Optimal regular differential operators with variable coefficients and applications

R E S E A R C H

Open Access

Optimal regular differential operators with

variable coefficients and applications

Veli 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 high-order elliptic differential-operator equations withVMOcoefficients are studied. For the linear case, the uniform coercivity property of parameter-dependent boundary value problems is obtained inLpspaces. Then, the existence and uniqueness of a strong solution of the boundary value problem for a high-order nonlinear equation are established. In application, the maximal regularity properties of the anisotropic elliptic equation and the system of equations withVMOcoefficients are derived.

AMS Subject Classification: 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(m)(x) +A(x)u(x) +

m–

k=

skm_A

k(x)u(k)(x) +λu(x) =f(x), ()

and the nonlinear equation

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

where a is a complex-valued function, s is a positive and λ is a complex parameter; A=A(x),Ak=Ak(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(vanishing mean

os-cillation). Sarason classVMOwas at first defined in []. In the recent years, there has been considerable interest to elliptic and parabolic equations withVMOcoefficients. This is mainly due to the fact thatVMOspaces contain as a subspaceC(¯) that ensures the extension ofLp-theory of operators with continuous coefficients to discontinuous

coef-ficients (see,e.g., [–]). On the other hand, the Sobolev spacesW,n_{() andW}σ,σn_(),

 <σ< , are also contained inVMO. Global regularity of the Dirichlet problem for ellip-tic equations withVMOcoefficients has been studied in [–]. We refer to the survey [], where excellent presentation and relations with 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 (differ-ential equations) 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 coef-ficients were studied,e.g., in [, , , ].

Here the equation with top-orderVMO-operator coefficients is considered in abstract spaces. We will 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 onlyp,E,mandAsuch that

m

i=

smi |_λ|–im_u(i)

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

Note that the principal part of a corresponding differential operator is non self-adjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent and Fredholmness are established. Then, the existence and uniqueness of the above nonlinear problem are de-rived. In application, we study maximal regularity properties of anisotropic elliptic equa-tions in mixedLp_{spaces and 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 the estimate of solutions. Therefore we use the modern harmonic analy-sis elements, e.g., the Hilbert operators and the commutator estimates in E-valued Lp

spaces, embedding theorems of Sobolev-Lions spaces and semigroup 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 main assertions.

2 Notations and background

Throughout the paper, we setEa 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_{,|B|is the Lebesgue measure ofBandf} Bis the

Forf ∈BMO(E) andr> , we set

sup

ρ≤r

B

f(x) –fB_Edx=η(r),

whereBranges in the class of balls with radiusρ.

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

η(r) theVMOmodulus off.

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

The Banach spaceEis called aUMD-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,lp spaces

and Lorentz spacesLpq,p,q∈(,∞).

Let

Sϕ=

λ∈C,|argλ| ≤ϕ∪ {}, ≤ϕ<π.

A linear operatorAis said to beϕ-positive (or positive) in a Banach spaceEwith bound M>  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. SometimesA+λIwill be written asA+λand denoted byAλ. It is known [, §..] that there exist fractional powersAθ _{of the positive operatorA. LetE(A}θ₎ denote the spaceD(Aθ_{) with the graphical norm}

uE(Aθ₎=

up+Aθup



p_{, }≤_p<∞_{, –}∞_<θ<∞_.

LetEandEbe two Banach spaces. A setW⊂L(E,E) is calledR-bounded (see [,

]) if there is a positive constantCsuch that for allT,T, . . . ,Tm∈Wandu,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

[, ].

LetS(Rn_{;E) denote the Schwartz class,i.e., the space of allE-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

E=E=E, it will be denoted byMpp(E).

Let

Un=

β= (β,β, . . . ,βn)∈Nn:βk∈ {, }

.

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

LA=

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 by σ_∞(E). Assume E and E are two Banach spaces and E is

continu-ously 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

)

possess-ing the generalized derivativesDm ku=

∂mu

∂xm_k such thatD m

ku∈Lp(;E) is endowed with the

norm

uWm,p_(;E

,E)=uLp(;E)+

n

k=

Dm_ku_Lp_(;E)<∞.

For E =E the space Wm,p(;E,E) will be denoted by Wm,p(;E). It is clear to see

that

Wm,p(;E,E) =Wm,p(;E)∩Lp(;E).

Letsbe a positive parameter. We define inWm,p_(;E

,E) the following parameterized

norm:

u_Wm,p

s (;E,E)=uLp(;E)+

n

k= _sDm

kuLp_(;E).

Functionu∈W,p_{(, ;E(A),E,L}

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

Theorem A Suppose the following conditions are satisfied:

() Eis a Banach space satisfying the multiplier condition with respect top∈(,∞)and Ais anR-positive operator inE;

() α= (α,α, . . . ,αn)aren-tuples of nonnegative integer numbers such that

κ=|α|

m ≤ 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

DαWm,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 conditions of TheoremAare satisfied.Assumeis 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;

() limy→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.

Proof Indeed, we observe that iff ∈VMO(E) withVMOmodulusη, there exists a constant Csuch thatf(x–y) –f(x)∗,E≤Cη(r) fory ≤rso that theE-valued usual mollifiers

converge tof in theBMOnorm. More precisely, givenf ∈VMO(E) withVMOmodulus

η(r), we can find a sequence ofE-valuedC∞ functions{fh}converging tof inE-valued

BMOspaces ash→ withVMOmoduliηh such thatηh≤η(r). In a similar way, other

cases are derived.

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:

Theorem A Assume all 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

H[a,f]_Lp_(,r;E)≤MεfLp_(,r;E), r∈(,δ).

Theorem A Let E be a UMD space,p∈(,∞)and A(·)uniformly R-positive in E.

More-over,let A(·)A–_(x

)∈L∞(R;L(E))∩BMO(L(E)),x∈R.Then the following commutator

operator is bounded in Lp_(R;E):

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.

Note that singular integral operators inE-valuedLp_{spaces were studied,e.g., in [].}

Theorem A Assume all conditions of TheoremAare satisfied andηis a VMO modulus

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 parameter-dependent DOE with constant coefficients

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

Lku=

νk

i=

sμi_α

kiu(i)() +βkiu(i)()

=fk, k= , , . . . , m, ()

whereνk∈ {, , . . . , m– },a,αki,βkiare complex numbers,μi=_mi +_mp ,θk=_ν_mk +_mp ,

sis a positive andλis a complex parameter;Aλ=A+λandAis a linear operator inE. Let

ω,ω, . . . ,ωmbe roots of the equationaωm+  = , [υij] be a m-dimensional matrix and η=|[υij]|be a determinant of the matrix [υij], where

υij=αj(–ωi)νj, i= , , . . . ,m, υij=βjω

νj

i , i=m+ ,m+ , . . . , m,

αk=αkmk, βk=βkmk, k,j= , , . . . , m.

It is known that (see,e.g., [, §.]) if the operatorAisϕ-positive inE, then operators

ωks–

 mA

 m

λ ,k= , , . . . , mgenerate the following analytic semigroups:

Uλs(x) =exp

xωs–m_A

 m λ

, Uλs(x) =exp

xωs–m_A

 m λ

, . . . ,

Umλs(x) =exp

xωms–

 m_A

 m λ

, Um+λs(x) =exp

–( –x)ωm+s–

 m_A

 m λ

Um+λs(x) =exp

–( –x)ωm+s–

 m_A

 m λ

, . . . ,

Umλs(x) =exp

–( –x)ωms–

 m_A

 m λ

forλ∈S(ϕ).

Let

Ek=

E(A),E_θ k,p.

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

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<πandη= ;

() |argωj–π| ≤π_ –ϕ,j= , , . . . ,m,|argωj| ≤π_ –ϕ,j=m+ , . . . , mand ωλj ∈S(ϕ).

Then

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

a unique solutionu∈Wm,p(, ;E(A),E). Moreover, the following coercive uniform esti-mate holds:

m

i=

smi |_λ|–im_u(i)

Lp_(,;E)+AuLp_(,;E)≤C

fLp_(,;E)+

m

k= fkEk

.

() Forfk= , the solution is represented as

u(x) =





Gλs(x,y)f(y)dy,Gλs(x,y)

=

m

k= m

j=

νk

i=

Bkij(λ)

s–Aλ

– 

m(m+νk–i–)

Ujλs(x)Ukλs( –y)

+Uλs(x–y), ()

whereBkij(λ) are uniformly bounded operators inEand

Uλs(x–y) = ⎧ ⎨ ⎩

a–_{s–__mA–(–m) λ

m

i=(–)m+iPi–Uiλs(x–y),x≥y},

–a–{s–m_A–(–

 m) λ

m

i=m+(–)m+iP–i Uiλs(x–y),x≤y},

where

Pi= (ωi–ω)· · ·(ωi–ωi–)(ωi+–ωi)· · ·(ωm–ωi), i= , , . . . , m.

Consider the BVP for DOE with variable coefficients

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

Lku=

νk

i=

sθk_α

kiu(i)() +βkiu(i)()

wherea=a(x) is a complex-valued function,mk∈ {, , . . . , m– },αki,βkiare complex

numbers,sis a positive andλis a complex parameter,θk=_ν_mk +_mp andA(x) is a linear

operator inE.

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

[υij] be a m-dimensional matrix andη(x) =|[υij]|be a determinant of the function matrix

[υij], where

υij=υij(x) =αj(–ωi)νj, i= , , . . . ,m, υij=βjω

νj

i , i=m+ ,m+ , . . . , m,

αk=αkmk, βk=βkmk, k,j= , , . . . , m.

In the next theorem, we consider the case when principal coefficients are continuous. The well-posedness of this problem occurs in studying of equations withVMO coeffi-cients. From [, Theorem ] and [, Theorem .], we get the following.

Theorem A_ Suppose the following conditions are satisfied:

() Eis a Banach space satisfying the multiplier condition with respect top∈(,∞); () |argωj–π| ≤π_ –ϕ,j= , , . . . ,m,|argωj| ≤π_ –ϕ,j=m+ , . . . , mand ωλj ∈S(ϕ)

a.e.x∈(, );

() a∈C[, ],a() =a()andη(x)= for a.e.x∈[, ]; () A(x)is a uniformlyR-positive operator inEand

A(·)A–(x)∈C

[, ];L(E), x∈(, ), A() =A().

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

solu-tionu∈W,p(, ;E(A),E) of the problem (). Moreover, the following coercive uniform estimate holds:

m

i=

smi |_λ|–im_u(i)

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

3 DOEs withVMOcoefficients

Consider the principal part of the problem ()

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

Lku= mk

i=

sμi_α

kiu(i)() +βkiu(i)()

= , k= , , . . . , m. ()

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

() a∈VMO∩L∞(R),ηis aVMOmodulus ofa;

() |argωj–π| ≤π_ –ϕ,j= , , . . . ,m,|argωj| ≤π_ –ϕ,j=m+ , . . . , mand ωλj∈S(ϕ) for≤ϕ<π,η(x)= a.e.x∈[, ];

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

A(·)A–(x)∈L∞

() 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m,p_{(, ;E(A),E),we have}

u(ν)_{(x) =} 



νλs(x,x–y)

a(x) –a(y)u(m)(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

νλs(x,x–y)

=

m

k= m

j=

νk

i=

B_kνij(λ)s–Aλ

– 

m(m+νk+ν–i–)

Ujλs(x)Uiλs( –y)

+Uνλs(x–y).

Here B_kij(λ)are uniformly bounded operators and

Uλs(x–y) = ⎧ ⎨ ⎩

a–_{s–+_mν_A–(––mν) λ

m

i=(–)m+i(ωi)νPi–Uiλs(x–y),x≥y},

–a–_{s–+mν_A–(–

–ν

m) λ

m

i=m+(–)m+ν+i(ωi)νP–i Uiλs(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(m)(x) +

A(x) +λ

u(x) =f(x), x∈(, ),

Lku=

νk

i=

sμi_α

kiu(i)() +βkiu(i)()

= , k= , , . . . , m. ()

Letu∈C(m)_{([, ];E(A)) be a solution of the problem (). Taking into account the}

equal-ityLu= (L–L)u+Luand Theorem A, we get

u(ν)(x) =





νλs(x,x–y)

a(x) –a(y)

u(m)(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 the above, we get () foru∈C(m)([, ];E(A)). Then a density argument

and Theorem Agive the conclusion for

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

(Lb+λ)u=sa(x)u(m)(x) +

A(x) +λu(x) =f(x), x∈(,b),

Lbku=

ν_k

i=

sμi_α

kiu(i)() +βkiu(i)(b)

= , k= , , . . . , m. ()

Theorem  Suppose Conditionis satisfied.Then there exists a number b∈(, )such

that the following uniform coercive estimate holds:

m

i=

smi |_λ|–im_u(i)

Lp_(,b;E)+AuLp_(,b;E)≤C(L_b+λ)u

Lp_(,b;E) ()

for u∈Wm,p_{(,b;E(A),E),λ∈S(ϕ)with large enough|λ|.}

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

u(ν)(x) =

b



νbλs(x,x–y)

a(x) –a(y)u()(y)

+A(x) –A(y)u(y) +f(y)dy+f(x), ()

where

νbλs(x,x–y)

=

m

k= m

j=

νk

i=

Bνkij(λ)

s–Aλ

– 

m(+νk+ν–i–)

Ujλ(x)Uiλ(b–y)

+Uνλs(x–y); ()

hereBkij(λ) are uniformly bounded operators, and

Uνλs(x–y) = ⎧ ⎨ ⎩

a–_{s–+mν_A–(–

– m) λ

m

i=(–)m+iP–i Uiλs(x–y),x≥y},

–a–_{s–__mA–(–m) λ

m

i=m+(–)m+iP–i Uiλs(x–y),x≤y}.

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)

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 proper-ties ofAand the analyticity of semigroupsUkλs(x) in a similar way as in [, Theorem .],

we get

BiλfLp_(,b;E)≤M|λ|–

m–i

m _f

Lp_(,b;E), i= , . ()

Since the Hilbert operator is bounded inLp(R;E) for aUMDspaceE, 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|–

m–i

m _u()

Lp_(,b;E),

DiλuLp_(,b;E)≤Mε|λ|–

m–i

m _A(x)u

Lp_(,b;E), i= , , , ()

λuLp_(,b;E)≤Mεu(m)

Lp_(,b;E), λuLp_(,b;E)≤MεA(x_)u

Lp_(,b;E).

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

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

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

m

i=

smi |_λ|–im_u(i)

Lp_(,;E)+AuLp_(,;E)≤C(L+λ)u

Lp_(,;E)+uLp_(,;E). ()

Proof This fact is shown by covering and flattening argument, in a similar way as in The-orem A. Particularly, by partition of unity, the problem is localized. Choosing diameters

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

coeffi-cients [, ]), we obtain the assertion.

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

D(Qs) =Wm,p

, ;E(A),E,Lk

, Qsu=sa(x)u(m)+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m,p_{(, ;E(A),E).Moreover,the}

following coercive uniform estimate holds:

m

i=

smi |_λ|–im_u(i)

Proof First, let us show that the operatorQ+λhas a left inverse. Really, it is clear to see that

uLp_(,;E)= 

|λ|(Qs+λ)uLp_(,;E)+QsuLp_(,;E).

By Theorem Aforu∈Wm,p(, ;E(A),E), we have

QsuLp_(,;E)≤Cu

Wsm,p(,;E(A),E).

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

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

m

i=

smi |_λ|–im_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)

. ()

In view of () foru∈Wm,p(, ;E(A),E), we get

m

i=

smi |_λ|–im_u(i)

Lp(,;E)+AuLp(,;E)≤C(Qs+λ)uLp(,;E). ()

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

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

allf∈Lp(, ;E). This fact can be derived in a standard way, approximating the equation with a similar one with smooth coefficients [, ]. More precisely, by virtue of [, Theo-rem .],UMDspaces satisfy the multiplier condition. Moreover, by part () of Lemma A,

givena∈VMOwithVMOmodulesη(r), we can find a sequence of mollifiers functions

{ah}converging toainBMOash→ withVMOmodulusηhsuch thatηh(r)≤η(r). In a

similar way, it can be derived for the operator functionA(x)A–(x)∈VMO(L(E)).

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

uni-form estimate:

m

i=

sim|_λ|–imd

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 an analytic semigroup forϕ∈(}π

,π) (see,e.g., [, §..]).

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

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

Theorem  Assume the following: () Conditionholds;

() for anyε> ,there isC(ε) > such that for a.e.x∈(, )and

Ak(x)u_E≤εu(E(A),E) k

m,∞

+C(ε)u, u∈E(A),E ,∞.

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

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

m

i=

smi |_λ|–im_u(i)

Lp_(,;E)+AuLp(,;E)≤CfLp(,;E). ()

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

fromLp(, ;E) toWm,p(, ;E(A),E). PutOsu=Qsu+Qu, where

Qu= m–

k=

skm_A

k(x)u(k)(x), u∈Wm,p

, ;E(A),E,Lk

.

By the second assumption and Theorem A, there is a smallεandC(ε) such that _s k

m_A

ku(k)_Lp_(,;E)≤Cs k

m_A–km_u

Lp_(,;E)

≤εu_Wm,p

s (,;E(A),E)+C(ε)uL

p_(,;E). ()

By Theorem , the operatorQs+λhas a bounded inverse (Qs+λ)–fromLp(, ;E) to

Wm,p_{(, ;E(A),E) for sufficiently large|λ|. So, () implies the following uniform}

esti-mate:

Q(Qs+λ)–_L(Lp_(,;E))< .

It is clear to see that

(Os+λ) =

I+Q(Qs+λ)–

(Qs+λ), (Os+λ)–= (Q+λ)–

I+Q(Qs+λ)– –

.

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

Theorem  implies the following result.

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

of the operatorOssatisfies the following sharp uniform estimate:

m

i=

sim|_λ|–imd

i

dxi(Os+λ) –

L(Lp_(,;E))

+A(Os+λ)–_L(Lp_(,;E))≤C

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

Lu=sa(x)u(m)(x) +A(x)u(x) +

m–

k=

skm_A

k(x)u(k)(x) =f(x), x∈(, ),

Lku=

νk

i=

sμi_α

kiu(i)() +βkiu(i)()

= , k= , , . . . , m. ()

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

∞(E).Then the problem ()is Fredholm from Wm,p_{(, ;E(A),E)into L}p_{(, ;E).}

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

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

s+λis

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

and by

per-turbation theory of linear operators, we obtain the assertion.

4 Nonlinear DOEs withVMOcoefficients

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

Lu=a(x)u(m)(x) +A(x)u(x) +

m–

k=

Ak(x)u(k)(x) =f(x), x∈(, ),

Lku=

νk

i=

αkiu(i)() +βkiu(i)(b)

= , k= , , . . . , m, ()

whereais a complex-valued function andA=A(x),Ak=Ak(x) are linear operators in a

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

Result  Let all conditions of Theorem  be satisfied. Then the problem () has a unique solutionu∈Wm,p_{(,b;E(A),E) forf ∈L}p_{(,b(s);E),p∈(,∞),λ∈S}

ϕwith large enough

|λ|, and the following coercive uniform estimate holds:

m

i=

|λ|–im_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 parameter in the fixed domain (, ):

b–m(s)a˜(τ)u(m)(τ) + (A˜+λ)u(τ) +

m–

k=

b–k(s)A˜k(τ)u(k)(τ) =f˜(τ),

mj

i=

b–i(s)αjiu(i)() +βjiu(i)()

= , j= , , . . . , m,

where

τ∈(, ), A˜ =Aτb–(s), A˜k=Ak

τb–(s),f˜(τ) =fτb–(s).

Consider the following nonlinear problem:

a(x)u(m)(x) +Bx,u,u(), . . . ,u(m–)u(x) =Fx,u,u(), . . . ,u(m–), ()

νk

i=

αkiu(i)() +βkiu(i)(b)

= , k= , , . . . , m,

whereνk∈ {, , . . . , m– },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m,p,b;E(A),E,

Ei=

E(A),E_σ

i,p, σi=  +ip

mp, X=

m–

i=

Ej.

Remark  By using [, §..], we obtain that the embeddingDjY∈Ejis continuous and

there exists a constantCsuch that forw∈Y,W={w,w, . . . ,wm–},wj=Djw(·),

wX,∞= m–

i=

Diw_C([,b],E j)= sup

x∈[,b] 

i=

Diw(x)_E

j≤CwY.

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

() a∈VMO∩L∞(R),a() =a(b);

() |argωj–π| ≤π_ –ϕ,j= , , . . . ,m,|argωj| ≤π_ –ϕ,j=m+ , . . . , mand _ωλ

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

() F(x,υ,υ, . . . ,υm–) : [,b]×X→Eis a measurable function for eachυi∈Ei,

i= , , . . . , m– ;F(x,·,·)is continuous with respect tox∈[,b]and f(x) =F(x, )∈X. Moreover, for eachR> , there existsμRsuch that

F(x,U) –F(x,U¯)_E≤μRU–U¯X,

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

and

UX≤R, ¯UX≤R.

() forU∈X, the operatorB(x,U)isR-positive inEuniformly with respect to

x∈[,b];B(x,U)B–_(x_,U)∈L

∞(, ;L(E))∩VMO(L(E)),x∈(, ), where domain

definitionD(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, . . . ,wm–} ∈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,

Theorem  Let Conditionhold.Then there is b∈(,b]such that the problem()has

a unique solution belonging to space Wm,p_{(,b;E(A),E).}

Proof Consider the linear problem

–a(x)w(m)(x) +A(x) +dw(x) =f(x), ()

Lkw=

νk

i=

αkiw(i)() +βkiw(i)(b) = , k= , , . . . , m,

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= , , . . . , m,υ–wY≤r

.

Forυ∈Br, consider the linear problem

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

Lku=

νk

i=

αkiu(i)() +βkiu(i)(b) = , k= , , . . . , m,

where

V=υ,υ(), . . . ,υ(m–), W=w,w(), . . . ,w(m–).

Define a mapQonBr byQυ=u, where uis 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(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).

Bear 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 withVMO coefficients

The Fredholm property of BVPs for elliptic equations with parameters in smooth domains were studied,e.g., in [, , ]; also, for non-smooth domains, these questions were investigated,e.g., in [].

Let⊂Rn_{be an open connected set with compactC}l_{-boundary∂. Let us consider}

fol-lowing anisotropic elliptic equation withVMOtop-order coefficients:

(L+λ)u=sa(x)∂

m_u

∂xm + m–

k=

skm_dk(x,y)∂

k_u

∂xk

+

|α|≤l

aα(y)Dα

yu+λu=f(x,y), x∈(, ),y∈, ()

Lku=

νk

i=

sμi_α

kiu(xi)(,y) +βkiu(xi)(,y)

= , k= , , . . . , m, ()

Bju=

|β|≤lj

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

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

numbers,

Dj= –i ∂ ∂yj

, νk∈ {, , . . . , m– }, y= (y, . . . ,yn), μi=

i m+

 mp.

ForG= (, )×,p= (p,p), Lp(G) will denote the space of all p-summable

scalar-valued functions with a mixed norm (see,e.g., [, §]),i.e., the space of all measurable functionsf defined onG, for which

fLp_(G)=





f(x,y)pdy

p

p_ dx



p <∞.

Analogously,Wm,l,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();

() |argωj–π| ≤π_ –ϕ,j= , , . . . ,m,|argωj| ≤π_ –ϕ,j=m+ , . . . , mand _ωjλ ∈S(ϕ)

forλ∈S(ϕ),≤ϕ<π,η(x)= ,a.e.x∈[, ]; () dk∈L∞,dk(·,y)d

–_k_m–σk

 (·)∈L∞(, )for a.e.y∈and <σk<  –_k_m;

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

k;

() bjβ∈Cl–lj(∂)for eachj,βandmj< l, l

j=bjβ(y)σj= for|β|=lj,y∈∂G,where σ= (σ,σ, . . . ,σn)∈Rnis a normal to∂G;

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

|α|=laα(y)ξα= ;

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

ν+ |α|=l

aα(y)Dαϑ(y) = ,

Bjϑ=

|β|=lj

bjβ(y)Dβu(y) =hj, j= , , . . . ,l

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