Parameter-dependent estimates for mixed-order boundary value problems

Universität Konstanz

Parameter-dependent estimates for mixed-order boundary value problems

Robert Denk Melvin Faierman

Konstanzer Schriften in Mathematik

(vormals: Konstanzer Schriften in Mathematik und Informatik)

Nr. 261, Februar 2010 ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-103206

URL: http://kops.ub.uni-konstanz.de/volltexte/2010/10320/

BOUNDARY VALUE PROBLEMS

R. DENK AND M. FAIERMAN

Abstract. In this paper we prove parameter-dependent a priori estimates for mixed-order boundary value problems of rather general structure. In partic- ular, the diagonal operators are not assumed to be of the same order. Our assumptions on the structure of the boundary value problem covers the case of Dirichlet type boundary conditions.

1. Introduction

Let us consider the model problem for a general mixed-order system of the form A(D)u(x)−λu(x) =f(x) inRⁿ+,

(1.1)

B_j(D)u(x) =g_j(x) onRⁿ⁻¹ forj= 1, . . . ,N ,e (1.2)

Here Rⁿ+ := {x ∈ Rⁿ : xn > 0} denotes the half-space with boundary ∂Rⁿ+ = Rⁿ⁻¹, u(x) = (u₁(x), . . . , u_N(x))^T andf(x) = (f₁(x), . . . , f_N(x))^T are N-dimen- sional vector functions defined inRⁿ+, gj(x) are scalar functions defined onRⁿ⁻¹. The matrixA(D) = A_jk(D)

j,k=1,...,N is a mixed-order system of differential op- erators, and B_j(D) is a 1×N-matrix operator for 1 ≤ j ≤ Ne. The Douglis- Nirenberg structure ofAis given by integers {s_j}_j=1,...,N and{t_j}_j=1,...,N satisfy- ings₁≥ · · · ≥s_n ≥0 andt₁≥ · · · ≥t_N ≥0. Settingm_j =s_j+t_j, we assume, for simplicity, thatm₁>· · ·> m_N >0.

In the sequel we will impose conditions which will ensure that mj is even for j = 1, . . . , N and set N_j := ¹₂(m₁+· · ·+m_j) for j = 1, . . . , N and Ne := N_N. Then the mixed-order structure of the boundary conditions is given by a sequence {σ_j}_j=1,...,

Ne of integers with σ_j <0 forj = 1, . . . ,N. We then assume orde A_jk≤ s_j +t_k, j, k = 1, . . . , N, and ordB_jk ≤ σ_j +t_k, j = 1, . . . ,N , ke = 1, . . . , N. As we will study the model problem only, we further assume that A_jk = 0 if ordA_jk < s_j+t_k andB_jk= 0 if ordB_jk < σ_j+t_k. Further the operatorsA and B_j are assumed to have constant coefficients,

Ajk(D) = X

|α|=sj+t_k

a^jk_αD^α, j, k= 1, . . . , N, B_jk(D) = X

|α|=σ_j+t_k

b^jk_αD^α, j= 1, . . . ,N , ke = 1, . . . , N.

Date: January 15, 2010.

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

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

1

Here we used the standard multi-index notation D^α = (−i)^|α|∂_x^α₁¹. . . ∂_x^α_nⁿ, α = (α1, . . . , αn). We will show that under suitable parameter-ellipticity conditions, unique solvability and a priori estimates for the solution hold. First results in this direction were obtained by Kozhevnikov in [K1]. In this paper the author deals with a system of pseudodifferential operators on a closed manifold; by introducing the so-called Kozhevnikov conditions, the author is able to establish a priori estimates and spectral results. In subsequent works [K2], [K3], the author deals with genuine boundary value problems, imposing, however, conditions on the dimension of the system (N = 2 in [K2]) or assuming triangular form of the boundary matrix.

In the paper [DMV] by Denk, Mennicken, and Volevich, and [DV] by Denk and Volevich, the Newton polygon method was used to establish a priori-estimates for rather general systems. However, also in the paper [DV] the authors impose severe restrictions on the order of the boundary operators. Both papers [K3] and [DV] are not able to deal with the important problem wheresj =tj =t⁰_j for j = 1, . . . , N (here{t⁰_j}^N_j=1denotes a monotonic decreasing sequence of positive integers) and the boundary conditions are of Dirichlet type (see [ADN, Section 2], [G, p.448]). In the present paper, we establish a priori estimates for solutions of (1.1), (1.2) under boundary conditions which include Dirichlet type conditions. In this way, we also generalize results from Agranovich, Denk, and Faierman [ADF] concerning scalar problems.

Let us mention that we restrict ourselves to the model problem (1.1), (1.2). The general case of boundary value problems in bounded domains with coefficients with limited smoothness, as well as less restrictive assumptions on the order structure, are treated in our paper [DF]; the present note should be seen as a simplified and shortened version of [DF].

The structure of the paper is as follows. In Section 2 we introduce some ter- minology and definitions concerning the boundary value problem (1.1), (1.2) and present the main result, Theorem 2.6 below. The proof of Theorem 2.6 can be found in Sections 3 and 4.

2. Assumptions and main results

Let us first introduce some notation. For 1 < p < ∞ and s ∈ N∪ {0}, let W_p^s(Rⁿ+) stand for the standardLp-Sobolev space with norm

kuks,p,Rⁿ+= X

|α|≤s

Z

Rⁿ₊

|D^αu(x)|^pdx^1/p .

For 1≤j≤N andλ∈C\ {0}, we define the parameter-dependent norm

|||u|||^(j)_s,p,

Rⁿ+=kuk_s,p,Rⁿ

++|λ|^s/mjkuk_0,p,Rⁿ

+ foru∈W_p^s(Rⁿ+).

We will also deal with the Bessel potential spaces H_p^s(Rⁿ+) for s ∈ Z, s < 0, and the related parameter-dependent norm|||u|||^(j)_s,p,Rn =kF⁻¹hξ, λi^s_jF uk0,p,Rⁿ and

|||u|||^(j)_s,p,Rn

+= inf|||v|||^(j)_s,p,Rn, where the infimum is taken over allv∈H_p^s(Rⁿ) for which u = v

Rⁿ+, F denotes the Fourier transformation in Rⁿ(x → ξ) and hξ, λij = (|ξ|²+|λ|^2/mj)^1/2 (see [GK, Section 1], [T, p. 177]).

On the boundary Rⁿ⁻¹, the trace spaces Wp^s−1/p(Rⁿ⁻¹), s ∈N, are defined in a standard way (see, e.g., [ADF, Section 2] and [Gr, p.20]). For λ∈C\{0} and

1≤j ≤N, we set

|||u|||^(j)_s−1/p,p,

Rⁿ⁻¹ =kuk_s−1/p,p,Rn−1+|λ|^(s−1/p)/mjkuk_0,p,Rn−1 foru∈W_p^s−1/p(Rⁿ⁻¹).

To formulate the parameter-ellipticity conditions on (A, B), we set forξ∈Rⁿ and 1≤r≤N

A^(r)₁₁(ξ) = (A_jk(ξ))j,k=1,...,r, A^(r)₁₂(ξ) = (A_jk(ξ)) j=1,...,r k=r+1,...,N

, A^(r)₂₁(ξ) = (A_jk(ξ))j=r+1,...,N

k=1,...,r

, and A^(r)₂₂(ξ) = (A_jk(ξ))j,k=r+1,...,N. Also forξ∈Rⁿ and 1≤r≤`<sub>1</sub>, `≤N, let

B<sup>(r,`)</sup>(ξ) = (B<sub>jk</sub>(ξ))<sub>j=N`−1(1−δr,`)+1,...,N`</sub>

k=1,...,N

, B<sup>(r,`)</sup><sub>`</sub>

1,1(ξ) = (B_jk(ξ))_{j=N`−1(1−δr,`)+1,...,N`}

k=1,...,`₁

, B<sup>(r,`)</sup><sub>`</sub>

1,2(ξ) = (Bjk(ξ))j=N<sub>`−1</sub>(1−δr,`)+1,...,N<sub>`</sub> k=`₁+1,...,N

,

where δ_r,` is the Kronecker delta. In addition we letI_r denote the r-dimensional unit matrix andI_r,0= diag(0, . . . ,0,1)∈R^r×rforr= 1, . . . , N.

Definition 2.1([K1], [DMV]). LetLbe a closed sector in the complex plane with vertex at the origin. Then the operatorA(D)−λ IN will be called parameter-elliptic inL if det A^(r)₁₁(ξ)−λI_r,0

6= 0 forξ∈Rⁿ\ {0}andλ∈ L, r= 1, . . . , N. In the sequel we letC±={z∈C, Imz ><0}.

Definition 2.2. Suppose that the operatorA(D)−λ IN is parameter-elliptic in the sectorL. Then the operatorA(D)−λ I_N will be called properly parameter-elliptic inL if the following conditions are satisfied.

(1) The polynomial det A^(r)₁₁(ξ⁰, z)−λI_r,0

has preciselyN_r zeros lying inC+

forξ⁰∈Rⁿ⁻¹\ {0}andλ∈ L, r= 1, . . . , N. (2) The polynomial det A^(r)₁₁(0, z)−λI_r,0

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

Remark 2.3.Referring to Condition (1) of Definition 2.2, we know from [AV, Section 2] that det(A^(r)₁₁(ξ⁰, z)−λI_r,0) has precisely N_r zeros in C+ if r = 1 or if r > 1 and n > 2. Turning next to Condition (2) of the definition, it is clear that the number of zeros of the determinant in C+ (resp. C−) does not depend upon λ.

Hence it follows from an expansion of the determinant in powers ofz and λ that Condition (2) always holds ifm_ris 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(D) is essentially upper triangular (see Definition 2.5 below).

Definition 2.4. We say that the boundary problem (1.1), (1.2) is parameter- elliptic inLifA(x, D)−λ I_N is properly parameter-elliptic inLand the following

conditions are satisfied. According to the notation introduced above, let B^(r,r)_r,1 (ξ⁰, Dn) = (Bjk(ξ⁰, Dn))j=1,...,N_r

k=1,...,r

, 1≤r≤N, B<sup>(r,`)</sup><sub>`,1</sub> (ξ⁰, D_n) = (B_jk(ξ⁰, D_n))_{j=N`−1+1,...,N`}

k=1,...,`

, 1≤r < `≤N.

Then

(1) the boundary problem on the half-line

A^(r)₁₁(ξ⁰, Dn)v(xn)−λIr,0v(xn) = 0 forxn >0, B_r,1^(r,r)(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≤N; (2) the boundary problem on the half-line

A^{(`)</sup><sub>11</sub>(0, Dn)v(xn)−λ I`,0v(xn) = 0 forxn >0, B<sub>`,1</sub><sup>(r,`)}(0, Dn)v(xn) = 0 atxn= 0,

|v(xn)| →0 asxn→ ∞, has only the trivial solution forλ∈ L \ {0}, 1≤r < `≤N.

For our purposes we need to introduce some further terminology. For 1≤j ≤ Ne, let π(j) = r if Nr−1 < j ≤ Nr, where N0 = 0. In addition we let hξi =

1 +|ξ|²1/2

,hξ⁰i= 1 +|ξ⁰|²1/2

,andhξ⁰, λij= |ξ⁰|²+|λ|^2/mj1/2

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

Definition 2.5. We say that the operatorA(D) is essentially upper triangular if a^jk_α = 0 for |α| =sj+tk, 1 ≤k ≤j−1, `= 2, . . . , N. Likewise we say that the operator B(D) = (Bjk(D))_j=1,...,

Ne k=1...,N

is essentially upper triangular if b^jk_α = 0 for

|α|=σj+tk, N`−1< j≤N`,1≤k≤`−1, `= 2, . . . , N.

We are now in a position to state the main result of this paper, namely Theorem 2.6 below, which will be proved in Sections 3 and 4. In this theorem we will require the further assumption, which will be made precise in Definition 4.4 below, that the operators A(D) and B(D) are compatible. Hence for the moment let us state that this condition will always be satisfied if B(D) is of Dirichlet type or if the operatorsA(D) andB(D) are essentially upper triangular.

Theorem 2.6. Suppose that the boundary problem (1.1),(1.2)is parameter-elliptic inL. Suppose also that the operatorsA(D)andB(D)are compatible. In addition, suppose that B(D) is essentially upper triangular. Then there exists a constant λ⁰=λ⁰(p)>1such that forλ∈ Lwith|λ| ≥λ⁰, the boundary problem(1.1),(1.2) has a unique solution u ∈ QN

j=1Wp^tj(Rⁿ+) for every f ∈ QN

j=1Hp^−sj(Rⁿ+)andg = g1, . . . , g

Ne

T

∈QNe

j=1Wp^−σj−1/p(Rⁿ⁻¹), and the a priori estimate (2.1)

N

X

j=1

|||uj|||^(j)_t

j,p,Rⁿ₊ ≤C





N

X

j=1

|||fj|||^(j)_−s

j,p,Rⁿ₊+

Ne

X

j=1

|||gj|||^(π(j))_−σ

j−1/p,p,Rⁿ⁻¹





holds, where the constantC does not depend upon thef_j, g_j, andλ.

Remark 2.7. It can be shown by standard arguments (cf. [AV]) and the results of Section 3 that in fact the estimate (2.1) is 2-sided, i.e., an estimate reverse to (2.1) holds.

3. Proof of the main theorem

Following a standard approach in elliptic theory, we will prove Theorem 2.6 by studying first the whole space equation and then the equation in the half-space. The main technical issue, the proof of Proposition 3.4 below, can be found in Section 4.

In the following,C, C1, C2, . . . ,Ce1,Ce2, . . . stand for unspecific constants which may vary at each time of their appearance.

Let us first consider the whole-space equation

(3.1) A(D)u(x)−λ u(x) =f(x) forx∈Rⁿ andλ∈ L \ {0}.

We then have the following two results.

Proposition 3.1. Suppose that u ∈ QN

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

j=1Hp^−sj(Rⁿ) and PN

j=1|||fj|||^(j)_−s

j,p,Rⁿ≤CPN

j=1|||uj|||^(j)_t

j,p,Rⁿ.

Proposition 3.2. Suppose that the operatorA(D)−λ I_N is parameter-elliptic in L and that f ∈QN

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

j=1Wp^tj(Rⁿ) andPN

j=1|||uj|||^(j)_t

j,p,Rⁿ≤CPN

j=1|||fj|||^(j)_−s

j,p,Rⁿ.

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(ξ)−λ I_N is invertible and

det (A(ξ)−λ IN) ≥C

N

Y

j=1

hξ, λi^m_j^j.

Furthermore, if we put (A(ξ)−λ IN)⁻¹ = (eaj,k(ξ, λ))^N_j,k=1, then theeaj,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_ξ^αeaj,k(ξ, λ)| ≤Chξi^sj+tkhξ, λi^−m_j ^jhξ, λi^−m_k ^k for allξ∈Rⁿ whose components are all non-zero.

Now observe that under Fourier transformation (3.1) becomes A(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 on Rⁿ given by F u(ξ) = (A(ξ)−λ I_N)⁻¹F 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 next fix our attention upon the boundary problem A(D)u(x)−λ u(x) = 0 forx∈Rⁿ+,

(3.2)

B_j(D)u(x) =g_j(x⁰) atx_n = 0, j= 1, . . . ,N ,e (3.3)

withλ∈C\ {0}.

Proposition 3.3. Suppose that u ∈ QN

j=1Wp^tj(Rⁿ+), that (3.3) holds, and that B(x, D)is essentially upper triangular. Then

g= (g1, . . . , g

Ne)^T ∈

Ne

Y

j=1

W_p^−σj−1/p(Rⁿ⁻¹) and

(3.4)

Ne

X

j=1

|||gj|||^(π(j))_−σ

j−¹_p,p,Rⁿ⁻¹≤C

N

X

j=1

|||uj|||^(j)_t

j,p,Rⁿ₊.

Proof. Let 1≤j≤Ne. Then it follows from [ADF, Proposition 2.2] that

|||gj|||^(π(j))_−σ

j−¹_p,p,Rⁿ⁻¹≤C1

N

X

k=1

X

|α|=σj+tk

b^jk_αD^αuk

(π(j))

−σj,p,Rⁿ+

≤C₂

N

X

k=π(j)

X

|α|=σj+tk

kD^αu_kk_−σj,p,Rⁿ

++|λ|

−σj

mπ(j)kD^αu_kk_0,p,Rⁿ

+

≤C₃

N

X

k=π(j)

kukktk,p,Rⁿ++|||uk|||^(π(j))_t

k,p,Rⁿ₊

≤2C₃

N

X

k=π(j)

|||uk|||^(π(j))_t

k,p,Rⁿ+.

Hence all the assertions of the proposition follow from this last result.

We now come to the main result of this section.

Proposition 3.4. Suppose that the boundary problem (1.1), (1.2) is parameter- elliptic in L. Suppose also that the operators A(D) and B(D) are compatible.

Then there exists a constant λ⁰=λ⁰(p)>1 such that forλ∈ L with|λ| ≥λ⁰, the boundary problem (3.2), (3.3) has a unique solutionu∈QN

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

Ne)^T ∈QNe

j=1Wp^−σj−1/p(Rⁿ⁻¹), and the a priori estimate

N

X

j=1

|||uj|||^(j)_t

j,p,Rⁿ+≤C

Ne

X

j=1

|||gj|||^(π(j))_−σ

j−1/p,p,Rⁿ⁻¹

holds.

The proof of this proposition will be given in Section 4. The proof of the main result Theorem 2.6 is now a consequence of Propositions 3.2, 3.3, and 3.4.

Proof of Theorem 2.6. Let us first fix our attention upon the problem in the half- space

(3.5) A(D)u(x)−λ u(x) =f(x) forx∈Rⁿ+andλ∈ L \ {0}.

Then from a consideration of the pairing between Hp^−sj(Rⁿ+), equipped with the norm||| · |||^(j)_−s

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

j,p⁰,Rⁿ+,1≤

j ≤N, p⁰ =p/(p−1) (see [GK, Theorem 1.1]), we see that there exists a λ⁰ >0 such that for λ ∈ L with |λ| ≥ λ⁰, the differential equation (3.5) has a solution u∈QN

j=1Wp^tj(Rⁿ+) and PN

j=1|||uj|||^(j)_t

j,p,Rⁿ+≤CPN

j=1|||fj|||^(j)_−s

j,p,Rⁿ+.

Here we used the existence of an extension operator from the half-space to the whole space. In fact, it follows from [T, Lemma 2.9.3, p.218] and [GK, Eqn.(1.27)]

that there is afe∈QN

j=1Hp^−sj(Rⁿ) such thatfe

Rⁿ+=f and

N

X

j=1

|||fej|||^(j)_−s

j,p,Rⁿ≤C

N

X

j=1

|||fj|||^(j)_−s

j,p,Rⁿ+.

Hence ifeudenotes the solution of (3.1) whenf there is replaced byfeandu=eu Rⁿ+, then the assertion follows directly from Proposition 3.2.

In this way we reduced problem (1.1), (1.2) to problem (3.2), (3.3), modifying the right-hand sides g_j. Now the proof of Theorem 2.6 follows by standard arguments

from Propositions 3.3 and 3.4.

4. Proof of Proposition 3.4

For the proof of Proposition 3.4, we use a partition of unity in the spaceRⁿ⁻¹of the covariableξ⁰. More precisely, let us fixλ∈ Lwith|λ|>1 sufficiently large and let {_j}^N₁ denote a sequence of numbers satisfying 0 < _j < 1, j = 1, . . . , N (the magnitudes ofλand thej will be specified below). We will establish estimates of the solution of (3.2), (3.3) forξ⁰ belonging to one of the regions

U0:={ξ⁰∈Rⁿ⁻¹:|ξ⁰| ≤ ⁷₈1|λ|^1/m1},

U_r:={ξ⁰∈Rⁿ⁻¹: ¹_8r|λ|^1/mr ≤ |ξ⁰| ≤ ⁷_8r+1|λ|^1/mr+1}, r= 1, . . . , N−1, UN :={ξ⁰∈Rⁿ⁻¹: ¹₈N|λ|^1/mN ≤ |ξ⁰|}.

As the proof of the a priori estimates in the regionsU0 and UN are similar to the corresponding proof in Ur, 1 ≤ r ≤ N−1, we will only consider the latter case. Therefore, throughout this section we will assume that the conditions of Proposition 3.4 hold, thatλ∈ Lwith|λ|sufficiently large, and that 1≤r≤N−1 is fixed.

Our first result concerns the zeros of det(A(ξ⁰, z)−λI_N) as a polynomial inz.

Proposition 4.1. For every fixed r we can choose numbers ⁰ sufficiently small and λ⁰ sufficiently large so that for r+1 ≤ ⁰ and |λ| ≥ λ⁰, det (A(ξ⁰, z)−λ IN) has preciselyNe zeros, say

z_j^(r)(ξ⁰, λ) ^N_j=1^e , lying in C+ and satisfying Imz_j^(r)(ξ⁰, λ)≥C1hξ⁰, λir, |z^(r)_j (ξ⁰, λ)| ≤C2hξ⁰, λir, j= 1, . . . , Nr, Imz_j^(r)(ξ⁰, λ)≥C₁hξ⁰, λi`, |z<sup>(r)</sup><sub>j</sub> (ξ<sup>0</sup>, λ)| ≤C<sub>2</sub>hξ<sup>0</sup>, λi`, j=N_{`−1</sub>, . . . , N<sub>`} for`=r+ 1, . . . , N, whereC2hξ<sup>0</sup>, λi`< C1hξ⁰, λi`+1 for`=r, . . . , N−1.

Proof. To begin with let us observe that

A(ξ⁰, z)−λ IN = A^(r)₁₁(ξ⁰, z)−λ Ir A^(r)₁₂(ξ⁰, z) A^(r)₂₁(ξ⁰, z) A^(r)₂₂(ξ⁰, z)−λ IN−r

!

and that

A^(r)₁₁(ξ⁰, z)−λ I_r=A^(r)₁₁(ξ⁰, z)−λI_r,0−λ(I_r−I_r,0).

Then as a consequence of our hypotheses we know that det(A^(r)₁₁(ξ⁰, z)−λI_r,0) has precisely N_r zeros lying in C+ and that there is a closed contour γ⁺_r(ξ⁰, λ)⊂C+

containing all these zeros in its interior such that forz∈γ_r⁺(ξ⁰, λ) we have Imz≥ C1hξ⁰, λir,|z| ≤C2hξ⁰, λir,and

C₃hξ⁰, λi^2N_r ^r ≤ det

A^(r)₁₁(ξ⁰, z)−λI_r,0

≤C₄hξ⁰, λi^2N_r ^r. By expanding the determinant, it is easy to show that forz∈γ_r⁺(ξ⁰, λ),

det A^(r)₁₁(ξ⁰, z)−λ Ik_r

−det A^(r)₁₁(ξ⁰, z)−λIr,0

≤ C

kr−1

X

`=1

X

1≤i(1)<...<i(`)≤kr−1

`

Y

k=1

|λ|^1/mi(k) hξ⁰, λir

^mi(k)

hξ⁰, λi^2N_r ^r. Similarly, one can show by the Laplace method of expanding a determinant, that

det A(ξ⁰, z)−λ IN

−det A^(r)₁₁(ξ⁰, z)−λ Ir

λ^N−r

≤C(_r+1, λ)

det(A^(r)₁₁(ξ⁰, z)−λ I_r)

· |λ|^N−r

with a factorC(r+1, λ) which can be made arbitrarily small ifr+1is chosen small enough and |λ| is chosen large enough. Hence it follows from Rouch´e’s theorem that in this case, det (A(ξ⁰, z)−λ IN) has preciselyNrzeros contained inγ_r⁺(ξ⁰, λ).

For`=r+ 1, . . . , N, we write

(4.1) A(ξ⁰, z)−λ IN = A<sup>(`)</sup><sub>11</sub>(ξ<sup>0</sup>, z)−λ I` A^{(`)</sup><sub>12</sub>(ξ<sup>0</sup>, z) A<sup>(`)}₂₁(ξ⁰, z) A<sup>(`)</sup><sub>22</sub>(ξ<sup>0</sup>, z)−λ I<sub>N−`</sub>

!

and

A<sup>(`)</sup><sub>11</sub>(ξ<sup>0</sup>, z)−λ I<sub>`</sub>=A<sup>(`)</sup><sub>11</sub>(0, z)−λ I<sub>`,0</sub>+A<sup>(`)</sup><sub>11</sub>(ξ<sup>0</sup>, z)− A<sub>11</sub>(0,0, z)−λ(I<sub>`</sub>−I_`,0).

Then as a consequence of our hypotheses we know that det(A<sup>(`)</sup><sub>11</sub>(0, z)−λI`,0) has preciselyN_{`</sub>−N<sub>`−1}zeros lying inC+and that there is a closed contourγ_r,`⁺(λ)⊂C+

containing all these zeros in its interior such that for z ∈γ<sub>r,`</sub><sup>+</sup>(λ) we have Imz ≥ C<sub>1</sub><sup>0</sup>|λ|<sup>m`1</sup> ,|z| ≤C₂⁰|λ|^m`1 and

C₃⁰|λ|

2N`

m` ≤

det

A<sup>(`)</sup><sub>11</sub>(0, z)−λ I`,0

≤C₄⁰|λ|

2N`

m`. Furthermore, we can show that forz∈γ<sub>r,`</sub>⁺(λ),

|det(A<sup>(`)</sup><sub>11</sub>(ξ<sup>0</sup>, z)−λ I<sub>`</sub>)−det(A<sup>(`)</sup><sub>11</sub>(0, z)−λI<sub>`,0</sub>)|

≤C

|λ|^{m`−11 −m`1} +δ_`,r+1r+1

|λ|^2m`N`. In a similar way as before, we obtain

det(A(ξ⁰, z)−λ IN)−det(A<sup>(`)</sup><sub>11</sub>(ξ<sup>0</sup>, z)−λ I`)λ^N−`

≤C⁰(_r+1, λ)

det(A<sup>(`)</sup><sub>11</sub>(ξ<sup>0</sup>, z)−λ I<sub>`</sub>)

· |λ|^N−`

with a factorC⁰(r+1, λ) which can be made small for smallr+1 and large|λ|. It follows again from Rouch´e’s theorem that then det(A(0, ξ⁰, z)−λ IN) has precisely N`−N<sub>`−1</sub> zeros contained in γ<sub>r,`</sub><sup>+</sup>(λ). Hence since ` was chosen arbitrary, this

completes the proof of the proposition.

Still assuming thatξ⁰ ∈ Ur, 1≤r≤N−1, and⁰ has been chosen sufficiently small and λ⁰ sufficiently large, it follows from [V] that the set of solutions of the differential equation

(4.2) A(ξ⁰, D_n)u(x_n)−λ u(x_n) = 0 forx_n>0

which decay exponentially at∞form a vector space of dimensionNe. Furthermore, this vector space is precisely the direct sum of the vector space (of dimension Nr) spanned by the columns of the matrix

(4.3) Z

γr⁺(ξ⁰,λ)

e^ixnz(A(ξ⁰, z)−λ IN)⁻¹ IN, zIN, . . . , z^m1−1IN dz,

and the N−r vector spaces (of dimensionN`−N`−1, ` =r+ 1, . . . , N) spanned by the columns of each of the matrices

Z

γ_r,`⁺(λ)

e^ixnz(A(ξ⁰, z)−λ IN)⁻¹ IN, zIN, . . . , z^m1−1IN dz for`=r+ 1, . . . , N.

Proposition 4.2. We can choose the constants ⁰ and λ⁰ of Proposition 4.1 suf- ficiently small and sufficiently large, respectively, so that for ξ⁰∈ U_r we have

rank Z

γ_r⁺(ξ⁰,λ)

B^(r,r)(ξ⁰, z) (A(ξ⁰, z)−λ IN)⁻¹ IN, zIN, . . . , z^m1−1IN

dz=Nr, rank

Z

γ_r,`⁺(λ)

B^(r,`)(ξ⁰, z) (A(ξ⁰, z)−λ I_N)⁻¹ I_N, zI_N, . . . , z^m1−1I_N dz

=N_{`</sub>−N<sub>`−1} for`=r+ 1, . . . , N.

(4.4)

The proof of this proposition is based on a thorough study of the integrals (4.4) in comparison with the analog integrals corresponding to the boundary value problems appearing in Definition 2.4. For the details, we refer the reader to [DF, Section 3].

In light of Proposition 4.2 and from what was said in the text preceding that proposition, we are now in a position to present some results pertaining to the solutions of (4.2).

Proposition 4.3. Suppose that ⁰ and λ⁰ have been chosen small enough and large enough, respectively, so that the conclusions of Proposition 4.2 hold. Then for ξ⁰ ∈ Ur, the differential equation (4.2) has Ne linearly independent solutions, {w^(r,ν)(ξ⁰, xn, λ)}^N_ν=1^e , which decay exponentially at∞and satisfy

(4.5) B^{(r,`)</sup>(ξ<sup>0</sup>, Dn)W<sup>(r,`)}(ξ⁰,0, λ) =IN<sub>`</sub>−Nr,`−1 for`=r, . . . , N,

whereN<sub>r,`−1</sub>= (1−δr,`)N<sub>`−1</sub>,W<sup>(r,`)</sup>(ξ⁰, x_n, λ)denotes theN×(N`−Nr,`−1)matrix function whose columns are precisely thew^(r,ν)(ξ⁰, xn, λ)forν=Nr,`−1+1, . . . , N`. Furthermore, we have the representations

W^(r,r)(ξ⁰, xn, λ) = Z

γ⁺r(ξ⁰,λ)

e^ixnz A(ξ⁰, z)−λIN⁻¹

G^(r,r)(ξ⁰, z, λ)dz, W^(r,`)(ξ⁰, x_n, λ) =

Z

γ⁺_r,`(λ)

e^ixnz A(ξ⁰, z)−λI_N−1

G<sup>(r,`)</sup>(ξ<sup>0</sup>, z, λ)dz, for`=r+ 1, . . . , N,

(4.6)

where forr≤`≤N

G^(r,`)(ξ⁰, z, λ) =

Ge<sup>(r,`)</sup>(η<sub>r,`</sub>⁰ , ζ_{r,`</sub>, µ<sub>r,`}, ρ_{r,`</sub>) 0·I<sub>N−`}

,

Ge<sup>(r,`)</sup>(η<sub>r,`</sub>⁰ , ζ_{r,`</sub>, µ<sub>r,`}, ρ_{r,`</sub>) =ρ<sup>−1</sup><sub>r,`}diag(ρ^s_{r,`</sub><sup>1</sup>, . . . , ρ<sup>s</sup><sub>r,`}^{`</sup>)K<sup>(r,`)}(η_{r,`</sub><sup>0</sup> , ζ<sub>r,`}, µ_{r,`</sub>, ρ<sub>r,`})

×diag(ρ^−σ<sub>r,`</sub><sup>Nr,`−1 +1</sup>, . . . , ρ^−σ<sub>r,`</sub><sup>N`</sup>),

ρr,r =hξ⁰, λir,ρr,` =|λ|<sup>1/m`</sup> for` > r, η<sub>r,`</sub>⁰ =ρ⁻¹<sub>r,`</sub>ξ<sup>0</sup>,ζr,` =ρ⁻¹<sub>r,`</sub>z, µr,` =ρ^−m<sub>r,`</sub> <sup>`</sup>λ, and

K<sup>(r,`)</sup>(η<sup>0</sup><sub>r,`</sub>, ζr,`, µr,`, ρr,`) =

K_jk<sup>(r,`)</sup>(η<sup>0</sup><sub>r,`</sub>, ζr,`, µr,`, ρr,`)

j=1,...,k`

k=N_{r,`−1</sub>+1,...,N<sub>`}

,

such that for each pairj, k,K_jk<sup>(r,`)</sup>(η<sup>0</sup><sub>r,`</sub>, ζ_{r,`</sub>, µ<sub>r,`}, ρ<sub>r,`</sub>)is a finite sum of a product of a power of ζr,` and an expression of rational type inη<sub>r,`</sub><sup>0</sup> , ζr,`, µr,` andρr,` (i.e., a rational function of terms which are integrals of rational functions of the components of η<sub>r,`</sub><sup>0</sup> , ζr,`, µr,` andρr,`) and is bounded in modulus by a constant depending only uponη⁰_r,r andµr,r if `=rand only upon ρr,` if` > r.

Proof. We will only consider the case ` =r, the assertion for the case ` > r will follow in a similar way. From Proposition 4.2 we know that there exist matricesJ andZ(z) such that theNr×Nr matrix

Λ(ξ⁰, λ) :=

Z

γ_r⁺(ξ⁰,λ)

B^(r,r)(ξ⁰, z) (A(ξ⁰, z)−λ IN)⁻¹ J

0

Z(z)dz

is invertible where J denotes anr×Nr matrix with the property that each of its columns has precisely one non-zero component, namely 1, andZ(z) = diag(z^q(1), . . . z^q(Nr)), where theq(j) denote non-negative integers not exceedingm1−1. More- over, using the homogeneities of A and B, by scaling arguments we obtain that Λ(ξ⁰, λ) can be written in the form

(4.7) Λ(ξ⁰, λ) =ρdiag ρ^σ1, . . . , ρ^σNr

K1(η⁰, µ) +K2(η⁰, µ, ρ)

×diag ρ^q(1)−s1, . . . , ρ^q(Nr)−sNr ,

whereρ=hξ⁰, λir, η⁰=ρ⁻¹ξ⁰, µ=ρ^−mrλ. Additionally we have

|detK₁(η⁰, µ)| ≥c₁>0,

and the elements of the matrix K2(η⁰, µ, ρ) are expressions of rational type of the arguments and are bounded by

(4.8) ^m_r+1^r+1+|λ|^{mr+1mr −1}+|λ|^1−mr−1mr .

Therefore, for smallr+1 and large|λ|, the matrix Λ(ξ⁰, λ) is invertible and Λ(ξ⁰, λ)⁻¹=ρ⁻¹diag ρ^s1−q(1), . . . , ρ^sNr−qN(r)

×(IN_r+K3(η⁰, µ, ρ))K1(η⁰, µ)⁻¹diag ρ^−σ1, . . . , ρ^−σNr , whereK3(η⁰, µ, ρ) is anNr×Nr matrix function defined by

I_Nr+K₃(η⁰, µ, ρ) = I_Nr+K₁(η⁰, µ)⁻¹K₂(η⁰, µ, ρ)−1

and where each entry ofK3(η⁰, µ, ρ) is a function of its arguments of rational type and is bounded in modulus by the expression (4.8). Hence it follows that

Z

γ⁺r(ξ⁰,λ)

B^(r,r)(ξ⁰, z) A(ξ⁰, z)−λIN⁻¹

Ge⁽¹⁾(η⁰, ζ, µ, ρ) +Ge⁽²⁾(η⁰, ζ, µ, ρ) 0·IN−k_r

dz

=I_Nr, where

Ge⁽¹⁾(η⁰, ζ, µ, ρ) =ρ⁻¹diag(ρ^s1, . . . , ρ^sr)J Z(ζ)K1(η⁰, µ)⁻¹diag(ρ^−σ1, . . . , ρ^−σNr), Ge⁽²⁾(η⁰, ζ, µ, ρ) =ρ⁻¹diag(ρ^s1, . . . , ρ^sr)J Z(ζ)K3(η⁰, µ, ρ)K1(η⁰, µ)⁻¹

×diag(ρ^−σ1, . . . , ρ^−σNr).

(4.9)

Let us denote byW^(r)(ξ⁰, xn, λ) theN×Ne matrix function whose columns are precisely thew^(r,ν)(ξ⁰, x_n, λ). For r≤j≤N, let

Ir,j(ξ⁰, λ) =

B^{(r,`)</sup>(ξ<sup>0</sup>, Dn)W<sup>(r,`1)}(ξ⁰,0, λ)^N

`,`₁=j, Ier,j(ξ⁰, λ) =

Be^{(r,`)</sup>(ξ<sup>0</sup>, Dn)fW<sup>(r,`1)}(ξ⁰,0, λ)^N

`,`₁=j, (4.10)

where

Be<sup>(r,`)</sup>(ξ<sup>0</sup>, Dn) = diag ρ<sup>−σ</sup><sub>r,`</sub><sup>Nr,`−1 +1</sup>, . . . , ρ<sup>−σ</sup><sub>r,`</sub>^N`

B^{(r,`)</sup>(ξ<sup>0</sup>, Dn), fW<sup>(r,`1)}(ξ⁰,0, λ) =W<sup>(r,`1)</sup>(ξ<sup>0</sup>,0, λ) diag ρ<sup>σ</sup><sub>r,`</sub>^{Nr,`1−1 +1}

1 , . . . , ρ^σ<sub>r,`</sub><sup>N`1</sup>

1

. Then we can write

(4.11)

B(ξ⁰, Dn)W^(r)(ξ⁰,0, λ) =Ir,r(ξ⁰, λ)

= diag eρ^σ₁¹, . . . ,ρe^σNf

Ne

Ier,r(ξ⁰, λ) diag ρe^−σ₁ ¹, . . . ,ρe^−σNf

Ne

, where

ρeν=

(ρr,r for 1≤ν≤Nr,

ρ_{r,`</sub> forN<sub>`−1}< ν ≤N<sub>`</sub>, `=r+ 1, . . . , N.

We remark at this point that as a consequence of Proposition 4.5 below, it will be seen that for`6=`1,Be^{(r,`)</sup>(ξ<sup>0</sup>, Dn)fW<sup>(r,`1)}(ξ⁰,0, λ) is an (N`−N<sub>r,`−1</sub>)×(N`<sub>1</sub>−Nr,`₁−1) matrix function whose entries are products of powers ofρr,`/ρr,`₁ and an expression of rational type in the components ofη_r,`⁰

1, µr,`<sub>1</sub>, andρr,`₁.

Definition 4.4. Suppose that the boundary problem (1.1), (1.2) is parameter- elliptic inLand thatλ∈ L. In addition, suppose that with respect to this boundary problem all hypotheses of Proposition 4.3 hold. Then bearing in mind the defini- tions of the various terms introduced above, we say that the operators A(D) and B(D) = (B₁(D), . . . , B

Ne(D))^T are compatible if for eachr satisfying 0≤ r < N and for each pair of integers`, `₁ satisfyingr≤`<sub>1</sub>< N,`₁+ 1≤`≤N, each entry of the matrix Be<sup>(r,`)</sup>(ξ⁰, Dn)fW<sup>(r,`1)</sup>(ξ<sup>0</sup>,0, λ) is bounded in modulus by a constant depending only upon (η<sup>0</sup><sub>r,r</sub>, µr,r) andµr,`for` > r.

Proposition 4.5. Suppose that the hypotheses of Proposition 4.3 hold and that the operators A(D) and B(D) are compatible. Suppose also that 0 ≤ r < N. Then for ⁰ sufficiently small and λ⁰ sufficiently large the matrix function Ier,r(ξ⁰, λ) of (4.11), and hence the matrix function I_r,r(ξ⁰, λ)are invertible and we have

Ir,r(ξ⁰, λ)⁻¹= diag ρe^σ₁¹, . . . ,ρe^σNf

Ne

Ier,r(ξ⁰, λ)⁻¹diag eρ^−σ₁ ¹, . . . ,ρe^−σNf

Ne

, where |deteIr,r(ξ⁰, λ)|> ¹₂, while the entries of eIr,r(ξ⁰, λ)⁻¹ are rational functions of the ρr,` and expressions of rational type in η<sub>r,`</sub>⁰ , µr,`, andρr,` for `≥rand are bounded in modulus by a constant depending only upon(η<sup>0</sup><sub>r,r</sub>, µr,r)andµr,`for` > r.

Lastly, the operatorsA(D)andB(D)are always compatible if (i) the boundary conditions (1.2)are of Dirichlet type or if

(ii) the operatorsA(D)andB(D)are both essentially upper triangular.

Proof. We will only prove the proposition for the case 1≤r < N; the caser= 0 can be similarly treated. Accordingly, let us fix our attention upon (4.10) and suppose that`6=`₁.

Suppose that` > `₁. Then by hypothesis the entries of the matrixBe^{(r,`)</sup>(ξ<sup>0</sup>, D<sub>n</sub>) Wf<sup>(r,`1)}(ξ⁰,0, λ) are bounded in modulus by a constant depending only on (η⁰_r,r, µr,`) and µr,` for ` > r. Let us now show that this boundedness condition is always satisfied if the boundary conditions (1.2) are of Dirichlet type. Indeed, if this latter condition holds, then every entry ofB<sup>(r,`)</sup>_`

1,1(η_r,`⁰

1, ζ_{r,`1</sub>) is 0, while the only non-zero entries ofB<sub>`}^(r,`)

1,2(η_r,`⁰

1, ζr,`<sub>1</sub>) are those lying in rowsN`−ν,ν = 0, . . . , N`−N<sub>`−1</sub>−1, and in column`. Then recalling from Section 1 that we now haves<sub>j</sub> =t<sub>j</sub> =t<sup>0</sup><sub>j</sub> for j= 1, . . . , N, it follows from the foregoing results that the entry in the (N`−ν)-th row and the`-th column ofBe<sup>(r,`)</sup>(ξ⁰, Dn)fW^(r,`1)(ξ⁰,0, λ) is bounded in modulus by

C|λ|

−¹₂−^t

0 k`

t0 k`1

−^ν₂^†( ¹

t0 k`

− ¹

t0 k`1

)

|λ|

t0 k`

t0

k`1 +δ<sub>r,`1r+1</sub> , where 0≤ν^†≤t⁰_k

`−1 and the constantCdepends only upon (η<sub>r,r</sub><sup>0</sup> , µ<sub>r,r</sub>) andµ<sub>r,`</sub>. Our claim concerning Dirichlet boundary conditions are an immediate consequence of this last result.

Note also that whenA(D) andB(D) are essentially upper triangular, then every entry ofB<sup>(r,`)</sup><sub>`</sub>

1,1(η_r,`⁰

1, ζr,`<sub>1</sub>) is 0, and hence the entries ofBe<sup>(r,`)</sup>(ξ⁰, Dn)fW^(r,`1)(ξ⁰,0, λ) are all 0. Thus the boundedness condition also holds under the cited conditions concerningA(D) andB(D).

Let us now consider the case`1 > `. Then it is a simple matter to deduce that for this case each entry of

Be^{(r,`)</sup>(ξ<sup>0</sup>, D<sub>n</sub>)fW<sup>(r,`)}(ξ⁰,0, λ) is bounded in modulus by

(4.12) C

|λ|

1 m`− ¹

m`1 +δr+1,`₁r+1+ (1−δ`₁,N)|λ|

m`1 +1 m`1

−1 , where the constantC depends only uponµ_r,`.