Estimates for solutions of a parameter-elliptic multi-order system of differential equations

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Universität Konstanz

Estimates for solutions of a parameter-elliptic multi-order system of differential equations

Robert Denk Melvin Faierman

Konstanzer Schriften in Mathematik

(vormals: Konstanzer Schriften in Mathematik und Informatik)

Nr. 257, September 2009 ISSN 1430-3558

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-86583

URL: http://kops.ub.uni-konstanz.de/volltexte/2009/8658/

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MULTI-ORDER SYSTEM OF DIFFERENTIAL EQUATIONS

R. DENK AND M. FAIERMAN

Abstract. This paper is concerned with a boundary value problem defined over a bounded region of Euclidean space, and in particular it is devoted to the establishment of a priori estimates for solutions of a parameter-elliptic multi- order system of differential equations under limited smoothness assumptions.

In this endeavour we extend the results of Agranovich, Denk, and Faierman pertaining to a priori estimates for solutions associated with a parameter- elliptic scalar problem, as well as the results of various other authors who have extended the results of Agranovich et. al. from the scalar case to parameter- elliptic systems of operators which are either of homogeneous type or have the property that the diagonal operators are all of the same order. In addition, we extend some results of Kozhevnikov and Denk and Volevich who have also dealt with sytems of the kind under consideration here, in that one of the works of Kozhevnikov deals only with 2×2 systems, while the other, as well as the work of the last two authors, do not cover Dirichlet boundary conditions.

1. Introduction

This paper is concerned with a boundary problem defined over a bounded region in Rⁿ, and in particular is devoted to the establishment of a priori estimates for solutions of a parameter-elliptic multi-order system of differential equations under limited smoothness assumptions. To elaborate on what was just said, let us now explain in more detail our two main objectives.

With regards to the first, we point out that very general results were derived in [ADF] pertaining to the spectral theory for scalar, non-selfadjoint elliptic boundary problems involving differential operators under limited smoothness assumptions and under a parameter-ellipticity condition. In particular, a method was developed there for deriving results pertaining to the eigenvalue asymptotics even under the limited conditions imposed. In [DFM] the techniques of [ADF] were used to extend the results for the scalar problem to that for a problem involving a parameter-elliptic system of differential operators of homogeneous type, and subsequently this result was extended in [F] to the case where only the diagonal operators of the system all had to be of the same order. It is important to mention at this point that all the spectral results derived in the above works depended fundamentally upon the establishment of a priori estimates for solutions of the boundary problem under consideration. Furthermore, in the above works, the a priori estimates could be established using standard methods, but when one attempts to extend the results of [ADF] to fully parameter-elliptic multi-order systems of differential operators,

Date: September 19, 2009.

1991Mathematics Subject Classification. Primary 35J55; Secondary 35S15.

Key words and phrases. Parameter-ellipticity, multi-order systems, a priori estimates.

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one finds that the standard methods are no longer adequate, and new techniques must be introduced.

Accordingly, with all of this in mind, let us turn to the problem under consideration here. Let N ∈Nwith N >1 and let {sj}^N₁ and{tj}^N₁ denote sequences of integers satisfyings₁≥s₂≥...≥s_N,t₁≥t₂≥...≥t_N ≥0, and putm_j =s_j+t_j forj = 1, ..., N. We suppose thatm₁=m₂=...=m_k₁ > m_k₁₊₁=...=m_k_d−1 >

m_k_d−1₊₁=...=m_k_d>0, wherek_d=N, putme_j =m_k_j forj= 1, ..., d, and let Ie_r denote the (k_r−kr−1)×(kr−kr−1) identity matrix forr= 1, ..., d, wherek₀= 0. In the sequel we will use the notationI_` to denote the`×`unit matrix for`∈Nand also impose conditions which will ensure that forr= 1, ..., d, the sum Pk_r

j=1m_j is even; we henceforth denote this sum by 2Nr. Then with{σj}^N₁^e,Ne =Nd, denoting a sequence of integers such that max{σj}^N₁^e < sN, we shall be concerned here with the boundary problem

A(x, D)u(x)−λu(x) =f(x) in Ω, (1.1)

Bj(x, D)u(x) =gj(x) on Γ forj= 1, . . . ,N ,e (1.2) where Ω is a bounded region inRⁿ, n≥2, with boundary Γ, u(x) = (u1(x), . . . , uN(x))^T, and f(x) = (f1(x), . . . , fN(x))^T are N×1 matrix functions defined in Ω, ^T denotes transpose, the gj(x) are scalar functions defined on Γ, A(x, D) is an N×N matrix operator whose entries Ajk(x, D) are linear differential operators defined on Ω of order not exceedingsj+tk and defined to be 0 ifsj+tk <0, and Bj(x, D),1≤j≤N, is a 1e ×N matrix operator whose entriesBj,k(x, D) are linear differential operators defined on Γ of order not exceedingσj+tk, and defined to be 0 ifσj+tk <0. Our assumptions concerning the problem (1.1), (1.2) will be made precise in Section 2.

To motivate the second objective of this paper, let us point out that the first investigation into the spectral theory for a fully parameter-elliptic multi-order system of operators was instigated by Kozhevnikov [K1]. In this paper the author deals with a system of pseudodifferential operators acting over a compact manifold without boundary; and by introducing the so-called Kozhevnikov conditions, the author is able to establish a priori estimates for solutions as well as various other spectral results. In subsequent works [K2], [K3] the author deals with a genuine boundary problem under infinitely smooth conditions involving a parameter-elliptic multi-order system of differential operators acting over a bounded region in Rⁿ and a system of differential operators defined on the boundary. In particular, in [K2] the author restricts himself to a 2×2 system of differential operators and a system of boundary operators which can be expressed in the form a a lower triangular block matrix, that is, in the terminology of this paper, we now haveN = 2, A(x, D) = (A_jk(x, D))²_j,k=1, B(x, D) = (B_jk(x, D))j=1,...,µ/2

k=1,2

= (T_jk(x, D))²_j,k=1, withT₁₂(x, D) = 0, whereµ=P2

j=1ordA_jj(x, D). By introducing various conditions related to parameter-ellipticity, a result pertaining to the resolvent operator is established. In [K3] the author removes the restriction of [K2] that N = 2, but now requires that the system of boundary operators admits a representation in the form of an upper triangular block matrix. Furthermore, he requires thatA(x, D) be of Petrovskii type. Then under various conditions, including those of [K1], the author derives some results pertaining to the resolvent operator.

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The problem considered in [K1] was also dealt with by Denk, Mennicken, and Volevich [DMV], but now in more detail. And by introducing conditions equivalent to those of Kozhevnikov and using the method of Newton polygons, the authors establish a priori estimates for solutions as well as various spectral results. In a subsequent work [DV], Denk and Volevich deal with a genuine boundary problem of the form (1.1), (1.2), but under the assumption that the operators B_jk(x, D), if not identically zero, contain only top order terms and that these orders must satisfy a special condition. Then by appealing to the Kozhevnikov conditions as well as to the conditions of Vishik- Lyusternik [VL], the authors establish a priori estimates for solutions of the boundary problem (1.1), (1.2) for the case where Ω =Rⁿ+and Γ =Rⁿ⁻¹. What is novel in this latter work is the introduction of the Vishik-Lyusternik conditions, as this enables one to deal with multi-order systems of operators in the traditional way, that is, by means of contour integration.

Unfortunately, because of the restrictions imposed, both papers [K3] and [DV]

are not even able to deal with the important problem where sj = tj = t⁰_j for j = 1, . . . , N, (here {t⁰_j}^N₁ denotes a monotonic decreasing sequence of positive integers) and the boundary conditions are of Dirichlet type (see [ADN, Section 2], [G, p.448]). Of course the paper [K2] does include Dirichlet boundary conditions, but the requirement thatN= 2 is very restrictive.

Thus the first objective of this paper is to extend the results of [ADF] concerning solutions of the boundary problem for the scalar case to the non-scalar problem under consideration here, while our second objective is to extend the results of [K3]

and [DV] by establishing a priori estimates for solutions of the boundary problem (1.1), (1.2) under boundary conditions which do include those of Dirichlet. And in this endeavour we shall make use of the conditions of both Kozhevnikov and Vishik- Lyusternik (see Definitions 2.4 and 2.6 below). Our main result is then given in Theorem 2.8 below.

Let us also mention at this point that from a consideration of length, we have limited ourselves in this paper solely to the proof of the sufficiency part of Theorem 2.8, that is to say, that the conditions cited in Definition 2.6 ensure the validity of this theorem. The converse of this result, that is, the proof of the necessity part of the theorem, will be left for a later work.

Finally, let us outline the contents of the paper. In Section 2 we introduce some terminology, definitions, and assumptions concerning the boundary problem (1.2), (1.2) which we require for our work and then present the main result of this work, namely Theorem 2.8 below. In Section 3 we restrict ourselves to the case where all the operators involved have constant coefficients and which act either over Rⁿ, without boundary conditions, or over Rⁿ+, with boundary conditions defined on Rⁿ⁻¹. All the preliminary results required for the proof of Theorem 2.8 are established here, with the main result being Proposition 3.6. These results are then used in Section 4 to prove Theorem 2.8.

2. Preliminaries

In this section we are going to introduce some terminology, definitions, and assumptions concerning the boundary problem (1.1), (1.2) which we require for our work and then state the main result of this paper, namely Theorem 2.8 below.

Accordingly, we letx= (x1, . . . , xn) = (x⁰, xn) denote a generic point inRⁿ and use the notationD_j=−i∂/∂xj,D= (D₁, . . . .D_n),D^α=D^α₁¹· · ·D_n^αⁿ =D⁰^α⁰D^α_nⁿ,

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andξ=ξ₁^α¹· · ·ξ_n^αⁿ forξ= (ξ1, . . . , ξn) = (ξ⁰, ξn)∈Rⁿ, whereα= (α1, . . . , αn) = (α⁰, αn) is a multi-index whose lengthPn

j=1αj is denoted by|α|. Differentiation with respect to another variable, say y ∈ Rⁿ, instead of x will be indicated by replacing D, D^α, D^0α⁰, and D_n^αⁿ by D_y, D^α_y, D^0α_y0⁰, and D_y^αⁿ

n, respectively. For 1 < p < ∞, s ∈ N0 =N∪ {0}, and G an open set in R^`, ` ∈ N, we let W_p^s(G) denote the Sobolev space of ordersrelated to L_p(G) and denote the norm in this space byk · k_s,p,G, wherekuk_s,p,G=

P

|α|≤s

R

G|D^αu(x)|^pdx1/p

foru∈W_p^s(G).

In addition we shall use norms depending upon a parameter λ∈C\{0}, namely for 1≤j≤d, we let

|||u|||^(j)_s,p,G=kuks,p,G+|λ|^s/^m^e^jkuk0,p,G foru∈W_p^s(G).

We also let ˚W_p^s(G) denote the closure ofC₀^∞(G) inW_p^s(G).

In the sequel we shall also at times deal with the Bessel-potential spaceH_p^s(G) for 0≥s∈Zand equipped with norms depending upon the parameterλ. Namely for u∈H_p^s(G) and 1≤j ≤d, we introduce the norms|||u|||^(j)_s,p,G=kF⁻¹hξ, λi^s_jF uk0,p,Rⁿ

if G = Rⁿ and |||u|||^(j)_s,p,G = inf|||v|||^(j)_s,p,

Rⁿ otherwise, where the infimum is taken is taken over allv∈H_p^s(Rⁿ) for whichu=v

G,Fdenotes the Fourier transformation inRⁿ(x→ξ) andhξ, λi_j =

|ξ|²+|λ|^f^mj² ¹₂

(see [GK, Section 1], [T, p. 77]).

Assume for the moment that the boundary Γ of Ω (see (1.1), (1.2)) is of class C^m−1,1 for some m ∈ N , and let s be an integer satisfying 1 ≤ s ≤ m. Then for G= Ω or G=Rⁿ+, the vectors u∈ W_p^s(G) have boundary values v =u

∂G

and we denote the space of these boundary values by Wp^s−1/p(∂G) and denote by k · k_{s−1/p,p,∂G} the norm in this space, where kvk_{s−1/p,p,∂G} = infkuks,p,G for v∈Wp^s−1/p(∂G) and the infimum is taken over thoseu∈W_p^s(G) for whichu

_∂G=v (see also [ADF, Section 2] and [Gr, p.20] for further definitions ofWp^s−1/p(∂G)). In addition we shall use norms depending upon a parameterλ∈C\{0}, namely for 1≤j ≤d,

|||v|||^(j)_{s−1/p,p,∂G}=kvk_{s−1/p,p,∂G}+|λ|^(s−1/p)/^m^e^jkvk0,p,∂G forv∈W_p^s−1/p(∂G), wherek · k0,p,∂Gdenotes the norm in Lp(∂G).

Turning now to the boundary problem (1.1), (1.2), let us write A_jk(x, D) = X

|α|≤s_j+t_k

a^jk_α(x)D^α forx∈Ω and 1≤j, k≤N, Bjk(x, D) = X

|α|≤σj+tk

b^jk_α(x)D^α forx∈Γ andk= 1, . . . , N, j= 1, . . . ,N .e (2.1) Observing that the orders of the operators A_jk(x, D), B_jk(x, D) remain un- changed if we replace the sequences{sj}^N₁,{tj}^N₁ ,and{σj}^N₁ê by{sj−σ^†}^N₁ ,{tj+ σ^†}^N₁, and {σ_j−σ^†}^N₁ê, respectively, where σ^† = max{σ_j}^N₁ê + 1, we see that by making such substitutions if necessary, there is no loss of generality in henceforth supposing that σj < 0 for j = 1, . . . ,Ne. Likewise, by replacing the sequences {sj}^N₁ ,{tj}^N₁, and {σj}^N₁ê by {sj−s_N}^N₁ ,{tj +s_N}^N₁ , and {σj −s_N}^N₁ê}, respectively, if necessary, we may also henceforth suppose thats_N ≥0.

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Assumption 2.1. It will henceforth be supposed that tj ≥ 0 and sj ≥ 0 for j= 1, . . . , N, and thatσj <0 forj= 1, . . . ,N.e

Assumption 2.2. It will henceforth be supposed that: (1) Γ is of classC^κ⁰^−1,1∩ C^s¹, where κ0 = max

t1,max{−σj}^N₁^e ; (2) for each pair j, k, a^jk_α ∈C^s^j(Ω) for

|α| ≤ sj +tk if sj > 0, while if sj = 0, then a^jk_α ∈ L_∞(Ω) for |α| < sj +tk

and a^jk_α ∈ C⁰(Ω) for |α| = sj+tk; (3) for each pair j, k, b^jk_α ∈ C^−σ^j^−1,1(Γ) for

|α| ≤σj+tk. Forξ∈Rⁿ let

A(x, ξ) =˚

A˚_jk(x, ξ)N j,k

forx∈Ω, B(x, ξ) =˚

B˚_jk(x, ξ)

j=1,...,Ne k=1,...,N

forx∈Γ,

and denote by ˚Bj(x, ξ) the j-th row of ˚B(x, ξ), where ˚Ajk(x, ξ) (resp. ˚Bjk(x, ξ)) consists of those terms inAjk(x, ξ) (resp. Bjk(x, ξ)), which are just of ordersj+tk

(resp. σj+tk). Then in the sequel we shall also require the following notation. For x∈Ω andξ∈Rⁿ, let

A^(r)₁₁(x, ξ) =

A˚jk(x, ξ)^kr

j,k=1 for 1≤r≤d, A^(r)₁₂(x, ξ) =

A˚_jk(x, ξ)

j=1,...,k_r k=k_r+1,...,N

, A^(r)₂₁(x, ξ) =

A˚_jk(x, ξ)

j=k_r+1,...,N k=1,...,k_r

, and A^(r)₂₂(x, ξ) =

A˚_jk(x, ξ)N j,k=k_r+1

for 1≤r≤d−1.

Also forx∈Γ, ξ∈Rⁿ, and 1≤r≤`₁, `≤d, let B^(r,`)(x, ξ) =

B˚jk(x, ξ)

j=N`−1(1−δr,`)+1,...,N`

k=1,...,N

, B_`^(r,`)

1,1(x, ξ) =

B˚jk(x, ξ)

j=N`−1(1−δr,`)+1,...,N`

k=1,...,k_`₁

, and

B_`^(r,`)

1,2(x, ξ) =

B˚jk(x, ξ)

j=N`−1(1−δ_r,`)+1,...,N_` k=k_`₁+1,...,N

,

whereδ_r,`is the Kronecker delta. In addition we letIe_r,0= diag(0·Ie₁, . . . ,0·Ie_r−1,Ie_r) forr= 2, . . . , dandIe1,0=Ie1.

Note that when x⁰ ∈ Γ we can rewrite the boundary problem (1.1), (1.2) in terms of a local coordinate system at x⁰ wherein x⁰ → 0 and ν → e_n, where ν denotes the exterior normal to Γ atx⁰ and (e₁, . . . , e_n) denotes the standard basis inRⁿ. Then supposing that this has been done, we shall in the sequel be concerned with the boundary problem

A(0, D)u(x)˚ −λ u(x) =f(x) forx∈Rⁿ+,

B˚_j(0, D)u(x) =g_j(x⁰) atx_n= 0 forj= 1, . . . ,N ,e

and corresponding to this boundary problem we define the associated symbols B(0, ξ),˚ B˚j,k(0, ξ),A˚j,k(0, ξ),A^(r)_j,k(0, ξ),B^(r,`)(0, ξ),andB_`^(r,`)

1,j(0, ξ) in precisely the same way their analogues were defined in the original coordinate system.

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Definition 2.3. LetL be a closed sector in the complex plane with vertex at the origin. Then the operator A(x, D)−λ IN will be called parameter-elliptic in L if det A^(r)₁₁(x, ξ)−λeI_r,0

6= 0 forx∈Ω, ξ∈Rⁿ\ {0}, andλ∈ L, r= 1, . . . , d.

In the sequel we letC±={z∈C, Imz ><0}.

Definition 2.4. Suppose that the operatorA(x, D)−λ IN is parameter-elliptic in the sectorLintroduced above. Letx⁰ be an arbitrary point of Γ and assume that the boundary problem (1.1), (1.2) has been rewritten in a local coordinate system associated with x⁰ in the manner just explained. Then the operator A(x, D)− λ I_N will be called properly parameter-elliptic inL if the following conditions are satisfied.

(1) The polynomial det

A^(r)₁₁(0, ξ⁰, z)−λeIr,0

has precisely Nr zeros lying in C+ forξ⁰ ∈Rⁿ⁻¹\ {0} and λ∈ L, r= 1, . . . , d.

(2) The polynomial det

A^(r)₁₁(0,0, z)−λeIr,0

has precisely Nr−N_r−1 zeros lying inC+ forλ∈ L \ {0}, r= 2, . . . , d.

Remark 2.5.Referring to Condition (1) of Definition 2.4, we know from [AV, Section 2] that det

A^(r)₁₁(0, ξ⁰, z)−λeI_r,0

has preciselyN_rzeros inC+ ifr= 1 or ifr >1 andn >2. In the sequel, when proper parameter-ellipticity is supposed, it will be assumed that this is also the case whenr >1 andn= 2. Turning next to Condition (2) of the definition, it is clear that the number of zeros of the determinant inC+

(resp. C−) does not depend upon λ. Hence it follows from an expansion of the determinant in powers ofzandλthat Condition (2) always holds ifmeris even or if meris odd,kr−kr−1is even, and there is aλ∈ L \ {0}such that−λ∈ L. Lastly we mention at this point that it is also clear from what was said above that Condition (2) is always satisfied if the operatorA(x, D) is essentially upper triangular at x⁰ (see Definition 2.7 below)

Definition 2.6. LetLdenote the sector introduced in Definition 2.3 above. Then we say that the boundary problem (1.1), (1.2) is parameter-elliptic inLifA(x, D)−

λ IN is properly parameter-elliptic inL and the following conditions are satisfied.

Letx⁰be an arbitrary point of Γ and suppose that the boundary problem (1.1), (1.2) has been rewritten in a local coordinate system associated with x⁰, as explained above. Then

(1) the boundary problem on the half-line

A^(r)₁₁(0, ξ⁰, Dn)v(xn)−λeIr,0v(xn) = 0 forxn>0, B_r,1^(r,r)(0, ξ⁰, D_n)v(x_n) = 0 atx_n = 0,

|v(xn)| →0 asxn → ∞,

has only the trivial solution forξ⁰ ∈Rⁿ⁻¹\ {0}, λ∈ L and 1≤r≤d;

(2) the boundary problem on the half-line

A^(`)₁₁(0,0, Dn)v(xn)−λIe`,0v(xn) = 0 forxn>0, B_`,1^(r,`)(0,0, Dn)v(xn) = 0 atxn = 0,

|v(x_n)| →0 asx_n→ ∞,

has only the trivial solution forλ∈ L \ {0}, 1≤r < dandr < `≤d.

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For our purposes we need to introduce some further terminology. To this end we henceforth let π1(j) = r if 1 ≤j ≤N and kr−1 < j ≤ kr, and π2(j) = r if 1 ≤j ≤Nd and Nr−1 < j ≤ Nr, where N0 = 0. In addition we let hξi=

1 +|ξ|²¹₂

,hξ⁰i= 1 +|ξ⁰|²¹₂

,andhξ⁰, λij =

|ξ⁰|²+|λ|^mj^f² ¹2

for 1≤j ≤d. We also require the following definition.

Definition 2.7. Letx⁰∈Γ. Then we say that the operatorA(x, D) is essentially upper triangular at x⁰ ifa^jk_α(x⁰) = 0 for |α| =sj +tk, k`−1 < j ≤k`,1 ≤ k≤ k`−1, `= 2, . . . , d. Likewise we say that the operatorB(x, D) = (Bjk(x, D))_j=1,...,

Ne k=1...,N

is essentially upper triangular atx⁰ ifb^jk_α(x⁰) = 0 for |α|=σj+tk, N`−1< j≤ N`,1≤k≤N`−1, `= 2, . . . , d.

We are now in a position to state the main result of this paper, namely Theorem 2.8 below, and which will be proved in Section 4. In this theorem we will require the further assumption, which will be made precise in Definition 3.13 below, that the operators A(x, D) andB(x, D) are compatible at each point of Γ. Hence for the moment let us state that this condition will always be satisfied ifB(x, D) is of Dirichlet type on Γ or if the operatorsA(x, D) and B(x, D) are essentially upper triangular at every point of Γ.

Theorem 2.8. Suppose that the boundary problem (1.1),(1.2)is parameter-elliptic in L. Suppose also that the operators A(x, D)andB(x, D)are compatible at every point ofΓ. In addition, suppose thatB(x, D)is essentially upper triangular at every point of Γ. Then there exists a constantλ⁰ =λ⁰(p)>1 such that for λ∈ L with

|λ| ≥λ⁰, the boundary problem (1.1),(1.2)has a unique solutionu∈QN

j=1Wp^t^j(Ω) for every f ∈QN

j=1Hp^−s^j(Ω)andg= g1, . . . , g

Ne

T

∈QNe

j=1W^−σ^j⁻

1

p p(Γ), and the a priori estimate

N

X

j=1

|||uj|||^(π_t ¹^(j))

j,p,Ω ≤C





N

X

j=1

|||fj|||^(π_−s¹^(j))

j,p,Ω+

Ne

X

j=1

|||gj|||^(π_−σ²^(j))

j−¹_p,p,Γ



 (2.2)

holds, where the constantC does not depend upon thefj, gj,andλ.

Remark 2.9. The proof of Theorem 2.8 will depend upon the results of Section 3 and those of [AV], and as a consequence of these results it will also follow that the estimate (2.2) is 2-sided, i.e., an estimate reverse to (2.2) holds. Indeed, we know from [AV] that we can cover Ω by a finite number of open sets{Uk}ⁿ₁¹, where Uk∩Γ6=∅ifk≤n0for somen0< n1, andUk⊂Ω fork > n0. If{φk}^N₁¹ denotes a partition of unity subordinate to the covering{Uk}ⁿ₁¹ such that suppφk∩Γ6=∅for k≤n0 and suppφk∩Γ =∅ otherwise, where supp denotes support, then a norm equivalent to the norm|||fj|||^(π_−s¹^(j))

j,p,Ω defined above is given by

n₀

X

k=1

|||φkfj|||^(π_−s¹^(j))

j,p,Rⁿ₊+

n₁

X

k=n0+1

|||φkfj|||^(π_−s¹^(j))

j,p,Rⁿ, where the norm |||φkfj|||^(π_−s¹^(j))

j,p,Rⁿ+ is taken in local coordinates. Since similar state- ments can be made for both|||u_j|||^(π_t ¹^(j))

j,p,Ω and|||g_j|||^(π²^(j))

−σj−¹_p,p,Γ, the assertion concerning the 2-sidedness of (2.2) follows directly from the results of Section 3.

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3. The Constant Coefficient Case

In this section we are going to establish some results concerning the existence of and a priori estimates for solutions of a boundary problem involving constant coefficient systems which is related to (1.1), (1.2). These results will then be used in Section 4 to prove Theorem 2.8. To this end, letx⁰∈Ω and let us fix our attention upon the differential equation

A(x˚ ⁰, D)u(x)−λ u(x) =f(x) forx∈Rⁿ andλ∈ L \ {0}. (3.1) Then we have the following two results, where here and below we letuj andfj,1≤ j≤N, denote the components ofuandf, respectively.

Proposition 3.1. Suppose that u ∈ QN

j=1Wp^t^j(Rⁿ) and that (3.1) holds. Then f ∈ QN

j=1Hp^−s^j(Rⁿ) and PN

j=1|||f_j|||^(π_−s¹^(j))

j,p,Rⁿ ≤ CPN

j=1|||u_j|||^(π_t ¹^(j))

j,p,Rⁿ, where the constant C does not depend uponuandλ.

Proposition 3.2. Suppose that the operator A(x, D)−λ IN is parameter-elliptic inLand thatf ∈QN

j=1Hp^−s^j(Rⁿ). Then there exists the constantλ⁰>0such that forλ∈ L with|λ| ≥λ⁰, the differential equation (3.1) has a unique solutionu∈ QN

j=1Wp^t^j(Rⁿ) and PN

j=1|||uj|||^(π_t ¹^(j))

j,p,Rⁿ ≤CPN

j=1|||fj|||^(π_−s¹^(j))

j,p,Rⁿ, where the constant C does not depend uponf and λ.

We will only prove Proposition 3.2 as the proof of Proposition 3.1 follows directly from the definition and the Mikhlin-Lizorkin multiplier theorem.

Proof of Proposition 3.2. Under our assumptions we know from [DMV] and [K1]

that forξ∈Rⁿ andλ∈ L with|λ| ≥λ⁰, ˚A(x⁰, ξ)−λ IN is invertible and

det

A(x˚ ⁰, ξ)−λ I_N ≥C

d

Y

j=1

hξ, λi^2(N_j ^j^−N^j−1⁾,

where the constant C does not depend upon ξ and λ. Furthermore, if we put A(x˚ ⁰, ξ)−λ I_N−1

= (ea_j,k(ξ, λ))^N_j,k=1, then the ea_j,k(ξ, λ) are rational functions of their arguments, while it also follows from the references just cited that for any multi-indexαwhose entries are either 0 or 1,

|ξ^αD^α_ξea_j,k(ξ, λ)| ≤Chξi^s^j^+t^khξ, λi^−m_π ^j

1(j)hξ, λi^−m_π ^k

1(k)

for allξ ∈Rⁿ whose components are all non-zero, where the constantC does not depend uponξandλ.

Now observe that under a Fourier transformation (3.1) becomes A(x˚ ⁰, ξ)F u(ξ)−λ F u(ξ) =F f(ξ).

Furthermore, in light of what was said above, we conclude that this equation has a unique solution in the space of tempered distributions onRⁿ given byF u(ξ) = A(x˚ ⁰, ξ)−λ I_N−1

F f(ξ). Hence all of the assertions of the proposition follow immediately from this last result, the definitions of the terms involved, and the

Mikhlin-Lizorkin multiplier theorem.

Let us suppose from now on that x⁰ ∈ Γ. Then assuming that the boundary problem (1.1), (1.2) has been rewritten in terms of the local coordinates at x⁰ as

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explained in the text preceding Definition 2.3, let us fix our attention upon the problem in the half-space

A(0, D)u(x)˚ −λ u(x) =f(x) forx∈Rⁿ+andλ∈ L \ {0}. (3.2) Then from a consideration of the pairing between Hp^−s^j(Rⁿ+), equipped with the norm||| · |||^(π_−s¹^(j))

j,p,Rⁿ+, and its dual ˚W_p^s0^j(Rⁿ+), equipped with the norm||| · |||^(π_s ¹^(j))

j,p⁰,Rⁿ+,1≤ j ≤d, p⁰ =p/(p−1) (see [GK, Theorem 1.1]), we can easily derive the following analogue of Proposition 3.1.

j=1Wp^t^j(Rⁿ+) and that (3.2) holds. Then f ∈ QN

j=1Hp^−s^j(Rⁿ+) and PN

j=1|||fj|||^(π_−s¹^(j))

j,p,Rⁿ+ ≤ CPN

j=1|||uj|||^(π_t ¹^(j))

j,p,Rⁿ+, where the constant C does not depend uponuandλ.

Proposition 3.4. Suppose that the operator A(x, D)−λ IN is parameter-elliptic inLand thatf ∈QN

j=1Hp^−s^j(Rⁿ+). Then there exists aλ⁰>0such that forλ∈ L with|λ| ≥λ⁰, the differential equation(3.2)has a solutionu∈QN

j=1Wp^t^j(Rⁿ+) and PN

j=1|||u_j|||^(π_t ¹^(j))

j,p,Rⁿ+ ≤ CPN

j=1|||f_j|||^(π_−s¹^(j))

j,p,Rⁿ+, where the constant C does not depend uponf andλ.

Proof. It follows from [T, Lemma 2.9.3, p.218] and [GK, Eqn.(1.27)] that there is afe∈QN

j=1Hp^−s^j(Rⁿ) such thatfe

Rⁿ+=f and

N

X

j=1

|||fej|||^(π_−s¹^(j))

j,p,Rⁿ≤C

N

X

j=1

|||fj|||^(π_−s¹^(j))

j,p,Rⁿ₊,

where the constant C does not depend upon f and λ. Hence if eu denotes the solution of (3.1) when f there is replaced byfeandu=eu

Rⁿ+, then the assertion of the proposition follows directly from Proposition 3.2.

Let us next fix our attention upon the boundary problem

A(0, D)u(x)˚ −λ u(x) = 0 forx∈Rⁿ+, (3.3) B˚j(0, D)u(x) =gj(x⁰) atxn = 0, j= 1, . . . ,N ,e (3.4) withλ∈C\ {0}.

j=1Wp^t^j(Rⁿ+) and that (3.4) holds. Then g= g1, . . . , g

Ne

^T

∈QNe

j=1W^−σ^j⁻

1

p p(Rⁿ⁻¹) and

Ne

X

j=1

|||gj|||^(π_−σ²^(j))

j−¹_p,p,Rⁿ⁻¹≤C

N

X

j=1 N

X

k=1

|||uk|||^(π_t ¹^(j))

k,p,Rⁿ₊, (3.5) where the constant C does not depend upon u and λ. Furthermore, if B(x, D) is essentially upper triangular atx⁰, then we may replace (3.5)by

Ne

X

j=1

|||g_j|||^(π²^(j))

−σj−¹_p,p,Rⁿ⁻¹≤C

N

X

j=1

|||u_j|||^(π_t ¹^(j))

j,p,Rⁿ+.

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Proof. Let 1≤j≤Ne and letµ(j) =π2(j) ifB(x, D) is essentially upper triangular atx⁰and letµ(j) = 1 otherwise. Then it follows from [ADF, Proposition 2.2] that

|||gj|||^(π_−σ²^(j))

j−¹_p,p,Rⁿ⁻¹ ≤C1

N

X

k=1

X

|α|=σj+t_k

b^jk_α(x⁰)D^αuk

(π₂(j))

−σj,p,Rⁿ₊

≤C2 N

X

k=µ(j)

X

|α|=σj+t_k

kD^αukk_−σ_j,p,Rⁿ₊+|λ|

−σj

mπf 2 (j)kD^αukk0,p,Rⁿ₊

≤C3 N

X

k=µ(j)

kukkt_k,p,Rⁿ++|||uk|||^(π_t ²^(j))

k,p,Rⁿ+

≤2C3 N

X

k=µ(j)

|||uk|||^(π_t ²^(j))

k,p,Rⁿ,

where the constants Cj do not depend uponuand λ. Hence all the assertions of

the proposition follow from this last result.

We now come to the main result of this section.

Proposition 3.6. Suppose that the boundary problem (1.1), (1.2) is parameter- elliptic inL. Suppose also that the operatorsA(x, D)andB(x, D)are compatible at x⁰. Then there exists a constant λ⁰=λ⁰(p)>1such that forλ∈ L with|λ| ≥λ⁰, the boundary problem (3.3), (3.4) has a unique solution u ∈ QN

j=1Wp^t^j(Rⁿ+) for every g= (g1, . . . , g

Ne)^T ∈QNe

j=1W^−σ^j⁻

1 p

p (Rⁿ⁻¹), and the a priori estimate

N

X

j=1

|||uj|||^(π_t ¹^(j))

j,p,Rⁿ₊ ≤C

Ne

X

j=1

|||gj|||^(π_−σ²^(j))

j−¹_p,p,Rⁿ⁻¹

holds, where the constantC does not depend upon thegj andλ.

As a consequence of Propositions 3.4, 3.5, and 3.6 as well as from a standard argument, we obtain the following result.

Proposition 3.7. Suppose that the hypotheses of Proposition 3.6 hold. Suppose also that the operatorB(x, D)is essentially upper triangular atx⁰. Then the exists a constantλ⁰=λ⁰(p)>1such that forλ∈ Lwith |λ| ≥λ⁰, the boundary problem (3.2),(3.4)has a unique solutionu∈QN

j=1Wp^t^j(Rⁿ+)for everyf ∈QN

j=1Hp^−s^j(Rⁿ+) andg∈QNe

j=1W^−σ^j⁻

1

p p(Rⁿ⁻¹), and the a priori estimate

N

X

j=1

|||u_j|||^(π_t ¹^(j))

j,p,Rⁿ+ ≤C





N

X

j=1

|||f_j|||^(π_−s¹^(j))

j,p,Rⁿ++

Ne

X

j=1

|||g_j|||^(π²^(j))

−σj−¹_p,j,p,Rⁿ⁻¹





holds, where the constantC does not depend upon f, g, andλ.

In order to prove Proposition 3.6, some preliminary results are required. Ac- cordingly, let us fix λ ∈ L with |λ| > 1 sufficiently large and let {j}^d₁ denote a sequence of numbers satisfying 0< _j <1, j = 1, . . . , d (the magnitudes of λand

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the j will be specified below). Also for r = 0, . . . , d, let us introduce functions ψr∈C^∞(Rⁿ⁻¹) satisfying 0≤ψr(ξ⁰)≤1,

suppψ0∈

ξ⁰∈Rⁿ⁻¹:|ξ⁰| ≤³₄1|λ|^1/^m^e¹ andψ0(ξ⁰) = 1 for|ξ⁰| ≤ ¹₄1|λ|^1/^m^e¹, suppψr∈

ξ⁰∈Rⁿ⁻¹: ¹₄r|λ|^1/^m^e^r ≤ |ξ⁰| ≤ ³₄r+1|λ|^1/^m^e^r+1

andψr(ξ⁰) = 1 for ³₄r|λ|^1/^m^e^r ≤ |ξ⁰| ≤¹₄r+1|λ|^1/^m^e^r+1, 1≤r < d, suppψd∈

ξ⁰∈Rⁿ⁻¹:|ξ⁰| ≥¹₄d|λ|^1/^m^e^d andψd(ξ⁰) = 1 for|ξ⁰| ≥ ³₄d|λ|^1/^m^e^d, while in additionPd

r=0ψ_r(ξ⁰) = 1 and eachψ_r is a Fourier multiplier of type (p, p) whose norm is bounded by a constant not depending uponλand the _j. Then in the sequel we will require the following three results.

Proposition 3.8. Suppose that the boundary problem (1.1),(1.2) is parameter- elliptic in L. Suppose also that λ∈ L with |λ| >1 and that 0≤ |ξ⁰| ≤ ⁷₈1|λ|^f^m¹¹. Then we can choose the numbers ⁰ sufficiently small and λ⁰ sufficiently large so that for₁≤⁰ and|λ| ≥λ⁰,det

A(0, ξ˚ ⁰, z)−λ I_N

has precisely N_d(=Ne)zeros, say

z_j⁽⁰⁾(ξ⁰, λ) ^N_j=1^d , lying in C+, and satisfying

Imz_j⁽⁰⁾(ξ⁰, λ)≥C1hξ⁰, λi1, |z⁽⁰⁾_j (ξ⁰, λ)| ≤C2hξ⁰, λi1, j = 1, . . . , N1, Imz_j⁽⁰⁾(ξ⁰, λ)≥C₁hξ⁰, λi_`, |z⁽⁰⁾_j (ξ⁰, λ)| ≤C₂hξ⁰, λi_`, j =N_`−1+ 1, . . . , N_` for`= 2, . . . , d, and whereC₂hξ⁰, λi_`< C₁hξ⁰, λi_`+1 for`= 1. . . , d−1,and theC_j denote constants not depending upon ξ⁰ andλ.

Proposition 3.9. Suppose that the boundary problem (1.1), (1.2) is parameter- elliptic in L. Suppose also that 1 ≤ r < d, that λ ∈ L with |λ| > 1, and that

1

8r|λ|^f^mr¹ ≤ |ξ⁰| ≤ ⁷₈r+1|λ|^f^mr+1¹ . Then for fixed r we can choose the numbers ⁰ sufficiently small and λ⁰ sufficiently large so that for r+1 ≤ ⁰ and |λ| ≥λ⁰, det

A(0, ξ˚ ⁰, z)−λ I_N

has precisely N_d zeros, say

z_j^(r)(ξ⁰, λ) ^N_j=1^d , lying in C+

and satisfying

Imz_j^(r)(ξ⁰, λ)≥C1hξ⁰, λir, |z_j^(r)(ξ⁰, λ)| ≤C2hξ⁰, λir, j= 1, . . . , Nr, Imz_j^(r)(ξ⁰, λ)| ≥C₁hξ⁰, λi`, |z_j^(r)(ξ⁰, λ)| ≤C₂hξ⁰, λi`, j=N_`−1, . . . , N_` for` =r+ 1, . . . , d, and where C₂hξ⁰, λi_` < C₁hξ⁰, λi_`+1 for` =r, . . . , d−1, and the C_j denote positive constants not depending upon ξ⁰ andλ.

Proposition 3.10. Suppose that the boundary problem (1.1), (1.2)is parameter- elliptic inL. Suppose also that λ∈ L with|λ|>1 and that ¹₈d|λ|^f^md¹ ≤ |ξ⁰|<∞.

Then for fixedd we can choose the numberλ⁰sufficiently large so that for|λ| ≥λ⁰, det

A(0, ξ˚ ⁰, z)−λ I_N

has preciselyN_dzeros, say

z^(d)_j (ξ⁰, λ) ^N_j=1^d , lying inC+and satisfying

Imz^(d)_j (ξ⁰, λ)≥C1hξ⁰, λid, |z^(d)_j (ξ⁰, λ)| ≤C2hξ⁰, λid for j= 1, . . . , Nd, where theCj denote constants not depending upon ξ⁰ andλ.

Since the proofs of Propositions 3.8 and 3.10 are similar to that of Proposition 3.9, we will only prove this latter proposition.

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Proof of Proposition 3.9. To begin with let us observe that

A(0, ξ˚ ⁰, z)−λ I_N = A^(r)₁₁(0, ξ⁰, z)−λ I_k_r A^(r)₁₂(0, ξ⁰, z) A^(r)₂₁(0, ξ⁰, z) A^(r)₂₂(0, ξ⁰, z)−λ I_N−k_r

!

and that

A^(r)₁₁(0, ξ⁰, z)−λ I_k_r =A^(r)₁₁(0, ξ⁰, z)−λeI_r,0−λ(I_k_r−Ie_r,0).

Then as a consequence of our hypotheses we know that det

A^(r)₁₁(0, ξ⁰, z)−λIer,0

has precisely Nr zeros lying in C+ and that there is a closed contourγ_r⁺(ξ⁰, λ)⊂ C+ containing all these zeros in its interior such that for z ∈ γ_r⁺(ξ⁰, λ), Imz ≥ C1hξ⁰, λir,|z| ≤C2hξ⁰, λir,and

C₃hξ⁰, λi^2N_r ^r ≤ det

A^(r)₁₁(0, ξ⁰, z)−λeI_r,0

≤C₄hξ⁰, λi^2N_r ^r,

where the constantsCj do not depend uponξ⁰, z,andλ. Furthermore, it is easy to show that forz∈γ_r⁺(ξ⁰, λ),

det

A^(r)₁₁(0, ξ⁰, z)−λ Ik_r

−det

A^(r)₁₁(0, ξ⁰, z)−λeIr,0 ≤ C

kr−1

X

`=1

X

1≤i(1)<...<i(`)≤kr−1

`

Y

k=1

|λ|^1/m^i(k) hξ⁰, λir

^mi(k)

hξ⁰, λi^2N_r ^r,

where the constant C does not depend uponξ⁰, z, andλ, while by employing the Laplace method of expanding a determinant, we can also show that, apart from a constant not depending upon ξ⁰, z,andλ, the term

det ˚A(0, ξ⁰, z)−λ I_N - det A^(r)₁₁(0, ξ⁰, z)−λ Ik_r

λ^N−k^r

is bounded by the sum det

A^(r)₁₁(0, ξ⁰, z)−λ I_k_r

N−kr−1

X

µ=0

hξ⁰, zi^m^e^r+1N−k_r−µ

|λ|^µ+

X

q



hξ⁰, zi^P^q^µ=1^m^iµ X

j1<...<jν

hξ⁰, zi^P^ν^µ=1^m^jµ|λ|^k^r^−q−ν



×



hξ⁰, zi^P^q^µ=1^m^e^iµ X

ej₁<...<ej_ν

hξ⁰, zi^P^ν^µ=1^m^e^jµ|λ|^N^−k^r^−q−ν



.

(3.6)

Here hξ⁰, zi = |ξ⁰|²+|z|²¹₂ , P

q indicates that the summation is over those q satisfying 1 ≤q ≤kr (resp. 1 ≤q ≤N −kr) if 2kr ≤ N (resp. 2kr > N), and {iµ}^q_µ=1 and{jµ}^ν_µ=1 denote distinct sequences of integers satisfying 1≤i1< . . . <

i_q ≤k_r and 1≤j₁< . . . < j_ν≤k_r, 1≤ν≤k_r−q, respectively, while{ei_µ}^q_µ=1and {ejµ}^ν_µ=1 denote distinct sequences of integers satisfyingkr+ 1≤ei1< . . . <eiq ≤N andkr+ 1≤ej1< . . . <ejν ≤N,1 ≤ν ≤N−kr−q, respectively, and where the summationP

j₁<...<j_ν in (3.6) is to be replaced by 1 ifq=k_r and the summation P

ej1<...<ejν in (3.6) is to be replaced by 1 if q = N −kr. Hence it follows from Rouch´e’s theorem that if_r+1 is chosen sufficiently small and|λ|sufficiently large, then det

A(0, ξ˚ ⁰, z)−λ IN

has preciselyNr zeros contained inγ_r⁺(ξ⁰, λ).