(+mk–k–i–)

Ujλs(x)Uλs( –y)

+ 

i,j=

m

k=

D_ij(λ)s–Aλ

–_(+mk–k–i–)

Ujλs(x)Uλs(y)

here B_ij(λ),D_ij(λ)are uniformly bounded operators,

Uνλs(x–y) =

⎧ ⎨ ⎩

ωi_a–_s(–ν)/_A–(–ν)/

λ Uλs(x–y), x≥y,

–ωi_a–_s(–ν)/_A–(–ν)/

λ Uλs(x–y), x≤y,

and the expression_λ(x,x–y)is a scalar multiple ofλ(x,x–y). Proof Consider the problem () fora(x) =a(x) andA(x) =A(x),i.e.,

(L+λ)u=sa(x)u()(x) +

A(x) +λ

u(x) =f(x), x∈(, ),

Lku= mk

i= sμi_α

kiu(i)() +βkiu(i)()

= , k= , . ()

Letu∈C()_{([, ];E(A)) be a solution of the problem (). Taking into account the equality} Lu= (L–L)u+Luand Theorem A, we get

u(i)(x) = 



iλs(x,x–y)

a(x) –a(y)u()(y)

+A(x) –A(y)

u(y) +f(y)dy+f(x),

A(x)u(x) = 



_λs(x,x–y)a(x) –a(y)

u()(y)

+A(x) –A(y)u(y) +f(y)dy+f(x).

Settingx=xin above, we get () foru∈C()([, ];E(A)). Then the density argument, using Theorem A, gives the conclusion for

u∈W,p, ;E(A),E, Lku= .

Consider the problem () on (,b),i.e.,

(Lb+λ)u=sa(x)u()(x) +

A(x) +λu(x) =f(x), x∈(,b), Lbku=

mk

i= sμi_α

kiu(i)() +βkiu(i)(b)

= , k= , . ()

Theorem  Suppose Conditionis satisfied.Then there exists a number b∈(, )such that the uniform coercive estimate



i=

si|_λ|–i_u(i)

Lp_(,b;E)+AuLp_(,b;E)≤C(L_b+λ)u

Lp_(,b;E) ()

holds for large enough|λ|and u∈W,p_{(,b;E(A),E),L}

bku= ,λ∈S(ϕ),where C is a

Proof By Lemma , for any solutionu∈W,p_{(,b;E(A),E) of the problem (), we have}

u(i)(x) = b



ibλs(x,x–y)

a(x) –a(y)u()(y)

+A(x) –A(y)u(y) +f(y)dy+f(x), ()

where

ibλs(x,x–y) =



i,j=

m

k=

B_ij(λ)s–Aλ

–_(+mk–k–i–)

Ujλ(x)Uλ(b–y)

+ 

i,j=

m

k=

D_ij(λ)s–Aλ

–(+mk–k–i–)_U

jλ(x)Uλ(y)

+Uνλs(x–y), ν= , , , ()

hereB_ij(λ),D_ij(λ) are uniformly bounded operators,

Uλ(x) =eωs – _A

λx_{, Uλ(x) =e}–ωs– A  

λ(b–x)

and

U_νλs(x–y) = ⎧ ⎨ ⎩

ωi

a–s(–ν)/A –(–ν)/

λ Uλs(x–y), x≥y,

–ωi_a–s(–ν)/_A–(–ν)/

λ Uλs(x–y), x≤y,

ν= , , .

Moreover, from () and (), clearly, we get

Au(x) = b



_bλs(x,x–y)a(x) –a(y)u()(y) +A(x) –A(y)u(y) +f(y)dy, ()

where the expression_bλ(x,x–y) differs frombλ(x,x–y) only by a constant.

Consider the operators

Bλf = 



Gbλs(x,y)f(y)dy, Biλsf =

b



ibλs(x,x–y)f(y)dy,

Siλu= b



ibλ(x,x–y)

a(x) –a(y)u()(y)dy,

Diλsu=

b



ibλs(x,x–y)

A(x) –A(y)u(y)dy, i= , , ,

λsu=

b



_bλs(x,x–y)a(x) –a(y)u()(y)dy,

λu= b



_bλ(x,x–y)A(x) –A(y)u(y)dy.

Since the operatorsBλandBλare regular onLp(,b;E), by using the positivity

we get

BiλfLp_(,b;E)≤M|λ|–

–i

 f_Lp_(,b;E), i= , . ()

Since the Hilbert operator is bounded inLp_{(R;E) for a UMD spaceE, we have}

BλfLp_(,b;E)≤Mf_Lp_(,b;E). ()

Thus, by virtue of Theorems A, Aand in view of ()-() for anyε> , there exists a positive numberb=b(ε,η,η) such that

SiλuLp_(,b;E)≤Mε|λ|–

–i

 _u()

Lp_(,b;E),

DiλuLp_(,b;E)≤Mε|λ|–

–i

 _A(x)u

Lp_(,b;E), i= , , , ()

λuLp_(,b;E)≤Mεu()

Lp_(,b;E), λuLp_(,b;E)≤MεA(x)u Lp_(,b;E).

Hence, the estimates ()-() imply ().

Theorem  Assume Conditionholds.Let u∈W,p_{(, ;E(A),E)be a solution of().Then,}

for sufficiently large|λ|,λ∈S(ϕ),the following coercive uniform estimate holds: 

i=

si|_λ|–i_u(i)

Lp_(,;E)+AuLp_(,;E)≤C(L+λ)u

Lp_(,;E)+uLp_(,;E). ()

Proof This fact is shown by a covering and flattening argument, in a similar way as in Theorem A. Particularly, by partition of unity, the problem is localized. Choosing diam-eters of supports for corresponding finite functions, by using Theorem , Theorems A, A, A and embedding Theorem A (see the same technique for DOEs with continuous coefficients [, ]), we obtain the assertion.

LetQsdenote the operator inLp(, ;E) generated by the problem () forλ= ,i.e.,

D(Qs) =W,p

, ;E(A),E,Lk

, Qsu=sa(x)u()+A(x)u.

Theorem  Assume Conditionholds.Then,for all f ∈Lp_{(, ;E),λ∈S(ϕ)and for large}

enough|λ|,the problem()has a unique solution u∈W,p_{(, ;E(A),E).Moreover,the}

fol-lowing coercive uniform estimate holds:



i=

si|_λ|–i_u(i)

Lp_(,;E)+AuLp_(,;E)≤Cf_Lp_(,;E). ()

Proof First, let us show that the operatorQ+λhas a left inverse. Really, it is clear that uLp(,;E)=



|λ|(Qs+λ)uLp_(,;E)+QsuLp(,;E)

.

By Theorem Aforu∈W,p_{(, ;E(A),E), we have}

QsuLp_(,;E)≤Cu

Then, by virtue of () and in view of the above relations, we infer, for allu∈W,p_{(, ;}

E(A),E) and sufficiently large|λ|, that there is a smallεandC(ε) such that 

i=

si|_λ|–i_u(i)

Lp_(,;E)+AuLp(,;E)

≤C(Qs+λ)u_Lp_(,;E)+εuW,p_{(,;E(A),E)}+C(ε)(Q_s+λ)u Lp_(,;E)

. ()

From the estimate () foru∈W,p_{(, ;E(A),E), we get}



i=

si|_λ|–i_u(i)

Lp_(,;E)+AuLp_(,;E)≤C(Q+λ)u

Lp_(,;E). ()

The estimate () implies that () has a unique solution and the operatorQs+λhas a

bounded inverse in its rank space. We need to show the rank space coincides with all the spaceLp_{(, ;E). It suffices to prove that there is a solutionu∈W},p_{(, ;E(A),E) for}

all f ∈Lp_{(, ;E). This fact can be derived in a standard way, approximating the}

equa-tion with a similar one with smooth coefficients [, ]. More precisely, by virtue of [, Theorem .],UMDspaces satisfy the multiplier condition. Moreover, by the part

() of Lemma A, given a∈VMO with VMO modules η(r), we can find a sequence

of mollifiers functions {ah} converging to a in BMOas h→ with a VMOmodulus

ηh such thatηh(r)≤η(r). In a similar way, it can be derived for the operator function

A(x)A–_(x

)∈VMO(L(E)).

Result  Theorem  implies that the resolvent (Qs+λ)–satisfies the following sharp

uni-form estimate:



i=

si|_λ|–id

i

dxi(Qs+λ)

–

L(Lp_(,;E))

+A(Qs+λ)–_L(Lp_(,;E))≤C ()

for|argλ| ≤ϕ,ϕ∈(,π) ands> .

The estimate () particularly implies that the operator Q is uniformly positive in Lp(, ;E) and generates analytic semigroups forϕ∈(π_,π) (see,e.g., [, §..]).

Remark  Conditionsa() =a(),A() =A() arise due to nonlocality of the boundary conditions (). If the boundary conditions are local, then the conditions mentioned above are not required any more.

Consider the problem (), whereLkuis the same boundary condition as in (). LetOs

denote a differential operator generated by the problem (). We will show the separability and Fredholmness of ().

Theorem  Assume () Conditionholds;

() for anyε> ,there isC(ε) > such that for a.e.x∈(, )and for <ν< , <ν<_, A(x)u_E≤εAuE+C(ε)uE, u∈E(A),

A(x)u_E≤εu(E(A),E)  ,∞

Then,for all f ∈Lp_{(, ;E)and for large enough|λ|,λ∈S(ϕ),there is a unique solution}

u∈W,p_{(, ;E(A),E)of the problem()and the following coercive uniform estimate holds:}



i=

si|_λ|–i_u(i)

Lp_(,;E)+AuLp_(,;E)≤Cf_Lp_(,;E). ()

Proof It is sufficient to show that the operatorOs+λhas a bounded inverse (Os+λ)–

fromLp_{(, ;E) toW},p_{(, ;E(A),E). PutO}

su=Qsu+Qu, where

Qu=s  _A

u()+Au,u∈W,p

, ;E(A),E,Lk

.

By the second assumption and Theorem A, there is a smallεandC(ε) such that AuLp_(,;E)≤CA–νu

Lp_(,;E)

≤εu

Ws,p(,;E(A),E)+C(ε)uL p_(,;E),

_s _A

u()_Lp_(,;E)≤Cs



_A–ν_u

Lp_(,;E)

≤εu_W,p

s (,;E(A),E)+C(ε)uL p_(,;E).

()

In view of estimates () and (), we have

AuLp_(,;E)<_δQ_su_Lp_(,;E),

s_Au()

Lp(,;E)<δQsuLp(,;E)

()

foru∈W,p_{(, ;E(A),E) andδ}

k< . By Theorem , the operatorQs+λhas a bounded

inverse (Qs+λ)– fromLp(, ;E) toW,p(, ;E(A),E) for sufficiently large|λ|. So, ()

implies the following uniform estimate:

Q(Qs+λ)–_L(Lp_(,;E))< .

It is clear that

(Os+λ) =

I+Q(Qs+λ)–

(Qs+λ),

(Os+λ)–= (Q+λ)–

I+Q(Qs+λ)–

– .

Then, by above relation and by virtue of Theorem , we get the assertion.

Theorem  implies the following result.

Result  Suppose all the conditions of Theorem  are satisfied. Then the resolvent (Os+

λ)–_{of the operatorO}

ssatisfies the following sharp uniform estimate:



i=

si|_λ|–id

i

dxi(Os+λ)

–

L(Lp_(,;E))

+A(Os+λ)–_L(Lp_(,;E))≤C

Consider the problem () forλ= ,i.e.,

Lu=sa(x)u()(x) +A(x)u(x) +A(x)u()+A(x)u=f(x), x∈(, ),

Lku= mk

i= sμi_α

kiu(i)() +βkiu(i)()

= , k= , . ()

Theorem  Assume all the conditions of Theoremhold and A–_∈σ

∞(E).Then the

prob-lem()is Fredholm from W,p(, ;E(A),E)into Lp(, ;E).

Proof Theorem  implies that the operatorOs+λhas a bounded inverse (Os+λ)–from

Lp_{(, ;E) toW},p_{(, ;E(A),E) for large enough|λ|; that is, the operatorO}

s+λis Fredholm

fromW,p_{(, ;E(A),E) intoL}p_{(, ;E). Then, by virtue of Theorem A}

and by perturbation

theory of linear operators, we obtain the assertion.

4 Nonlinear DOEs with VMO coefficients

Let, at first, consider the linear BVP in a moving domain (,b(s)),

a(x)u()(x) +A(x)u(x) +A(x)u()(x) +A(x)u(x) =f(x), ()

Lju= mj

i=

αjiu(i)() +βjiu(i)(b)

= , j= , ,

whereais a complex valued function, andA=A(x),Ai=Ai(x) are possible operators in a

Banach spaceE, whereb(s) is a positive continuous independent ofu. Theorem  implies the following result.

Result  Let all the conditions of Theorem  be satisfied. Then the problem (), for f ∈Lp_{(,b(s);E),p∈(,∞), λ∈S}

ϕ and for large enough|λ|, has a unique solutionu∈ W,p(,b;E(A),E) and the following coercive uniform estimate holds:



i=

|λ|–i_u(i)

Lp_(,b;E)+AuLp_(,b;E)≤ f_Lp_(,b;E).

Proof Really, under the substitutionτ =xb(s), the moving boundary problem () maps to the following BVP with a parameter in a fixed domain (, ):

b–(s)a˜(τ)u()(τ) + (A˜ +λ)u(τ) +b–(s)A˜(τ)u()(τ) +A˜(τ)u(τ) =f(τ),

mj

i=

b–i(s)αjiu(i)() +βjiu(i)()

= , j= , ,

where

τ∈(, ), A˜ =Aτb–(s), A˜i=Ai

τb–(s).

Consider the nonlinear problem

a(x)u()(x) +B(x,u,ux)u(x) =F(x,u,ux), () mk

i=

αkiu(i)() +βkiu(i)(a)

= , k= , ,

wheremk∈ {, },αki,βkiare complex numbers,x∈(,b), wherebis a positive number in

(,b].

In this section, we will prove the existence and uniqueness of a maximal regular solution of the nonlinear problem (). AssumeAis aϕ-positive operator in a Banach spaceE. Let

X=Lp(,b;E), Y=W,p,b;E(A),E,

Ej=

E(A),E_σ

j,p, σj=

 +jp

p , X=



j= Ej.

Remark  By using [, §.], we obtain that the embeddingDj_Y∈E

jis continuous and

there exists the constantCsuch that forw∈Y,W={w,w},wj=Djw(·),j= , ,

wX,∞= 

j=

Djw_{C([,b],Ej)}= sup

x∈[,b] 

j=

Djw(x)_Ej≤CwY.

Condition  Assume the following are satisfied: () η= (–)m_α

β– (–)mαβ= anda(x)is a positive continuous function on[,b], a() =a(b);

() Eis a UMD space andp∈(,∞);

() F(x,υ,υ) : [,b]×X→Eis a measurable function for eachυi∈Ei,i= , ;

F(x,·,·)is continuous with respect tox∈[,b]andf(x) =F(x, )∈X. Moreover, for eachR> , there existsμRsuch that

F(x,U) –F(x,U¯)_E≤μRU–U¯ X,

whereU={u,u}andU¯ ={¯u,u¯}for a.a.x∈[,b],ui,u¯i∈Eiand

UX≤R, ¯UX≤R.

() forU={u,u} ∈X, the operatorB(x,U)isR-positive inEuniformly with respect tox∈[,b];B(x,U)B–_(x_{,U)∈C([,b];B(E)), where the domain definition} D(B(x,U))does not depend onxandU;B(x,W) : (,b)×X→B(E(A),E)is continuous, whereA=A(x) =B(x,W)for fixedW={w,w} ∈X;

() for eachR> , there is a positive constantL(R)such that

[B(x,U) –B(x,U¯)]υE≤L(R)U–U¯ XAυEforx∈(,b),U,U¯ ∈X,

UX, ¯UX≤Randυ∈D(A)andA() =A(b).

Theorem  Let Conditionhold.Then there is b∈(,b]such that the problem()has a unique solution that belongs to the space W

Proof Consider the linear problem

–a(x)w()(x) +A(x) +dw(x) =f(x), ()

Lkw= mk

i=

αkiw(i)() +βkiw(i)(b) = ,

where

f(x) =F(x, ), x∈(,b).

By virtue of Result , the problem () has a unique solution for allf ∈Xand for sufficiently larged>  that satisfies the following

wY≤CfX,

where the constantC does not depend onf ∈Xandb∈(,b]. We want to solve the problem () locally by means of maximal regularity of the linear problem () via the contraction mapping theorem. For this purpose, letwbe a solution of the linear BVP (). Consider a ball

Br=

υ∈Y,Lkυ= ,k= , ,υ–wY ≤r

.

Forυ∈Br, consider the linear problem

–a(x)u()(x) +Au(x) +du=F(x,V) +B(,W) –B(x,V)υ, () Lku=

mk

i=

αkiu(i)() +βkiu(i)(b) = ,

where

V=υ,υ(), W=w,w().

Define a mapQonBr byQυ=u, whereu is a solution of the problem (). We want

to show thatQ(Br)⊂Br and thatQis a contraction operator provided bis sufficiently

small andris chosen properly. For this aim, by using maximal regularity properties of the problem (), we have

Qυ–wY=u–wY ≤CF(x,V) –F(x, )_X+B(,W) –B(x,V)

υ_X. By assumption (), we have

B(,W)υ–B(x,V)υ_X

≤ sup

x∈[,b]

B(,W) –B(x,W)υ_B(X

,X)+B(x,W) –B(x,V)B(X,X)υY

≤δ(b) +L(R)W–V∞,X

υ–wY+wY

≤δ(b) +L(R)Cυ–wY+υ–wY

υ–wY+wY

≤δ(b) +L(R)[Cr+r]

r+wY

,

where

δ(b) = sup

x∈[,b]

B(,W) –B(x,W)_B(X ,X).

Bearing in mind

F(x,V) –F(x, , )_E≤δ(b) +F(x,V) –F(x,W)_E+F(x,W) –F(x, )_E ≤δ(b) +μR

υ–wY+wY

μRC

υ–wY+wY

≤μR

Cr+wY

,

whereR=Cr+wYis a fixed number. In view of the above estimates, by a suitable choice

ofμR,LRand for sufficiently smallb∈[;b), we have

Qυ–wY≤r,

i.e.,

Q(Br)⊂Br.

Moreover, in a similar way, we obtain

Qυ–Qυ¯Y ≤C

μRC+Ma+L(R)

υ–wY+Cr

+L(R)C

r+wY

υ–υ¯Y

+δ(b).

By a suitable choice ofμR, LR and for sufficiently smallb∈(,b), we obtain Qυ– Qυ¯Y<ηυ–υ¯Y,η< ,i.e.,Qis a contraction operator. Eventually, the contraction

map-ping principle implies a unique fixed point ofQinBrwhich is the unique strong solution

u∈Y.

5 Boundary value problems for anisotropic elliptic equations with VMO coefficients

The Fredholm property of BVPs for elliptic equations with parameters in smooth domains were studied,e.g., in [, ], also, for nonsmooth domains, these questions were investi-gated,e.g., in [].

Let⊂Rnbe an open connected set with a compactCm-boundary∂. Let us con-sider the nonlocal boundary value problems on a cylindrical domainG= (, )×for the following anisotropic elliptic equation withVMOtop-order coefficients:

(L+λ)u=sa(x)∂ _u

∂x +s 

_d(x,y)∂u

∂x+d(x,y)u

+

|α|≤m

Lku= mk

i= sμi_α

kiu(xi)(,y) +βkiu(xi)(,y)

= , k= , , ()

Bju=

|β|≤mj

bjβ(y)Dβyu(x,y) = , x∈(, ),y∈∂,j= , , . . . ,m, ()

wheresis a positive parameter,a,diare complex valued functions,αkiandβkiare complex

numbers,

Dj= –i

∂ ∂yj

, mk∈ {, }, y= (y, . . . ,yn), μi=

i +

 p.

IfG= (, )×,p= (p,p),Lp(G) will denote the space of allp-summable scalar-valued functions with a mixed norm (see,e.g., [, §]),i.e., the space of all measurable functions f defined onG, for which

fLp_(G)= 



_f(x,y)p dy

p p

dx 

p

<∞.

Analogously,W,m,p_{(G) denotes the anisotropic Sobolev space with the corresponding}

mixed norm [, §].

Theorem  Let the following conditions be satisfied;

() a,d∈VMO∩L∞(R),a() =a(),–a∈S(ϕ),μ(x)= ,a.e.x∈[, ]; () Reωk= and _ωλ

k ∈S(ϕ)forλ∈S(ϕ),≤ϕ<π,k= , a.e.x∈[, ];

() d∈L∞,d(·,y)d  –ν

 (·)∈L∞(, )for a.e.y∈and <ν<;

() aα∈VMO∩L∞(Rn)for each|α|= mandaα∈[L∞+Lγk]()for each|α|=k< m withrk≥qandm–k>_rl

k;

() bjβ∈Cm–mj(∂)for eachj,βandmj< m,

m j=bjβ(y

)σj= ,for|β|=mj,y

∈∂G, whereσ= (σ,σ, . . . ,σn)∈Rnis a normal to∂G;

() fory∈ ¯,ξ∈Rn_{,ν∈S(ϕ),ϕ∈(,π),|ξ|+|ν| = letν+}

|α|=maα(y)ξα= ;

() for eachy∈∂,a local BVP in local coordinates corresponding toy

ν+

|α|=m

aα(y)Dαϑ(y) = ,

Bjϑ=

|β|=mj

bjβ(y)Dβu(y) =hj, j= , , . . . ,m

has a unique solutionϑ∈C(R+)for allh= (h,h, . . . ,hn)∈Rn,and forξ∈Rn–

with|ξ|+|ν| = . Then

(a) for allf ∈Lp_{(G),λ∈S(ϕ)and sufficiently large|λ|,the problem()-()has a}

unique solutionubelonging toW,m,p_{(G)and the following coercive uniform}

estimate holds:



i=

si|_λ|–i∂

i_u

∂i_x

Lp_(G)

+

|β|=m

Dβ

yuLp_(G)≤CfLp_(G);

Proof LetE=Lp_{(). Then by virtue of [], the part () of Condition  is satisfied.} Con-sider the operatorAacting inLp_{() defined by}

D(A) =Wm,p_(;B

ju= ), Au=

|α|≤m

aα(y)Dαu(y).

Forx∈, also consider operators inLp₍₎ D(Ai) =Wm,p(;Bju= ),

A(x)u=d(x,y)u(y), A(x)u=d(x,y)u(y).

The problem ()-() can be rewritten in the form (), whereu(x) =u(x,·),f(x) =f(x,·) are functions with values inE=Lp_{(). By virtue of [], the problem}

νu(y) +

|α|≤m

aα(y)Dα_yu(y) =f(y),

Bju=

|β|≤mj

bjβ(y)Dβ_yu(y) = , j= , , . . . ,m

has a unique solution forf ∈Lp_{() and argν}∈_S(ϕ),|_ν| → ∞_{. Moreover, in view of [,} Theorem .], the operatorAisR-positive inLp_{(),i.e., Condition  holds. Moreover, it is} known that the embeddingWm,p₍₎⊂_Lp_{() is compact (see,e.g., [, Theorem ..]).} Then, by using interpolation properties of Sobolev spaces (see,e.g., [, §]), it is clear that the condition () of Theorem  is fulfilled too. Then from Theorems , , the assertions

are obtained.

6 Systems of differential equations with VMO coefficients

Consider the nonlocal BVPs for infinity systems of parameter-differential equations with

principalVMOcoefficients,

sa(x)u()_m(x) +

N

j= s_b

mj(x)u()j (x) + N

j=

dmj(x)uj(x) +

dm(x) +λ

um(x)

=fm(x), x∈(, ),m= , , . . . ,N, ()

Lku= mk

i= sμi_α

kiu(mı)() +βkiu(mi)()

= , k= , , ()

wheresis a positive parameter,a,bmj,dmjare complex valued functions,Nis a finite or

infinite natural number,αkiandβkiare complex numbers,μi=_i +__p.

Let

d(x) =dm(x)

, dm> , u={um}, Du={dmum}, m= , , . . . ,∞,

lq(D) =

u:u∈lq(N),ulq(D)=Dulq=

_N

m=

|dmum|q



q

<∞

,