NORMAL SOLVABILITY OF GENERAL LINEAR ELLIPTIC PROBLEMS

NORMAL SOLVABILITY OF GENERAL LINEAR ELLIPTIC PROBLEMS

A. VOLPERT AND V. VOLPERT Received 13 June 2004

The paper is devoted to general elliptic problems in the Douglis-Nirenberg sense. We obtain a necessary and sufficient condition of normal solvability in the case of unbounded domains. Along with the ellipticity condition, proper ellipticity and Lopatinsky condition that determine normal solvability of elliptic problems in bounded domains, one more condition formulated in terms of limiting problems should be imposed in the case of unbounded domains.

1. Introduction

In this work we study normal solvability of general elliptic problems in the Douglis- Nirenberg sense. If in the case of bounded domains with a sufficiently smooth boundary the normal solvability is completely determined by the conditions of ellipticity, proper el- lipticity, and Lopatinsky condition (see [2,3,19,20]), then in the case of unbounded do- mains one more condition related to behavior of solutions at infinity should be imposed.

If the coefficients of the operator have limits at infinity, and the domain is cylindrical or conical at infinity, then the additional condition is determined by the invertibility of limiting operators, that is of the operators with the limiting coefficients in the limiting domain. This situation is studied in a number of works for differential [4,11,21,25,26]

and pseudodifferential operators [15,17,16].

If the coefficients do not have limits at infinity but the domain is the wholeRⁿ, the notion of limiting operators was used in [12,13]. Previously it was used in the one- dimensional case to study differential equations with quasi-periodic coefficients [6,8,9, 14] (see also [18]).

In the case of arbitrary domains, we need to introduce the notion of limiting domains and limiting problems. In the case of general elliptic problems and H¨older spaces it is done in [23]. In [22] we study scalar elliptic problems in Sobolev spaces (see below). We obtain conditions of normal solvability in terms of uniqueness of solutions of limiting problems. In this work we generalize these results to the case of systems.

We should note that the choice of function spaces plays important role. We intro- duce a generalization of Sobolev-Slobodetskii spaces that will be essentially used in the

Copyright©2005 Hindawi Publishing Corporation Abstract and Applied Analysis 2005:7 (2005) 733–756 DOI:10.1155/AAA.2005.733

subsequent works where we consider nonhomogeneous adjoint problems, obtain a priori estimates for them and prove their normal solvability. These results are used to prove the Fredholm property of general elliptic problems. For the scalar equation the solvability conditions will be also formulated in terms of formally adjoint problems. We will use the approach developed in [10] for scalar elliptic problems in bounded domains.

1.1. Function spaces. Sobolev spacesW^s,pproved to be very convenient in the study of elliptic problems in bounded domains. But more flexible spaces are needed for elliptic problems in unbounded domains. We need some generalization of the spaceW^s,p. More exactly, we mean such spaces which coincide withW^s,pin bounded domains but have a prescribed behavior at infinity in unbounded domains. It turns out that such spaces can be constructed for arbitrary Banach spaces of distributions (not only Sobolev spaces) as follows.

Consider first functions defined onRⁿ. As usual we denote byDthe space of infinitely differentiable functions with compact support and byDits dual. LetE⊂Dbe a Banach space, the inclusion is understood both in algebraic and topological sense. Denote byEloc

the collection of allu∈Dsuch that f u∈Efor all f ∈D. Letω(x)∈D, 0≤ω(x)≤1, ω(x)=1 for|x| ≤1/2,ω(x)=0 for|x| ≥1.

Definition 1.1. Eq(1≤q≤ ∞) is the space of allu∈Elocsuch that uEq:=

Rⁿ

u(·)ω(· −y)^q_Ed y 1/q

<∞, 1≤q <∞, uE∞:=sup

y∈Rⁿ

u(·)ω(· −y)_E<∞.

(1.1)

It is proved thatEqis a Banach space. IfΩis a domain inRⁿ, then by definitionEq(Ω) is the space of restrictions ofE_qtoΩwith the usual norm of restrictions. It is easy to see that ifΩis a bounded domain, then

E_q(Ω)=E(Ω), 1≤q≤ ∞. (1.2)

In particular, ifE=W^s,p, then we denoteWq^s,p=Eq(1≤q≤ ∞). It is proved that W^s,pp =W^s,p (s≥0, 1< p <∞). (1.3) Hence the spacesWq^s,p generalize the Sobolev spaces (q <∞) and the Stepanov spaces (q= ∞) (see [8,9,14]).

IfE=L^p, thenLq^p=Eq. It can be proved that ifubelongs toL^p locally and|u(x)| ≤ K|x|^−αfor|x|sufficiently large, whereKis a positive constant, andαq > n, thenu∈Lq^p. Unlike the spacesL^p for which there is no embeddingL^p(Rⁿ) inL^p1(Rⁿ) for any 1< p, p1<∞,p=p1, it is easy to prove that

Lq^p

Rⁿ⊂Lq^p1¹

Rⁿ p≥p1,q≤q1

. (1.4)

1.2. Elliptic problems. Consider the operators Aiu=

N k=1

|α|≤αik

a^α_ik(x)D^αuk, i=1,. . .,N,x∈Ω, Bju=

N k=1

|β|≤βjk

b^β_jk(x)D^βuk, i=1,. . .,m,x∈∂Ω,

(1.5)

whereu=(u1,. . .,u_N),Ω⊂Rⁿis an unbounded domain that satisfy certain conditions given below. According to the definition of elliptic operators in the Douglis-Nirenberg sense [5] we suppose that

α_ik≤s_i+t_k, i,k=1,. . .,N, β_jk≤σ_j+t_k, j=1,. . .,m,k=1,. . .,N (1.6) for some integerssi,tk,σjsuch thatsi≤0, maxsi=0,tk≥0.

Denote byEthe space of vector-valued functionsu=(u1,. . .,uN), whereujbelongs to the Sobolev spaceW^l+tj,p(Ω), j=1,. . .,N, 1< p <∞,lis an integer,l≥max(0,σj+ 1), E=Π^Nj=1W^l+tj,p(Ω). The norm in this space is defined as

uE= N j=1

uj

W^{l+t j,p}(Ω). (1.7)

The operatorA_iacts fromEtoW^l−si,p(Ω), the operatorB_jacts fromEtoW^{l−σj−1/ p,p}(∂Ω).

Denote

L=

A1,. . .,AN,B1,. . .,Bm ,

F=Π^N_i=1W^l−si,p(Ω)×Π^m_j=1W^{l−σj−1/ p,p}(∂Ω). (1.8) We will consider the operatorLas acting fromE_∞toF_∞.

Throughout the paper we assume that the operatorLsatisfies the condition of uniform ellipticity.

1.3. Limiting problems. We recall that the operator is normally solvable with a finite dimensional kernel if and only if it is proper, that is the inverse image of a compact set is compact in any closed bounded set. In this work we obtain necessary and sufficient conditions for a general elliptic operator to satisfy this property. Consider as example the following operator

Lu=a(x)u+b(x)u+c(x)u (1.9) acting fromH²(R) toL²(R). If we assume that there exist limits of the coefficients of the operator at infinity, then we can define the operators

L_±u=a_±u+b_±u+c_±u, (1.10) where the subscripts + and−denote the limiting values at +∞and−∞, respectively. As it

is well known, the operatorLsatisfies the Fredholm property if and only if the equations L_±u=0 do not have nonzero bounded solutions. We can easily write down this condition explicitly:

−a_±ξ²+b_±iξ+c_±=0 (1.11)

if we look for solutions of this equations in the formu=exp(iξ).

This simple approach is not applicable for general elliptic problems where limits of the coefficients may not exist and the domain can be arbitrary. In the next section we will define limiting problems in the general case. Construction of limiting domains can be briefly described as follows. Letxk∈Ωbe a sequence, which tends to infinity. Consider the shifted domainsΩk corresponding to the shifted characteristic functionsχ(x+xk), whereχ(x) is the characteristic function of the domainΩ. Consider a ballB_r⊂Rⁿwith the center at the origin and with the radiusr. Suppose that for allkthere are points of the boundaries∂Ωk insideBr. If the boundaries are sufficiently smooth, we can expect that from the sequence Ωk∩B_r we can choose a subsequence that converges to some limiting domainΩ∗. After that we take a larger ball and choose a convergent subsequence of the previous subsequence. The usual diagonal process allows us to extend the limiting domain to the whole space.

To define limiting operators we consider shifted coefficientsa^α(x+x_k),b^α_j(x+x_k) and choose subsequences that converge to some limiting functions ˆa^α(x), ˆb^α_j(x) uniformly in every bounded set. The limiting operator is the operator with the limiting coefficients.

Limiting operators and limiting domains constitute limiting problems. It is clear that the same problem can have a family of limiting problems depending on the choice of the sequencexkand on the choice of both converging subsequences of domains and coeffi- cients.

We note that in the case whereΩ=Rⁿthe limiting domain is alsoRⁿ. In this case the limiting operators were introduced and used in [12,13,17,18].

1.4. Normal solvability. The following condition determines normal solvability of ellip- tic problems.

Condition NS. Any limiting problem

Luˆ =0, x∈Ω∗,u∈E_∞Ω∗

(1.12) has only zero solution.

It is a necessary and sufficient condition for general elliptic operators considered in H¨older spaces to be normally solvable with a finite dimensional kernel [23]. For scalar elliptic problems in Sobolev spaces it was proved in [22]. In this work we generalize these results for elliptic systems. More precisely, we prove that the elliptic operatorLis normally solvable and has a finite-dimensional kernel in the spaceW∞^l,p(1< p <∞) if and only if Condition NS is satisfied. Using this result it can be proved that the elliptic operatorL is Fredholm (if the limiting operators are invertible) in the spaceWq^l,pfor 1< p <∞and someq. This result will be published elsewhere.

It is easy to see how this condition is related to the condition formulated in terms of the Fourier transform. In fact, for operator (1.9) the nonzero solution of the limiting problemL_±u=0 has the formu0(x)=e^iξx, whereξ is the value for which the essential spectrum passes through 0. The functionu0(x) belongs obviously to the H¨older spaces and also to the spaceW∞^2,p(R). However it does not belong to the usual Sobolev space W^2,p(R). So Condition NS cannot be obtained in terms of usual Sobolev spaces (see also [22] for counter-examples inRⁿ). This is one of the reasons why it is more convenient to work withWq^s,pspaces.

2. A priori estimates in the spacesW∞^s,p

In this section, we define the spacesW∞^s,pand obtain a priori estimates of solutions, which are similar to those in usual Sobolev spaces.

Denote byW∞^k,p(Ω) the space of functions defined as the closure of smooth functions in the norm

u_W∞^k,p_(Ω)=sup

y∈ΩuW^k,p(Ω∩Qy). (2.1) HereΩis a domain inRⁿ,Q_yis a unit ball with the center aty, · W^k,p is the Sobolev norm. We note that in bounded domainsΩthe norms of the spacesW^k,p(Ω) andW∞^k,p(Ω) are equivalent. In the one-dimensional case withk=0 similar spaces were used in [8,9, 14]. This definition is equivalent toDefinition 1.1.

We suppose that the boundary∂Ωbelongs to the H¨older spaceC^k+θ, 0< θ <1, and that the H¨older norms of the corresponding functions in local coordinates are bounded independently of the point of the boundary. Then we can define the spaceW∞^{k−1/ p,p}(∂Ω) of traces on the boundary∂Ωof the domainΩ,

φ_W∞^{k−1/ p,p}_(∂Ω)=infv_W∞^k,p_(Ω), (2.2)

where the infimum is taken with respect to all functionsv∈W∞^k,p(Ω) equal φ at the boundary, andk >1/ p.

The spaceW∞^k,p(Ω) withk=0 will be denoted byL∞^p(Ω). We will use also the notations E_∞=Π^Nj=1W_∞^l+tj,p(Ω),

F_∞=Π^N_i=1W∞^l−si,p(Ω)×Π^m_j=1W∞^{l−σj−1/ p,p}(∂Ω). (2.3) We consider the operatorLdefined by (1.8) and denotel1=max(0,σj+ 1). We sup- pose that the integerlin the definition of the spaces is such thatl_≥l1, and the boundary

∂Ωbelongs to the classC^r+θwithrspecified in Condition D below.

Theorem2.1. Letu∈Π^Nj=1W∞^l1+tj,p(Ω). Then for anyl≥l1we haveu∈E_∞and

uE∞≤cLuF∞+u_L^p_∞(Ω) , (2.4) where the constantcdoes not depend onu.

Proof. Letω(x) be an infinitely differentiable nonnegative function such that ω(x)=1, |x| ≤1

2, ω(x)=0, |x| ≥1. (2.5)

Denote ωy(x)=ω(x−y). Suppose u(x) is a function satisfying the conditions of the theorem. Thenω_yu∈Π^Nj=1W∞^l1+tj,p(Ω). Since the support of this function is bounded, we can use now a priori estimates of solutions [1]:

ω_yu_E≤cLω_yu_F+ω_yu_Lp(Ω) , ∀y∈Rⁿ, (2.6) where the constantcdoes not depend ony. We now estimate the right-hand side of the last inequality. We have

A_iω_yu=ω_yA_iu+T_i, (2.7) where

T_i= N k=1

|α|≤αik

a^α_ik

β+γ≤α,|β|>0

c_βγD^βω_yD^γu_k, (2.8) andc_βγare some constants. If|τ| ≤l−s_i, then

D^τωyAiu_Lp(Ω)≤MAiu_W^l−si,p

∞ (Ω). (2.9)

For any>0 we have the estimate Ti

W^l−si,p(Ω)≤ N k=1

uk

W^l+tk,p(Ω∩Qy)+C N k=1

uk

L^p(Ω∩Qy)

≤uE∞+Cu_L^p_∞(Ω),

(2.10)

whereQ_yis a unit ball with the center aty.

Thus

A_iω_yu_Wl−si,p(Ω)≤MA_iu_Wl−si,p

∞ (Ω)+uE∞+Cu_L∞^p_(Ω). (2.11) Consider next the boundary operators in the right-hand side of (2.6). We have

Bj

ωyu=ωyΦj+Sj, (2.12)

whereΦj=B_ju,

Sj= N k=1

|β|≤βjk

b^β_jk

α+γ≤β,|α|>0

λαγD^αωyD^γuk, (2.13) andλ_αγare some constants.

There exists a functionv∈W_∞^l−σj,p(Ω) such thatv=Φion∂Ωand v_W^l−σ j,p

∞ (Ω)≤2Φj

W∞^{l−σ j−1/ p,p}(∂Ω). (2.14)

Sincev∈W∞^l−σj,p(Ω), thenωyv∈W^l−σj,p(Ω) and ω_yv_Wl−σ j,p(Ω)≤Mv

W∞^{l−σ j,p}(Ω) (2.15) with a constantMindependent ofv. Sinceωyv=ωyΦjon∂Ω, then

ωjΦj

W^{l−σ j−1/ p,p}(∂Ω)≤M1Φj

W∞^{l−σ j−1/ p,p}(∂Ω). (2.16)

Further,

S_jWl−σ j⁻1/ p,p(∂Ω)≤S_jWl−σ j,p(Ω)≤ N k=1

u_kWl+tk,p(Ω∩Qy)+C N k=1

u_kLp(Ω∩Qy)

≤uE∞+Cu_L∞^p_(Ω).

(2.17)

Thus Bj

ωyu_W^l−σ j⁻1/ p,p(∂Ω)≤MΦj

W∞^{l−σ j−1/ p,p}(∂Ω)+uE∞+Cu_L^p_∞(Ω). (2.18) From (2.6), (2.11), and (2.18) we obtain the estimate

ωyu_E≤cM2LuF∞+κuE∞+Cu_L∞^p(Ω) (2.19)

with some constantsM2andκ. Taking>0 sufficiently small, we obtain (2.4). The theo-

rem is proved.

3. Limiting problems

In this section, we define limiting domains and limiting operators. They determine limit- ing problems. We will restrict ourselves to the definitions and to the result, which we give without proofs, that will be used below. More detailed presentation including the proofs can be found in [22].

3.1. Limiting domains. In this section, we define limiting domains for unbounded do- mains inRⁿ, show their existence and study some of their properties. We consider an unbounded domainΩ⊂Rⁿ, which satisfies the following condition.

Condition D. For eachx0∈∂Ωthere exists a neighborhoodU(x0) such that:

(1)U(x0) contains a sphere with the radiusδand the centerx0, whereδis indepen- dent ofx0,

(2) there exists a homeomorphismψ(x;x0) of the neighborhoodU(x0) on the unit sphereB= {y:|y|<1}inRⁿsuch that the images ofΩU(x0) and∂Ω∩U(x0) coincide withB+= {y:yn>0, |y|<1}andB0= {y:yn=0, |y|<1}, respec- tively,

(3) the functionψ(x;x0) and its inverse belong to the H¨older spaceC^r+θ, 0< θ <1.

Their · r+θ-norms are bounded uniformly inx0. For definiteness we suppose thatδ <1. We assume also that

r≥maxl+t_i,l−s_i,l−σ_j+ 1, i=1,. . .,N, j=1,. . .,m. (3.1) The first expression in the maximum is used for a priori estimates of solutions, the second and the third will allow us to extend the coefficients of the operator (seeSection 3.3).

To define convergence of domains we use the following Hausdorffmetric space. LetM andNdenote two nonempty closed sets inRⁿ. Denote

σ(M,N)=sup

a∈M

ρ(a,N), σ(N,M)=sup

b∈N

ρ(b,M), (3.2)

whereρ(a,N) denotes the distance from a pointato a setN, and let

ρ(M,N)=maxσ(M,N),σ(N,M). (3.3) We denote byΞa metric space of bounded closed nonempty sets inRⁿwith the dis- tance given by (3.3). We say that a sequence of domainsΩmconverges to a domainΩin Ξlocif

ρΩ¯m∩B¯_R, ¯Ω∩B¯_R−→0, m−→ ∞ (3.4) for anyR >0 andB_R= {x:|x|< R}. Here the bar denotes the closure of domains.

Definition 3.1. LetΩ⊂Rⁿbe an unbounded domain,x_m∈Ω,|x_m| → ∞asm→ ∞;χ(x) be the characteristic function ofΩ, andΩmbe a shifted domain defined by the character- istic functionχm(x)=χ(x+xm). We say thatΩ∗is alimiting domainof the domainΩif Ωm→Ω∗inΞlocasm→ ∞.

We denote byΛ(Ω) the set of all limiting domains of the domainΩ(for all sequences xm). We will show below that if Condition D is satisfied, then the limiting domains exist and also satisfy this condition.

Theorem3.2. If a domainΩsatisfies Condition D, then there exists a function f(x)defined inRⁿsuch that:

(1) f(x)∈C^k+θ(Rⁿ),k≥r, (2) f(x)>0if and only ifx∈Ω, (3)|∇f(x)| ≥1forx∈∂Ω,

(4) min(d(x), 1)≤ |f(x)|, whered(x)is the distance fromxto∂Ω.

LetΩbe an unbounded domain satisfying Condition D and f(x) be a function satis- fying conditions ofTheorem 3.2. Consider a sequencexm∈Ω,|xm| → ∞. Denote

fm(x)=fx+xm

. (3.5)

Theorem3.3. Let f_m(x)→f_∗(x)inC_loc^k (Rⁿ), wherekis not greater than that inTheorem 3.2. Denote

Ω∗=

x:x∈Rⁿ, f_∗(x)>0. (3.6) Then

(1) f_∗(x)∈C^k+θ(Rⁿ),

(2)Ω∗is an nonempty open set.

IfΩ∗=Rⁿ, then (3)|∇f_∗(x)|∂Ω∗≥1,

(4) min(d_∗(x), 1)≤ |f_∗(x)|, whered_∗(x)is the distance fromxto∂Ω∗.

Theorem3.4. If f_m(x)→ f_∗(x)inC_loc^k asm→ ∞, then∂Ωm→∂Ω∗inΞloc. Moreover, the limiting domainΩ∗either satisfies Condition D orΩ∗=Rⁿ.

Theorem3.5. LetΩbe an unbounded domain satisfying Condition D,x_m∈Ω,|x_m| → ∞, and f(x)be the function constructed inTheorem 3.2.

Then there exists a subsequencex_miand a function f_∗(x)such that

f_mi(x)≡fx+x_mi−→f_∗(x) (3.7) inC^k_loc(Rⁿ), and the domainΩ∗= {x:f_∗(x)>0}either satisfies Condition D orΩ∗=Rⁿ. Moreover,Ω¯mi→Ω¯∗inΞloc, whereΩmi= {x:f_mi(x)>0}.

3.2. Convergence. In the previous section we have introduced limiting domains. Here we define the corresponding limiting problems.

LetΩbe a domain satisfying Condition D andχ(x) be its characteristic function. Con- sider a sequencexm∈Ω,|xm| → ∞and the shifted domainsΩm defined by the shifted characteristic functionsχm(x)=χ(x+xm). We suppose that the sequence of domainsΩm

converge inΞlocto some limiting domainΩ∗. In this section we suppose that 0≤k≤r.

Definition 3.6. Letum∈W∞^k,p(Ωm), m=1, 2,. . . .We say thatum converges to a limit- ing function u_∗∈W_∞^k,p(Ω∗) inW_loc^k,p(Ωm→Ω∗) if there exists an extension v_m(x)∈ W∞^k,p(Rⁿ) ofu_m(x),m=1, 2,. . .and an extensionv_∗(x)∈W∞^k,p(Rⁿ) ofu_∗(x) such that vm→v_∗inW_loc^k,p(Rⁿ).

Definition 3.7. Letum∈W∞^{k−1/ p,p}(∂Ωm),k >1/ p,m=1, 2,. . . .We say thatumconverges to a limiting functionu_∗∈W_∞^{k−1/ p,p}(∂Ω∗) inW_loc^{k−1/ p,p}(∂Ωm→∂Ω∗) if there exists an extensionv_m(x)∈W∞^k,p(Rⁿ) ofu_m(x),m=1, 2,. . .and an extensionv_∗(x)∈W∞^k,p(Rⁿ) of u_∗(x) such thatv_m→v_∗inW_loc^k,p(Rⁿ).

Definition 3.8. Letu_m(x)_∈C^k(Ωm),m₌1, 2,. . . .We say thatu_m(x) converges to a lim- iting function u_∗(x)∈C^k(Ω∗) in C^k_loc(Ωm→Ω∗) if there exists an extensionv_m(x)∈ C^k(Rⁿ) ofum(x),m=1, 2,. . .and an extensionv_∗(x)∈C^k(Rⁿ) ofu_∗(x) such that

vm−→v_∗ inC^k_locRⁿ. (3.8)

Definition 3.9. Letu_m(x)∈C^k(∂Ωm),m=1, 2,. . . .We say thatu_m(x) converges to a lim- iting functionu_∗(x)∈C^k(∂Ω∗) inC_loc^k (∂Ωm→∂Ω∗) if there exists an extensionv_m(x)∈ C^k(Rⁿ) ofum(x),m=1, 2,. . .and an extensionv_∗(x)∈C^k(Rⁿ) ofu_∗(x) such that

vm−→v_∗ inC^k_locRⁿ. (3.9)

Theorem3.10. The limiting functionu_∗(x)in Definitions3.6–3.9does not depend on the choice of extensionsvm(x)andv_∗(x).

Theorem3.11. Suppose that0< k≤r−1. Let u_m∈W∞^k+1,p

Ωm

, u_mW∞^k+1,p_(Ω

m)≤M, (3.10)

where the constantMdoes not depend onm. Then there exists a functionu_∗∈W∞^k+1,p(Ω∗) and a subsequenceumisuch thatumi→u_∗inW_loc^k,p(Ωm→Ω∗).

Theorem3.12. Suppose that0< k≤r−1. Letu_m∈W∞^{k+1−1/ p,p}(∂Ωm), um

W∞^{k+1−1/ p,p}(∂Ωm)≤M, (3.11)

where the constantMdoes not depend onm. Then there exists a function

u_∗∈W∞^{k+1−1/ p,p}

∂Ω∗

(3.12) and a subsequenceumisuch that

u_mi−→u_∗ inW_loc^{k+1−−1/ p,p}∂Ωm−→∂Ω∗

, (3.13)

where0<< k+ 1−1/ p.

Theorem3.13. Letu_m∈C^k+θ(Ωm),u_mC^k+θ≤M, where the constantM is independent ofm. Then there exists a functionu_∗∈C^k+θ(Ω∗)and a subsequenceumksuch thatumk→u_∗ inC_loc^k (Ωmk→Ω∗).

Letu_m∈C^k+θ(∂Ωm),u_mC^k+θ≤M. Then there exists a functionu_∗∈C^k+θ(∂Ω∗)and a subsequenceu_mksuch thatu_mk→u_∗inC_loc^k (∂Ωmk→∂Ω∗).

3.3. Limiting operators. Suppose that we are given a sequence{x_ν},ν=1, 2,. . .,x_ν∈ Ω,|x_ν| → ∞. Consider the shifted domainsΩνwith the characteristic functionsχ_ν(x)= χ(x+x_ν) whereχ(x) is the characteristic function ofΩ, and the shifted coefficients of the operatorsAiandBj:

a^α_ik,ν(x)₌a^α_ikx+x_ν, b^β_jk,ν(x)₌b^β_jkx+x_ν. (3.14) We suppose that

a^α_ik(x)∈C^l−si+θΩ¯, b^β_jk(x)∈C^l−σj+θ(∂Ω), (3.15)

where 0< θ <1, and that these coefficients can be extended toRⁿ:

a^α_ik(x)∈C^l−si+θRⁿ, b^β_jk(x)∈C^l−σj+θRⁿ. (3.16) Therefore

a^α_ik,ν(x)_Cl−si+θ(Rⁿ)≤M, b^β_jk,ν(x)

C^{l−σ j+θ}(Rⁿ)≤M (3.17) with some constantM independent ofν. It follows fromTheorem 3.5that there exists a subsequence of the sequenceΩν, for which we keep the same notation, such that it converges to a limiting domainΩ∗. From (3.17) it follows that this subsequence can be chosen such that

a^α_ik,ν−→aˆ^α_ik inC^l−siRⁿlocally, b^β_jk,ν−→bˆ^β_jk inC^l−σjRⁿlocally, (3.18) where ˆa^α_ikand ˆb^β_jkare limiting coefficients,

ˆ

a^α_ik∈C^l−si+θRⁿ, bˆ^β_jk∈C^l−σj+θRⁿ. (3.19) We have constructed limiting operators:

Aˆ_iu= N k=1

|α|≤αik

ˆ

a^α_ik(x)D^αu_k, i=1,. . .,N,x∈Ω∗, Bˆju=

N k=1

|β|≤βjk

bˆ^β_jk(x)D^βuk, i=1,. . .,m,x∈∂Ω∗, Lˆ=Aˆ1,. . ., ˆAN, ˆB1,. . ., ˆBm

.

(3.20)

We consider them as acting fromE_∞(Ω∗) toF_∞(Ω∗).

4. A priori estimates with condition ns

InSection 5, we will prove that Condition NS (Section 1.4) is necessary and sufficient in order for the operatorLto be normal solvable with a finite dimensional kernel. In this section we will use it to obtain a priori estimates of solutions stronger than those given byTheorem 2.1. Estimates of this type are first obtained in [12,13] for elliptic operators in the wholeRⁿ.

Theorem4.1. Let Condition NS be satisfied. Then there exist numbersM0andR0such that the following estimate holds:

uE∞≤M0

LuF∞+uL^p(ΩR0) , ∀u∈E_∞. (4.1)

HereΩR0=Ω∩ {|x| ≤R0}.

Proof. Suppose that the assertion of the theorem is not right. LetM_k→ ∞andR_k→ ∞be given sequences. Then there existsu_k∈E_∞such that

u_kE∞> M_kLu_kF∞+u_kLp(Ω_Rk) . (4.2) We can suppose that

u_kE∞=1. (4.3)

Then

Lu_kF∞+u_kLp(Ω_Rk)< 1

M_k ^−→0 ask−→ ∞. (4.4)

FromTheorem 2.1we obtain

Luk

F∞+uk

L∞^p(Ω)≥1

c. (4.5)

It follows from (4.4) thatLukF∞→0. Hence u_kL^p_∞(Ω)> 1

2c fork≥k0 (4.6)

with somek0. Since

uk

L^p∞(Ω)=sup

y∈Ω

uk

L^p(Qy∩Ω), (4.7)

then it follows from (4.6) that there existsyk∈Ωsuch that u_kLp(Q_yk∩Ω)> 1

2c. (4.8)

From (4.4)

uk

L^p(Ω_Rk)−→0. (4.9)

This convergence and (4.8) imply that|y_k| → ∞. Denote

Lu_k=f_k. (4.10)

From (4.4) we get

fk

F∞−→0 ask−→ ∞. (4.11)

Denote nextx=y+y_k,

wk(y)=uk

y+yk

. (4.12)