Russian Journal of Mathematical Physics, Vol. 7, No. 4, 2000, pp. 288–318.
Original Russian Text Copyrightc 2000 by Denk, Volevich
English Translation Copyrightc 2000 by /Interperiodica Publishing (Russia)
A Priori Estimates for a Singularly Perturbed Mixed-Order Boundary Value Problem
R. Denk* and L. Volevich**
* NWF I-Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany
** Keldysh Institute of Applied Mathematics, Russian Acad. Sci., Miusskaya sqr. 4, 125047 Moscow, Russia
Received April 13, 2000
Abstract—In this paper we study mixed-order (Douglis–Nirenberg) boundary value prob- lems that depend on a real parameter but are not elliptic with parameter in the sense of Agmon–Agranovich–Vishik. Using the method of the Newton polygon, we provea priori es- timates for the solutions of such problems in the corresponding Sobolev spaces. For the related singularly perturbed problem, the boundary layer structure of the solutions is described. As an application of thea prioriestimate, we obtain new estimates for the transmission problem studied by Faierman [7].
1. INTRODUCTION
The aim of this paper is to study mixed-order systems of partial differential operators with block structure of the form
(1.1) A(x, D, λ) =
A11(x, D) A12(x, D) A21(x, D) A22(x, D)−λI
depending on the real parameterλand acting on a compact manifold M with boundary∂M. We assume that system (1.1) admits a Douglis–Nirenberg structure. In what follows, it is supplied with general boundary conditions. Note that the operator (1.1) can be regarded as a mixture of a parameter-independent Douglis–Nirenberg system and a λ-dependent system. If the matrix A11(x, D) were replaced byA11(x, D)−λI, we would obtain the parameter-dependent mixed-order system treated in [4, 12, 13] and other papers.
There are several reasons to study the (nonstandard) operator matrix (1.1). First of all, this study is related to the investigation of the resolvent of a Douglis–Nirenberg system on a manifold with boundary. In general, it is impossible to assign a parameter-dependent quasi-homogeneous principal symbol to such a system. For instance, for the 2×2 Douglis–Nirenberg system
(1.2)
A11(x, D)−λ A12(x, D) A21(x, D) A22(x, D)−λ
with ordA11 > ordA22, there is no definite “weight” for the parameter λ. Therefore, we cannot apply the theory of parameter ellipticity developed by Agmon [1] and Agranovich–Vishik [3]. The aim of the study of boundary value problems for system (1.2) is to prove the unique solvability and to obtainuniformestimates (with respect toλ) for the solution for largeλ. For scalar equations and for systems of constant order, the corresponding results follow from the Agmon–Agranovich–Vishik theory. For general mixed-order systems, this problem seems to be still open (see [4, 12, 13] for partial results).
To address this problem in its full generality, one can use the concept of Newton polygon, which proves to be very fruitful both in the theory of parameter-dependent Douglis–Nirenberg systems on closed manifolds and in the theory of singularly perturbed operator pencils (see, e.g., [4, 5, 6]).
The main idea of the approach based on the notion of Newton polygon is to assign various weights
** The work of the second author is supported by RFBR grant No. 0001-00387.
288
Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/5070/
to the parameter λ and to obtain, for each weight, a quasi-homogeneous symbol of the operator.
If in (1.2) we assign to λthe weight ordA22, then the quasi-homogeneous symbol with respect to this weight becomes
A11(x, ξ) A12(x, ξ) A21(x, ξ) A22(x, ξ)−λ
,
which is exactly the principal quasi-homogeneous symbol of (1.1). For each weight of the parame- ter, we obtain an operator with quasi-homogeneous principal symbol. Thus, we face the following question: Under what conditions can we prove uniform estimates for the solutions of (1.1)?
Another reason to study operators of the form (1.1) lies in the close connection between the parameter-dependent operator (1.1) and singularly perturbed boundary value problems (cf. [8, 14, 16]). Let us consider the equation A(x, D, λ)u = 0 with A given by (1.1) and endow it with appropriate boundary conditions. By settingλ=ε−1and multiplying the second block row in (1.1) byε, we obtain the system
(1.3) A11(x, D)u1+A22(x, D)u2 = 0,
ε A21(x, D)u1+A22(x, D)u2
−u2 = 0.
The problem is to find conditions on the boundary operators under which estimates for the solution can beuniformwith respect toεasε→0. Moreover, it is desirable to find an asymptotic expansion in ε for the solution and to construct solutions of boundary-layer type. Since equations (1.3) are only a reformulation of the equation A(x, D, λ)u = 0, it follows that the results concerning (1.1) immediately imply results on singularly perturbed boundary value problems of the form (1.3).
Finally, we can mention the question of spectral asymptotics for boundary value problems and transmission problems with indefinite weight functions (see [2] and the references therein). For the case in which the weight function vanishes identically in some subdomain, the model transmission problem that arises was recently studied by Faierman [7]. One of the main goals is again to obtain uniform estimates for the solutions of the transmission problem because this finally leads to the spectral asymptotics [2].
Summing up, we can see that, in all these applications, the question is to find conditions (in particular, on the boundary operators) ensuring uniform estimates (with respect to the parameter) for the solutions of the corresponding problems. Moreover, it is desirable that these conditions be similar to the classical conditions (parameter-independent or parameter-dependent in the sense of Agmon–Agranovich–Vishik) of Shapiro–Lopatinskii type. As in the theory of parameter ellipticity, the estimates must work in appropriate parameter-dependent Sobolev spaces, and hence the ad- ditional question of introducing spaces suitable for operators of the form (1.1) arises. To be more exact, we endow the standard Sobolev space with a parameter-dependent norm adjusted to our problems. Here the main difficulty is in the definition of the boundary Sobolev spaces (boundary parameter-dependent norms).
The first result in this direction is given in the paper [6], which treats scalar operator pencils such that their dependence on the parameterλis polynomial and the leading term with respect toλ contains an operator of positive order. (Note that, roughly speaking, the determinant of the symbol of system (1.1) is of this form.) The case of systems of the form (1.1) is more complicated than that of scalar pencils with polynomial dependence onλbecause, in our case, a solution consists of several components belonging to different parameter-dependent Sobolev spaces and connected by boundary conditions and by the operatorA(x, D, λ).
The present paper solves the above problem for system (1.1) by defining the notion of weak parameter-ellipticity and by proving ana priori estimate for the corresponding weakly parameter- elliptic boundary value problems (with general boundary conditions). The notion of weak parameter ellipticity and the norms in the a priori estimates are defined in terms of the Newton polygon corresponding to the boundary value problem, continuing the program originating from [4–6]. It seems that the approach based on the notion of Newton polygon provides an appropriate tool to study and understand nonstandard boundary value problems, including those of the form (1.1).
The paper is organized as follows. In Section 2 we formulate the precise assumptions on the struc- ture of the operator (1.1) and of the boundary operators and define the notion of weak parameter- ellipticity for related boundary value problems. In Section 3 we prove the basic ODE estimate for
the so-called fundamental system of solutions of the corresponding model problem in the half-space.
These estimates are rewritten in terms of the Newton polygon in Section 4, where the basic defini- tions for Sobolev spaces associated with the Newton polygon can also be found. The ODE estimates form a substantial step in the proof of a priori estimates presented in Section 5. In Section 6 we apply these results to transmission problems, and in Section 7 we discuss the formal asymptotic solution for related singularly perturbed problems.
2. NEWTON POLYGON AND WEAKLY
PARAMETER-ELLIPTIC BOUNDARY VALUE PROBLEMS
As in the introduction, let A(x, D, λ) = (aij(x, D))i,j=1,...N be anN×N operator matrix acting on a compact manifoldM with boundary∂M. We assume that this operator has Douglis–Nirenberg structure, i.e., there are nonnegative integerssj and tj,j= 1, . . . , N, such that
(2.1) ordaij(x, D)≤si+tj (i, j= 1, . . . , N). The operatoraij(x, D) is assumed to be of the form
aij(x, D) = X
|α|≤si+tj
aijα(x)Dα
with infinitely smooth scalar coefficients. The principal symbol ofaij(x, D) is defined as a(0)ij (x, ξ) = X
|α|=si+tj
aijα(x)ξα.
Note that a(0)ij = 0 if the order of aij is less than si+tj. Here and in the following, we use the standard multi-index notation
Dα= (−i)|α|( ∂
∂x1
)α1· · ·( ∂
∂xn
)αn, ξα=ξ1α1·. . .·ξαnn. We assume that there exist numbers N1 and N2 withN1+N2=N for which
(2.2) s1+t1=s2+t2=· · ·=sN1+tN1= 2µ , sN1+1+tN1+1=· · ·=sN +tN = 2m−2µ, with integers m > µ >0. Accordingly, we represent the operator matrix A(x, D, λ) in the form
A(x, D, λ) =
A11(x, D) A12(x, D) A21(x, D) A22(x, D)−λIN2
, where
A11 = (aij(x, D))i,j=1,...,N1, A12 = (aij(x, D))i=1,...,N1, j=N1+1,...,N, A21 = (aij(x, D))i=N1+1,...,N, j=1,...,N1, A22 = (aij(x, D))i,j=N1+1,...,N.
HereIN2 stands for theN2×N2 identity matrix. The principal symbol of the operatorA(x, D, λ) is given by
A(0)(x, ξ, λ) := A(0)11(x, ξ) A(0)12(x, ξ) A(0)21(x, ξ) A(0)22(x, ξ)−λIN2
!
whereA(0)ij stand for the principal symbols ofAij, respectively. In what follows, to make expressions homogeneous, we replaceλbyq2m−2µ. For brevity we writeA(x, D, q) instead ofA(x, D, q2m−2µ).
Assume that
(2.3) B(x, D) =
bij(x, D)
i=1,...,N1µ+N2(m−µ) j=1,...,N
is a matrix of boundary operators of the form bij(x, D)u=
X
|β|≤mi+tj
bijβ(x)Dβu
∂M
of order≤mi+tj with coefficientsbijβ ∈C∞(M). Here themi are integers. We also assume that (2.4) m1≤m2≤ · · · ≤mN1µ < mN1µ+1≤ · · · ≤mN1µ+N2(m−µ).
In the following, denote the number of boundary conditions by
(2.5) R:=N1µ+N2(m−µ).
We also use the abbreviations
(2.6) R1:=N1µ, R2:=N2(m−µ).
The aim of this paper is to find a priori estimates for the solutions
u=
u1
... uN
of the boundary value problem
(2.7) A(x, D, q)u=f =
f1
... fN
, B(x, D)u=g=
g1
... gR
.
Assume that this boundary value problem satisfies an ellipticity condition, the so-called weak parameter ellipticity. We first introduce this notion for the operatorA.
Definition 2.1. A symbol A(x0, ξ, q) is said to be weakly parameter elliptic (with parameter q∈[0,∞)) at a point x0∈M if the inequality
(2.8) |detA(0)(x0, ξ, q)| ≥C|ξ|2R1 (q+|ξ|)2R2 (ξ ∈Rn, q ∈[0,∞))
holds with some constantCthat does not depend onξandq. An operatorA(x, D, q) and its symbol are said to be weakly parameter elliptic inM if (2.8) holds for every x0∈M. A similar definition is related to operators acting onRn.
Remark 2.2. a) IfAis weakly parameter elliptic inM, then we may assume that the constant C in (2.8) does not depend on x0 as well (by the continuity and compactness).
b) The right-hand side of (2.8) is closely related to the so-called Newton polygon corresponding to the symbol detA(x0, ξ, q). This polygon is defined as the convex hull of the set{(0,0), (0,2R2), (2R1,2R2), (2R1+ 2R2,0)}(cf. Figure 1). The points (2R1,2R2) and (2R1+ 2R2,0) of this polygon correspond to the expressions|ξ|2R1q2R2 and |ξ|2R1+2R2. The sum of these two terms is equivalent to the right-hand side of (2.8). We discuss this point of view below in Section 4.
- 6
i k
@
@
@
@
@
@
• •
•
2R1+ 2R2
2R1
2R2
Fig. 1. The Newton polygonN(2R1,2R2). c) Write
P(x, ξ, q) := detA(0)(x, ξ, q).
For a chosenx, this is a polynomial in (ξ, q) of order 2R= 2µN1+ 2(m−µ)N2= 2R1+ 2R2that is of order 2R2 with respect toq. We can rewrite this polynomial in the form
P(x, ξ, q) =P2R(x, ξ) +q P2R−1(x, ξ) +· · ·+q2R2P2R1(x, ξ),
wherePj are polynomials inξ of orderj. It follows from the definition of the determinant that P2R(x, ξ) = detA(0)(x, ξ,0), P2R1(x, ξ) = (−1)N2detA(0)11(x, ξ).
Lemma 3.2 of [5] leads to the following equivalent conditions for the weak parameter ellipticity.
Lemma 2.3. An operator A is weakly parameter elliptic in M if and only if the following conditions hold:
(i) For anyx0∈M, the matrix operatorA(x, D,0)is elliptic in the sense of Douglis-Nirenberg.
(ii) For anyx0∈M, the matrix operator A11(x, D)is elliptic in the sense of Douglis-Nirenberg.
(iii) detA(0)(x0, ξ, q)6= 0 for allx0∈M, ξ∈Rn\ {0}, and q∈(0,∞).
If A is weakly parameter elliptic and if a point x0 ∈ ∂M is chosen, then we can rewrite A in local coordinates corresponding to x0, i.e., we choose a coordinate system in a neighborhood of x0 such that, locally, ∂M is given by the equation xn = 0 and M is given by the inequality xn >0. We write x = (x0, xn) for the variables and (ξ0, τ) for the dual variables. Due to 2.3 (iii), the polynomial detA(0)(x0, ξ0, τ, q) has no real rootsτ forξ0= (ξ1, . . . , ξn−1)6= 0. Hence, if n >2, then the polynomial has exactly R roots in C+ :={z ∈C: Imz >0} withR introduced in (2.5).
For n = 2, this condition is our additional assumption in what follows. In the local coordinates corresponding tox0∈∂M we set (cf. [4])
(2.9) Q(x0, τ) :=τ−2R1detA(0)(x0,0, τ,1) (τ ∈R).
As was mentioned in the introduction, the boundary value problem (2.7) can equivalently be formulated as a singularly perturbed boundary value problem. The above polynomialQ can seem to be somewhat unusual in the elliptic theory, but it is a basic notion in the singular perturbation theory. The condition of the next definition first appeared in the paper of Vishik–Lyusternik [16]
on singular perturbations under the title of the condition of regular degeneration. We stress that this is an example showing how useful it is to change the point of view between large parameter problems and singular perturbations. Note that, according to (2.8), the polynomial (2.9) has no real roots, and its roots belong toC+ and C−.
Definition 2.4. We say that A(x, ξ0, τ, q) satisfies the Vishik–Lyusternik condition if, for each x0∈∂M, the polynomialQ(x0, τ) has exactlyR2roots in C+.
Remark 2.5. For the important special case in which N = 2 and N1=N2= 1, the condition of Definition 2.4 is satisfied automatically if A is weakly parameter elliptic. Indeed, in this case, A(0)ij (x0,0, τ) is a (scalar) homogeneous polynomial in τ of degree si +tj. Thus, we can write A(0)ij (x0,0, τ) =aijτsi+tj with aij ∈C. Due to the weak parameter ellipticity, we have
detA(0)(x0,0, τ,1)6= 0 for all τ ∈R\ {0}.
Therefore,Q(x0, τ) = (a11a22−a12a21)τ2R2−a11 has exactly R2 roots in C+ and no real roots.
Now let us formulate an analog of the standard Shapiro–Lopatinskii condition for the boundary value problem (1.1), (2.3) in which we have a more delicate dependence on the parameter than in the classical case of the Agmon–Agranovich–Vishik theory (for results on the parameter elliptic Douglis–Nirenberg systems, see also [15]). To this end, we consider x0 ∈ ∂M and introduce the corresponding local reference system. According to the block form of the matrixA, we write
v(t) =
v(1)(t) v(2)(t)
,
where v(1) consists of the first N1 components of the vector v and v(2) consists of the other N2
components. We respectively write B(x, D) =
B11(x, D) B12(x, D) B21(x, D) B22(x, D)
,
where Bij are matrix differential operators of sizes R1×N1, R1×N2, R2×N1, and R2×N2, respectively. We also set
B1(x, D) := B11(x, D), B12(x, D)
, B2(x, D) := B21(x, D), B22(x, D) .
Definition 2.6. The boundary value problem (A, B) of the form (1.1), (2.3) is said to beweakly parameter elliptic (with parameter in [0,∞)) if the following conditions are satisfied.
(i) A(x, D, q) is weakly parameter elliptic in M.
(ii) For anyx0∈∂M,ξ0∈Rn−1\ {0},q∈[0,∞), andg= (g1, . . . , gR)∈CR, the problem A(0)(x0, ξ0, Dt, q)
w(1)(t) w(2)(t)
= 0
0
(t >0), (2.10)
B(0)(x0, ξ0, Dt)
w(1)(t) w(2)(t)
t=0
=g , (2.11)
w(i)(t)→0 as t→ ∞ (i= 1,2), whereDt=−i∂/∂t, has a unique solution.
(iii) For any x0∈∂M,ξ0∈Rn−1\ {0}, and h(1)∈CR1, the problem A(0)11(x0, ξ0, Dt)w(1)(t) = 0 (t >0), (2.12)
B11(0)(x0, ξ0, Dt)w(1)(t)
t=0=h(1), (2.13)
w(1)(t)→0 as t→ ∞,
has a unique solution.
(iv) For anyx0∈∂M and any vector h(2)∈CR2, the problem A(0)(x0,0, Dt,1)
v(1)(t) v(2)(t)
= 0
0
(t >0), (2.14)
B2(0)(x0,0, Dt)
v(1)(t) v(2)(t)
t=0
=h(2), (2.15)
v(i)(t)→0 as t→ ∞ (i= 1,2), has a unique solution.
Remark 2.7. a) Condition (iii) in Definition 2.6 is the ordinary Shapiro–Lopatinskii condition for the Douglis–Nirenberg systemA11(x, D) with boundary operators given byB11(x, D). Similarly, setting q = 0 in 2.6 (ii), we obtain the ordinary (parameter-independent) Shapiro–Lopatinskii condition for the entire system (A(x, D,0), B(x, D)).
b) For ξ0 = 0, condition 2.6 (ii) fails in general (even for positive q). Thus, our definition substantially differs from the Agmon–Agranovich–Vishik definition of ellipticity with parameter.
Due to the homogeneity, it suffices to consider (ii) on the hemisphere {(ξ0, q)∈Rn−1×[0,∞) :
|ξ0|2 +q2 = 1}. In contrast to the Agmon–Agranovich–Vishik condition (which is regarded at the points of the closed hemisphere), condition 2.6 (ii) deals with the punctured hemisphere with deleted point (0,1). In a sense, conditions (iii) and (iv) give a completion for this point. Let us discuss this point of view.
Consider the transformation
(ξ0, q)7−→ ξ0
(|ξ0|2+q2)1/2, q (|ξ0|2+q2)1/2
! ,
which maps every point (ξ0, q) ∈ Rn−1×[0,∞) with |ξ0|+q 6= 0 to a point of our hemisphere.
Under this transformation, the point (0,1) is not only the image of a point (0, q) for finiteq > 0 but also the limit of the images of points (ξ0, q) as q →+∞.
Condition 2.6 (iv), whereξ0= 0, corresponds to the first group of points, i.e., to the images of the points (0, q). Now letq→ ∞. By settingλ=ε−1and considering the homogeneous right-hand side, we come to system (1.3) with small parameter. Passing to the limit asε→0, we obtainu2= 0 in (1.3). Thus, we must solve the systemA11(x, D)u1= 0 with an overdetermined system of boundary conditions. If we take only the first R1 boundary conditions, then we obtain a problem for which condition 2.6 (iii) guarantees solvability. To satisfy all boundary conditions, we need condition 2.6 (iv). Due to the Vishik–Lyusternik theory, this condition allows us to add a combination of boundary layers to the above solution.
We continue the study of the relationship between condition (iv) and the existence of boundary layers in Section 7.
c) Conditions 2.6 (ii)–(iv) can be formulated algebraically, cf. [17]. For instance, 2.6 (iv) is equivalent to the following condition.
(iv0) Choosex0∈∂M. Let γ(2) be a contour inC+ enclosing all zeros ofQ(x0,·) whose imaginary parts are positive. Then the rank of the matrix
Z
γ(2)
B2(0)(x0,0, τ) h
A(0)(x0,0, τ,1) i−1
IN, τ IN, . . . , τ2m−1IN
dτ is equal toR2 (and thus the rank of this matrix is the largest possible).
3. AN ESTIMATE FOR THE SOLUTIONS OF THE SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS
The aim of this section is to find estimates for the solutions of (2.10)–(2.11) under the assumption of weak parameter ellipticity. Thus, throughout this section we assume thatAsatisfies the Vishik–
Lyusternik condition and that (A, B) satisfies the conditions of Definition 2.6. We choosex0∈M and write the boundary value problem in coordinates corresponding to x0. We write A(ξ, q) :=
A(0)(x0, ξ, q),B(ξ) :=B(0)(x0, ξ), andQ(τ) :=Q(x0, τ).
Lemma 3.1. Let γ(1) be a contour in C+ enclosing all the zeros of detA11(ξ0,·) for |ξ0| = 1 whose imaginary parts are positive and let γ(2) be a contour in C+ enclosing all the zeros of Q in C+. Then there exists a q0 > 0 and an enumeration τ1(ξ0, q), . . ., τR(ξ0, q) of the zeros of detA(ξ0,·, q) with positive imaginary parts such that, for all q ≥ q0 and |ξ0| = 1, the following conditions hold:
(i) γ(1) enclosesτ1(ξ0, q), . . . , τR1(ξ0, q),
(ii) γ(2) enclosesq−1τR1+1(ξ0, q), . . . , q−1τR(ξ0, q).
Proof. As was noted in Remark 2.2 c), the determinant P(ξ0, τ, q) = detA(ξ0, τ, q) is weakly parameter elliptic in the sense of [5]. The behavior of the zeros of this polynomial is described in Lemma 3.5 of [5]. As a result, under an appropriate enumeration, the set{τ1(ξ0, q),. . .,τR1(ξ0, q)}
tends to the set{τ10(ξ0),. . . ,τR01(ξ0, q)} of all zeros of detA11(ξ0,·) with positive imaginary parts, and q−1τj(ξ0, q) →τj1 forj =R1+ 1, . . . , R, where τR11+1,. . . ,τR1 are the zeros ofQ inC+. This implies (i) and (ii) as above.
Denote by M(ξ0, q) the finite-dimensional space of solutions of the homogeneous equation A(ξ0, Dt, q)v(t) = 0 (t >0),
and let M+(ξ0, q) be the subspace of M(ξ0, q) consisting of solutions tending to zero as t tends to +∞. The dimension of this space isR. By Lemma 3.1, this space can be represented as the direct sumM+(ξ0, q) =M(1)+ (ξ0, q) ˙+M(2)+ (ξ0, q),where the first space on the right-hand side corresponds to the zeros bounded as q→ ∞ and the other subspace corresponds to the other zeros. Note that dimM(1)+ (ξ0, q) =R1 and dimM(2)+ (ξ0, q) =R2 forq≥q0.
Lemma 3.2. Suppose that condition (iii) of Definition 2.6 is satisfied. Then there is a value q0>0 such that, for |ξ0|= 1 andq ≥q0, the following assertions hold.
(i) The problem on the half-line,
A(ξ0, Dt, q)v(t) = 0 (t >0), (3.1)
B1(ξ0, Dt)v(0) =h(1), (3.2)
v(t)∈M(1)+ (ξ0, q) (3.3)
is uniquely solvable for arbitrary h(1) ∈CR1.
(ii) There exists an N ×R1 rectangular matrix N(1)(ξ0, τ, q), polynomial with respect to τ and bounded for|ξ0|= 1, q≥q0, and τ ∈γ(1),for which
1 2πi
Z
γ(1)
B1(ξ0, τ)A−1(ξ0, τ, q)N(1)(ξ0, τ, q)dτ =IR1, where γ(1) is the same contour as in Lemma 3.1.
Proof. (i). As in condition (iv0) of Remark 2.7, a necessary and sufficient condition for the unique solvability of problem (3.1)–(3.2) can be written in the following form [17]: the rank of the matrix
Λ(ξ0, q) := 1 2πi
Z
γ(1)
B1(ξ0, τ)A−1(ξ0, τ, q)(IN, τ IN, . . . , τ2m−1IN)dτ is equal toR1. Let us show first that
(3.4) A−1(ξ0, τ, q) =
A−111(ξ0, τ) 0
0 0
+O(q−2(m−µ)). To prove (3.4), we rewrite the matrix A(ξ, q) in the form
A(ξ, q) =
A11(ξ) 0 0 −qm−µIN2
h
IN +H(ξ, q)i
IN1 0 0 qm−µIN2
, where
H(ξ, q) :=
0 q−(m−µ)A−111(ξ)A12(ξ)
−q−(m−µ)A21(ξ) −q−2(m−µ)A22(ξ)
.
It follows from this relation and from the condition m > µ that the inverse matrix A−1(ξ, q) =:
G(ξ, q) exists forq sufficiently large and is equal to G(ξ, q) =
IN1 0 0 q−(m−µ)IN2
h
IN −H+O q−2(m−µ)i
A−111(ξ) 0 0 −q−(m−µ)IN2
=
A−111(ξ) 0
0 0
+O(q−2(m−µ)). It follows from (3.4) that
q→∞lim Λ(ξ0, q) = 1 2πi
Z
γ(1)
B11(ξ0, τ)A−111(ξ0, τ),0
(IN, τ IN, . . . , τ2m−1IN)dτ .
Denote byM(ξ0, q) the R1×R1 submatrix of Λ(ξ0, q) that consists of the columns of Λ(ξ0, q) with the indices kN+j (k= 0, . . . , µ−1; j= 1, . . . , N1). Then
M(ξ0) := lim
q→∞M(ξ0, q) = 1 2πi
Z
γ(1)
B11(ξ0, τ)A−111(ξ0, τ)(IN1, . . . , τµ−1IN1)dτ .
By condition (iii) of Definition 2.6, the matrixM(ξ0) is nonsingular. It follows from (3.4) that the matrixM(ξ0, q) is also nonsingular forq ≥q0, where q0 is large enough, which proves (i).
To prove (ii), it suffices to define (cf. [17]) an N ×R1-dimensional matrix N(1)(ξ0, τ, q) by the formula
N(1)(ξ0, τ, q) :=
N1(1)(ξ0, τ, q) 0
with theN1×R1-dimensional matrix N1(1)(ξ0, τ, q) given by
N1(1)(ξ0, τ, q) = (IN1, τ IN1, . . . , τµ−1IN1)M−1(ξ0, q).
Lemma 3.3. Suppose that condition(iv)of Definition2.6is satisfied. Then there exists aq0>0 such that, for |ξ0|= 1 and q≥q0, the following statements hold.
(i) The problem on the half-line
A(ξ0, Dt, q)v(t) = 0 (t >0), (3.5)
B2(ξ0, Dt)v(0) =h(2), (3.6)
v(t)∈M(2)+ (ξ0, q) (3.7)
is uniquely solvable for arbitrary h(2)∈CR2.
(ii) There exists an N ×R2 rectangular matrix N(2)(ξ0, τ, q) polynomial in τ and bounded for all
|ξ0|= 1, q≥q0 and τ ∈γ(2) for which 1
2πi Z
γ(2)
B2
ξ0 q, τ
A−1
ξ0 q, τ,1
N(2)(ξ0, τ, q)dτ =IR2, where γ(2) is the same contour as in Lemma 3.1.
Proof. (i). As in Lemma 3.2, we use the criterion in [17] for the unique solvability of the problem on the half-line. This criterion can trivially be modified, namely, the matricesτlIN can be replaced by (cτ)lIN with arbitrary real c6= 0.
To prove (i), note that the rank of the matrix 1
2πi Z
γ(2)(q)
B2(ξ0, τ)A−1(ξ0, τ, q) IN,τ
qIN, . . . ,τ q
2m−1
IN
dτ
is equal to R2, where γ(2)(q) := {qτ : τ ∈ γ(2)}. As in the proof of Lemma 3.2, write G(ξ, q) = A−1(ξ, q). The element gij(ξ, q) of the matrix G is a homogeneous function in (ξ, q) of degree
−sj −ti. Therefore, making the change of variables τ = qτ˜ in the above integral, we obtain the matrix
qmR1 +1
. .. qmR
Λ(ξ0, q)
q1−s1
. ..
q1−sN
, where
Λ(ξ0, q) = 1 2πi
Z
γ(2)
B2
ξ0 q, τ
A−1ξ0 q, τ,1
(IN, τ IN, . . . , τ2m−1IN)dτ .
Obviously, we must show that, in a neighborhood of q = +∞, the rank of this matrix is R2. By condition (iv) of Definition 2.6, the rank of the matrix
Λ(0,1) := 1 2πi
Z
γ(2)
B2(0, τ)A−1(0, τ,1)(IN, τ IN, . . . , τ2m−1IN)dτ
is equal to R2. For τ ∈ γ(2) and |ξ0| = 1, the matrix-valued function B2(ξ0/q, τ) A−1(ξ0/q, τ,1) uniformly tends toB2(0, τ)A−1(0, τ,1) asq →+∞. This proves (i).
To prove (ii), we choose an invertible submatrix of Λ(0,1) of sizeR2×R2. Let the indices of the columns of this submatrix be j1, . . . , jR2. Then, for large q, the submatrix of Λ(ξ0, q) consisting of the same columnsj1, . . . , jR2 is invertible as well. In other words, there exists a 2mN×R2 matrix N˜(2)(ξ0, q) such that Λ(ξ0, q) ˜N(2)(ξ0, q) =IR2. Now it remains to set
N(2)(ξ0, τ, q) := (IN, τ IN, . . . , τ2m−1IN) ˜N(2)(ξ0, q).
Lemma 3.4. Let
w(t) =
w(1)(t, ξ0, q) w(2)(t, ξ0, q)
be the solution of (2.10)–(2.11). Then, for |ξ0| = 1 and sufficiently large q, the function w can be written in the form
(3.8)
w(t, ξ0, q) = 1 2πi
Z
γ(1)
A−1(ξ0, τ, q)N(1)(ξ0, τ, q)eitτ dτ ·ψ(1)(ξ0, q)
+
q−t1
. .. q−tN
1 2πi
Z
γ(2)
A−1ξ0 q, τ,1
N(2)(ξ0, τ, q)eitqτdτ·ψ(2)(ξ0, q)
whereγ(1) andγ(2)are as in Lemma3.1andN(1), N(2)are as in Lemmas3.2and3.3, respectively.
The vectorsψ(1)(ξ0, q)∈CR1 and ψ(2)(ξ0, q)∈CR2 are given by the formula ψ(1)(ξ0, q)
ψ(2)(ξ0, q)
=M(ξ0, q)g
for some matrixM(ξ0, q) =
Mij(ξ0, q)
i,j=1,...,R such that
(3.9) |Mij(ξ0, q)| ≤
C , i≤R1, j≤R1, C qmR1−mj, i≤R1, j > R1, C q−mR1 +1, i > R1, j≤R1, C q−mj, i > R1, j > R1.
Proof. We are following the scheme of [6], Lemmas 3.1–3.3, which uses an idea of Frank [8].
We definew by (3.8) with unknown vectors ψ(i)(ξ0, q). First we note that the ordinary differential operatorA(ξ0, Dt, q) sends each of the two integrals in the right-hand side of (3.8) to zero. For the first integral, this is obvious. As for the other integral, we note that
A(ξ0, Dt, q)
q−t1
. .. q−tN
=
qs1
. .. qsN
Aξ0 q ,1
qDt,1 .
To computeψ(i), we apply the boundary operators to (3.8) and obtain B1(ξ0, Dt)w(t, ξ0, q) =ψ(1)(ξ0, q) + ∆1
1 2πi
Z
γ(2)
B1
ξ0 q , τ
A−1
ξ0 q, τ,1
N(2)(ξ0, τ, q)dτ ψ(2)(ξ0, q) with
∆1:=
qm1
. .. qmR1
.
Here we used Lemma 3.2 and the homogeneity ofbij. In the same way, using Lemma 3.3, we obtain B2(ξ0, Dt)w(t, ξ0, q) = 1
2πi Z
γ(1)
B2(ξ0, τ)A−1(ξ0, τ, q)N(1)(ξ0, τ, q)dτ ψ(1)(ξ0, q) + ∆2ψ(2)(ξ0, q)
with
∆2:=
qmR1 +1
. .. qmR
. Therefore, condition (2.11) leads to the system of linear equations (3.10)
IR1 ∆1H12
H21 ∆2
ψ(1) ψ(2)
=g , where we used the notation
H12 := 1 2πi
Z
γ(2)
B1
ξ0 q, τ
A−1ξ0 q, τ,1
N(2)(ξ0, τ, q)dτ , H21 := 1
2πi Z
γ(1)
B2(ξ0, τ)A−1(ξ0, τ, q)N(1)(ξ0, τ, q)dτ .
As was shown in the proof of Lemma 3.3 in [6], for largeq, the matrix in (3.10) is invertible (here we use the inequalitymR1 < mR1+1, see (2.4)), and its inverse is given by
M =
G1 −G1∆1H12∆−12
−G2∆−12 H21 G2∆−12
with G1 := (IR1 −∆1H12∆−12 H21)−1 and G2 := (IR2 −∆−12 H21∆1H12)−1. Since the matrices G1 and G2 are bounded for sufficiently large q, the estimate (3.9) follows. (Note that in [6], in the situation of Lemma 3.3, the additional condition mj ≥ 0 was imposed. However, it readily follows from the proof of this lemma that the additional condition is not needed, and the condition mR1 < mR1+1is substantial.)
For j= 1, . . . , R, denote by
wj =
wj1(t, ξ0, q) ... wjN(t, ξ0, q)
the solution of (2.10)–(2.11) in which g is replaced by thejth unit vector ej ∈CR. Theorem 3.5. a)For i= 1, . . . , N1, j = 1, . . . , R, and l= 0,1,2, . . ., we have (3.11)
kDltwji(·, ξ0, q)kL2(R+)≤C
|ξ0|l−mj−ti−1/2, l≤mR1+1+ti, j≤R1,
|ξ0|mR1 +1−mj(q+|ξ0|)l−mR1 +1−ti−1/2, l > mR1+1+ti, j≤R1,
|ξ0|l−mR1−ti−1/2(q+|ξ0|)mR1−mj, l≤mR1+ti, j > R1, (q+|ξ0|)l−mj−ti−1/2, l > mR1+ti, j > R1, for allξ0∈Rn−1\ {0} and q∈[0,∞).
b) For i=N1+ 1, . . . , N, j= 1, . . . , R, andl= 0,1,2, . . ., we have (3.12)
kDtlwji(·, ξ0, q)kL2(R+) ≤C
|ξ0|l−mj−1/2(q+|ξ0|)−ti, l≤mR1+1, j≤R1,
|ξ0|mR1 +1−mj(q+|ξ0|)l−mR1 +1−ti−1/2, l > mR1+1, j≤R1,
|ξ0|l−mR1−1/2(q+|ξ0|)mR1−mj−ti, l≤mR1, j > R1, (q+|ξ0|)l−mj−ti−1/2, l > mR1, j > R1,
for allξ0∈Rn−1\ {0} and q∈[0,∞).
Proof. a) We assume first thati≤N1. Due to the unique solvability of (2.10)–(2.11), forξ06= 0 and q∈[0,∞) we have
wji(t, ξ0, q) =|ξ0|−mj−tiwji
|ξ0|t, ξ0
|ξ0|, q
|ξ0|
, and hence
kDnlwji(·, ξ0, q)kL2(R+)=|ξ0|l−mj−ti−1/2 Dnlwji
·, ξ0
|ξ0|, q
|ξ0|
L2(R+).
Therefore, it suffices to show that, for|ξ0|= 1 and for all q∈[0,∞), the following estimate holds:
(3.13) kDltwji(·, ξ0, q)kL2(R+)≤C
1, l≤mR1+1+ti, j≤R1,
˜
ql−mR1 +1−ti−1/2, l > mR1+1+ti, j≤R1,
˜
qmR1−mj, l≤mR1+ti, j > R1,
˜
ql−mj−ti−1/2, l > mR1+ti, j > R1,
where ˜q := max{1, q}. For |ξ0| = 1 and q ∈ [0, q0] this is true indeed due to continuity and compactness (the proof uses condition (ii) of Definition 2.6). Thus, let q be sufficiently large. We use the representation (3.8) for g = ej. Considering Dtlwj instead of wj, we obtain an additional factorτl in the integral overγ(1)and an additional factorqlτl in the integral overγ(2). Integrating Dnlwj with respect to t, we obtain
(3.14) kDtlwjikL2(R+)≤C
|ψ(1)(ξ0, q)|+ql−ti−1/2|ψ(2)(ξ0, q)|
where we used the fact that we may chooseγ(1) and γ(2) at a positive distance from the real axis, and | · |stands for the standard norm in the corresponding space (CR1 orCR2).
Now inequalities (3.9) lead to the relation kDltwjikL2(R+) ≤C
O(1) +O(ql−ti−1/2−mR1 +1), j≤R1, O(qmR1−mj) +O(ql−ti−1/2−mj), j > R1, which implies (3.13).
b) The proof for the case i > N1 is similar to that above with only one difference. In part a), we used the fact that the elements of G(ξ0, τ, q) = A−1(ξ0, τ, q) are O(1) for a chosen (ξ0, τ) and for large values of q. By (3.4), for i > N1, the elementsgij(ξ0, τ, q) of this matrix are O(q−2(m−µ)) = O(q−2(si+ti)) and, for such indices i, we can replace (3.14) by the stronger estimate
kDtlwjikL2(R+) ≤C
|q−tiψ(1)(ξ0, q)|+ql−ti−1/2|ψ(2)(ξ0, q)|
, which leads to the desired estimate.
4. NEWTON POLYGON AND THE CORRESPONDING SOBOLEV SPACES
As was already noticed in Remark 2.2 b), the definition of weak parameter ellipticity is closely connected to the Newton polygon that corresponds to the determinant of the symbol ofA. Moreover, thea priori estimates below use Sobolev spaces that arise by using the Newton polygon approach.
In the present section, we briefly recall the main concepts and results of this approach and rewrite the estimate of Theorem 3.5 in terms of the Newton polygon. For a more detailed exposition, the reader is referred to [9] and to [4] and [5].
Let r= (r1, r2) be a pair of reals. Define a weight function Ξr:Rn×R+→Ras follows:
(4.1) Ξr(ξ, q) := (1 +|ξ|)r1(q+|ξ|)r2. Similarly, define a function Φr,r∈R2, by
(4.2) Φr(ξ, q) :=|ξ|r1(q+|ξ|)r2.
If r1 and r2 are positive integers, then these functions allow a geometric interpretation. Let us describe it. For positive integersr1 andr2, define the Newton polygonNr as the convex hull of the set{(0,0),(0, r2),(r1, r2),(r1+r2,0)}(cf. Figure 1). We can readily see that there is an equivalence
Ξr(ξ, q)≈ X
(i,k)∈Nr∩Z2
|ξ|iqk,
where the sign≈means that the ratio of the left-hand side to the right-hand side is bounded above and below by positive constants that do not depend onξ and q. Similarly, Φr(ξ, q)≈P
(i,k)|ξ|iqk, where the sum is now taken over all integral points on the edge connecting (r1, r2) and (r1+r2,0).
(In a sense, this edge is the leading part of the Newton polygon Nr.)
Now we again assume that r1and r2 are arbitrary real numbers. In connection with the weight function Ξr, we endow the (classical) Sobolev spaceHr1+r2(Rn) with a parameter-dependent norm k · kr given by the formula
(4.3) kukr :=kF−1ΞrF ukL2(Rn),
where F stands for the Fourier transform. We write Hr(Rn) for Hr1+r2(Rn) endowed with this norm, omitting the symbol q for brevity in the notation for Hr and k · kr. If Rn is replaced by Rn−1 = {(x0, xn) ∈ Rn : xn = 0}, then we use the weight function Ξr(ξ0, q) := Ξr(ξ0,0, q) in the definition of the norm k · kr,Rn−1 on the space Hr1+r2(Rn−1). On the half-space Rn+ := {x = (x0, xn)∈Rn:xn>0}, on the manifoldM, and on its boundary∂M, similar norms can be defined in the standard way (cf. [18]); denote the resulting norms byk · kr,Rn
+, etc.
Let us consider the traces of functions in Hr(Rn+) on the boundary Rn−1. The question is how to define parameter-dependent norms on the Sobolev spaces on the boundary so that the trace operatorγ0:u7→u(·,0) is continuous and its norm is bounded by a constant independent of q. To answer this question, we define the functions
Ξ(−a)r (ξ, q) =
(1 +|ξ|)r1−a(q+|ξ|)r2, a≤r1, (q+|ξ|)r1+r2−a, a > r1, (4.4)
Φ(−a)r (ξ, q) =
|ξ|r1−a(q+|ξ|)r2, a≤r1, (q+|ξ|)r1+r2−a, a > r1, (4.5)
forr = (r1, r2) ∈R2 and a∈R. Denote the norm related to the function Ξ(−a)r by k · k(−a)r . The spaceHr1+r2−a endowed with this norm is denoted byHr(−a).
The functions in (4.4) and (4.5) can be interpreted geometrically as well. For positive integers r1 and r2 and for 0≤a≤r1+r2, the function Ξ(−a)r corresponds to the Newton polygon that is constructed fromNr by the left shift bya in parallel to the horizontal axis.
The following theorem describes the trace spaces of the Sobolev space Hr(Rn+) in the above sense. This is one of the main results of the theory of Sobolev spaces defined by Newton polygons;
for the case in which r1 and r2 are positive integers, this description follows from Theorem 2.9 in [5]. However, we can readily see that the proof in [5] works for arbitrary reals r1 and r2 (with r1+r2>1/2) as well.
