M. V. Keldysh Institute of Applied Mathematics Russian Academy of Sciences

R. Denk, R. Mennicken, and L. R. Volevich

ON ELLIPTIC OPERATOR PENCILS

WITH GENERAL BOUNDARY CONDITIONS

Preprint No. 37

Moscow 1999

Abstract

R. Denk, R. Mennicken, and L. Volevich^{1}. On elliptic operator pencils with general boundary conditions.

In this paper operator pencils A(x, D, λ) are investigated which depend polynomially on the parameter λ and act on a manifold with boundary. The operator A is assumed to satisfy the condition of N-ellipticity with parameter which is an ellipticity condition formulated with the use of the Newton polygon. We consider general boundary operators B_{1}(x, D), …, B_{m}(x, D) and define N-ellipticity for the boundary value problem (A, B_{1}, …, B_{m}) analogously to the Shapiro–Lopatinskii condition. It is shown that the boundary value problem is N-elliptic if and only if an a priori estimate holds, where the norms in the estimate are again defined in terms of the Newton polygon. These results are closely connected with singular perturbation theory and lead to uniform estimates for problems of Vishik–Lyusternik type containing a small parameter.

1Supported in part by the Deutsche Forschungsgemeinschaft and by Russian Foundation of Fundamental Research, Grant 97-01-00541

1. Introduction

In this paper we consider an operator pencil depending polynomially on the complex parameter λ and being of the form

A(x, D, λ) = A_{2m}(x, D)+λA_{2m−1}(x, D)+⋯+λ^{2m−2μ}A_{2μ}(x, D) , (1.1)

where m and μ are integer numbers with m > μ > 0 and A_{2μ}, …, A_{2m} are partial differential operators of the form

A_{j}(x, D) = ∑_{|α|≤j} a_{αj}(x)D^{α} (j = 2μ, 2μ + 1, …, 2m) .  (1.2)

We assume that the pencil (1.1) acts on a compact manifold M with boundary ∂M; the coefficients a_{αj} in (1.2) are complex-valued. The manifold, its boundary and the coefficients are assumed to be infinitely smooth. In (1.2) and in the following, we write x = (x_{1}, …, x_{n}) and use the standard multi-index notation

D^{α} = D_{1}^{α_{1}} ⋯ D_{n}^{α_{n}} , D_{j} = −i\frac{∂}{∂x_{j}} , |α| = α_{1} + … + α_{n} .

The Dirichlet boundary value problem corresponding to the pencil (1.1) was studied in detail in the paper [5]. The aim of the present paper is to obtain an a priori estimate for general boundary value problems connected with (1.1). So we assume that we have boundary operators, for simplicity independent of the complex parameter λ, of the form

B_{j}(x, D) = ∑_{|β|≤m_{j}} b_{jβ}(x)D^{β} (j = 1, …, m) ,  (1.3)

where the numbering is chosen such that for the orders of the operators B_{j} we have m_{1} ≤ m_{2} ≤ … ≤ m_{m}. Additionally, we assume that

m_{μ} < m_{μ+1} .  (1.4)

The coefficients of B_{j} are supposed to be defined in \overline{M} and to be infinitely smooth.

We will show that an a priori estimate holds if the boundary value problem (A, B_{1}, …, B_{m}) satisfies the condition of N-ellipticity with parameter which will be explained in Section 2. Moreover, we will prove that N-ellipticity is also necessary for the validity of the a priori estimate (see Section 6).

The principal symbol A^{(0)}(x, ξ, λ) of (1.1) is defined as

A^{(0)}(x, ξ, λ) := A_{2m}^{(0)}(x, ξ)+λA_{2m−1}^{(0)}(x, ξ)+…+λ^{2m−2μ}A_{2μ}^{(0)}(x, ξ) , (1.5)

where

A_{j}^{(0)}(x, ξ) := ∑_{|α|=j} a_{αj}(x)ξ^{α} (j = 2μ, …, 2m)  (1.6)

stands for the principal symbol of A_{j}. In (1.6) we have set ξ^{α} = ξ_{1}^{α_{1}} ⋯ ξ_{n}^{α_{n}} for ξ = (ξ_{1}, …, ξ_{n}). The principal symbols (1.5) and (1.6) are invariant under change of coordinates and thus globally defined on the cotangent bundle T^{∗}M\{0}. The principal symbols B_{j}^{(0)} of the boundary operators B_{j} are defined analogously.

In [5] the Newton polygon approach was used to formulate and prove an a priori estimate for the Dirichlet boundary value problem. This method (which was also applied to Douglis–Nirenberg systems in [4]) turns out to be suitable for general boundary conditions, too. The concept of the Newton polygon makes it possible to define the general notion of N-ellipticity with parameter which is a generalization of the classical definition of ellipticity with parameter given by Agmon [1] and Agranovich–Vishik [3]. For the connection to N-parabolic problems and Douglis–Nirenberg systems, the reader is referred to [5], Section 1.

Replacing in (1.1) λ by ε^{−1}, we obtain a problem of singular perturbation theory as it was studied, for instance, by Vishik and Lyusternik [10]. The a priori estimate stated below in Section 4 corresponds to a uniform (with respect to ε) estimate in the Vishik–Lyusternik theory (see also [6], [9]). We will show this close connection in the Appendix.

2. The Shapiro–Lopatinskii condition

As the manifold M is compact we may fix a finite number of coordinate systems. Locally in each of these coordinate systems the operator pencil A(x, D, λ) is of the form (1.1) and acts in ℝ^{n}. We can suppose without loss of generality that the coefficients of A(x, D, λ) are (in local coordinates) of the form

a_{αj}(x) = a_{αj} + a′_{αj}(x), a′_{αj} ∈ D(ℝ^{n}) .  (2.1)

Definition 2.1 Let x^{0} ∈ \overline{M} be fixed. The interior symbol A(x^{0}, ξ, λ) is called N-elliptic with parameter in [0, ∞) at x^{0} (cf. [5]) if the estimate

|A^{(0)}(x^{0}, ξ, λ)| ≥ C|ξ|^{2μ} (λ + |ξ|)^{2m−2μ} (ξ ∈ ℝ^{n}, λ ∈ [0, ∞)) (2.2)

holds with a constant C which does not depend on ξ or λ. If this is true for every x^{0} ∈ \overline{M}, the symbol A(x, ξ, λ) and the operator A(x, D, λ) are called N-elliptic with parameter in [0, ∞).

By continuity and compactness, for an N-ellliptic operator the constant C in (2.2) can be chosen independently of x^{0}.

Now we shall define the analogue of the Shapiro–Lopatinskii condition for our problem. For this, we fix a point x^{0} ∈ ∂M and a coordinate system in the neighbourhood of x^{0} such that in this system locally the boundary ∂M is given by the equation x_{n} = 0. We use in ℝ_{+}^{n} := {x ∈ ℝ^{n} : x_{n} > 0} the coordinates x = (x′, x_{n}) and the dual coordinates ξ = (ξ′, ξ_{n}). If A is N-elliptic with parameter, it follows from [5], Lemma 3.2, that for every x^{0} ∈ \overline{M} we have

A^{(0)}(x^{0}, ξ, λ) ≠ 0 (ξ ∈ ℝ^{n}\{0}, λ ∈ [0, ∞)) .  (2.3)

In the case n > 2 this implies that A^{(0)}, considered as a polynomial in ξ_{n}, has exactly m roots with positive imaginary part for every ξ′ ≠ 0. In the case n = 2 this is an additional condition which we assume to hold in the following. Similar considerations hold for A_{2μ}^{(0)}.

Let A be N-elliptic with parameter in [0, ∞), fix x^{0} ∈ ∂M and write A in local coordinates corresponding to x^{0} as considered above. Then we define the polynomial in τ ∈ ℂ

Q(x^{0}, τ) = τ^{−2μ}A^{(0)}(x^{0}, 0, τ, 1) .  (2.4)

Definition 2.2 The operator A(x, D, λ) degenerates regularly at the boundary ∂M if for every x^{0} ∈ ∂M the polynomial (2.4) has exactly m − μ roots in the upper half-plane of the complex plane.

Remark 2.3 a) It is easily seen that if for a fixed x^{0} ∈ ∂M and a fixed coordinate system polynomial (2.4) has m − μ roots in the upper halfplane, then this polynomial has this property for arbitrary x^{0} ∈ ∂M and for an arbitrary coordinate system.

b) The condition of regular degeneration has its direct counterpart in the theory of singular perturbations (see, e.g., [10], Section 6).

c) Some examples where the condition of regular degeneration (Definition 2.2) holds automatically can be found in [5], Remark 3.4.

If A is N-elliptic with parameter in [0, ∞), then for any fixed x^{0} ∈ \overline{M} and ξ′ ∈ ℝ^{n−1}\{0}, we see from (2.3) that we can factorize the principal symbol A^{(0)}(x^{0}, ξ, λ) in the form

A^{(0)}(x^{0}, ξ, λ) = A_{+}^{(0)}(x^{0}, ξ, λ) A_{−}^{(0)}(x^{0}, ξ, λ) .

Here

A_{+}^{(0)}(x^{0}, ξ′, τ, λ) := ∏_{j=1}^{m}(τ − τ_{j}(x^{0}, ξ′, λ)) ,  (2.5)

where τ_{1}, …, τ_{m} are the zeros of A^{(0)} with positive imaginary part.

Now let x^{0} ∈ ∂M and denote by B′_{j}(x^{0}, ξ′, ξ_{n}, λ) the remainder of B_{j}^{(0)}(x^{0}, ξ) after division by A_{+}^{(0)}(x^{0}, ξ, λ), where all polynomials are considered as polynomials in ξ_{n}. We write B′_{j} in the form

B′_{j}(x^{0}, ξ′, ξ_{n}, λ) = ∑_{k=1}^{m} b_{jk}(x^{0}, ξ′, λ)ξ_{n}^{k−1} .  (2.6)

and define the Lopatinskii determinant by

Lop(x^{0}, ξ′, λ) := det (b_{jk}(x^{0}, ξ′, λ))_{j,k=1,...,m} .  (2.7)

Then the condition

Lop(x^{0}, ξ′, λ) ≠ 0  (2.8)

means that the polynomials B_{j}^{(0)}(x^{0}, ξ′, ·) are linearly independent modulo A_{+}^{(0)}(x^{0}, ξ′, ·, λ). It is well-known that condition (2.8) is satisfied if and only if the ordinary differential equation on the half-line

A(ξ′, D_{t}, λ) w(t) = 0  (t > 0) ,  (2.9)
B_{k}(ξ′, D_{t}) w(t)|_{t=0} = h_{k} (k = 1, …, m) ,  (2.10)
w(t) → 0  (t → +∞) ,

is uniquely solvable for every (h_{1}, …, h_{m}) ∈ ℂ^{m}. Here D_{t} stands for −i\frac{∂}{∂t}.

Definition 2.4 Let A satisfy the regular degeneration condition. Then the boundary problem (A, B_{1}, …, B_{m}) is called N-elliptic with parameter λ ∈ [0, ∞) if the following conditions hold:

a) The interior symbol A(x, ξ, λ) is N-elliptic with parameter in [0, ∞) in the sense of Definition 2.1.

b) For every fixed x^{0} ∈ ∂M, every ξ′ ≠ 0 and every λ ∈ [0, ∞) the polynomials (B_{j}^{(0)}(x^{0}, ξ′, ·))_{j=1,...,m} are linearly independent modulo A_{+}^{(0)}(x^{0}, ξ′, ·, λ), i.e. (2.8) holds.

c) For every fixed x^{0} ∈ ∂M, the boundary problem

(A_{2μ}^{(0)}(x^{0}, D), B_{1}(x^{0}, D), …, B_{μ}(x^{0}, D))

fulfills the Shapiro–Lopatinskii condition, i.e. (B_{j}^{(0)}(x^{0}, ξ))_{j=1,...,μ} are linearly independent modulo (A_{2μ}^{(0)})_{+}(x^{0}, ξ). Here (A_{2μ}^{(0)})_{+} is defined in analogy to (2.5) with A replaced by A_{2μ}.

d) Let Q_{+}(x^{0}, τ) := ∏_{j=μ+1}^{m}(τ − τ_{j}^{1}(x^{0})) where τ_{μ+1}^{1}, …, τ_{m}^{1} denote the zeros of Q(x^{0}, τ) with positive imaginary part. Then the polynomials (B_{j}^{(0)}(x^{0}, 0, τ))_{j=μ+1,...,m} are linearly independent modulo Q_{+}(x^{0}, τ) for every x^{0} ∈ ∂M.

Remark 2.5 a) Note that the degree of B_{j}^{(0)}(x^{0}, 0, ·) is m_{j} which may be greater than 2m − 2μ.

b) Note that condition b) in Definition 2.4 differs from the Agmon– Agranovich–Vishik condition of ellipticity with parameter. If the symbols A(x, ξ, λ) and B_{j}(x, ξ) are homogeneous with respect to (ξ, λ), the Agmon–Agranovich–Vishik condition means that

Lop(x^{0}, ξ′, λ) ≠ 0 for |ξ′|^{2} + λ^{2} = 1, λ ≥ 0 .  (2.11)

In particular, in this case the inequality (2.8) holds for λ = 1 and ξ′ = 0. In the case of N-ellipticity, however, the Lopatinskii determinant is in general not defined for ξ′ = 0 and may tend to zero as ξ′ → 0.

c) Taking in 2.4 b) λ = 0 and |ξ′| = 1, we obtain the standard Shapiro– Lopatinskii condition for the boundary value problem (A_{2m}, B_{1}, …, B_{m}).

d) In the Vishik–Lyusternik theory, the analogue of condition 2.4 d) leads to the existence of solutions of boundary layer type (see [10], Section 6).

3. The basic ODE estimate

In a first step we consider the model problem in the half space. Let (A, B_{1}, …, B_{m}) be of the form (1.1), (1.3) and acting in ℝ_{+}^{n}. We suppose that A is homogeneous in (ξ, λ), i.e. has the form

A(ξ, λ) = A_{2m}(ξ) + λA_{2m−1}(ξ) + … + λ^{2m−2μ}A_{2μ}(ξ) ,  (3.1)

where A_{j}(ξ) is a homogeneous polynomial in ξ of degree j. Similarly we assume that B_{j} is given by

B_{j}(ξ) = ∑_{|β|=m_{j}} b_{jβ}ξ^{β} (j = 1, …, m) .  (3.2)

For fixed λ ≥ 0 and ξ′ ∈ ℝ^{n−1}\{0} we investigate the boundary problem

A(ξ′, D_{t}, λ) w_{j}(t) = 0  (t > 0) ,  (3.3)
B_{k}(ξ′, D_{t}) w_{j}(t)|_{t=0} = δ_{jk} (k = 1, …, m) ,  (3.4)
w_{j}(t) → 0  (t → +∞) .

In [5], the following lemma on the roots of the polynomial A(ξ′, ·, λ) is proved.

Lemma 3.1 Let the polynomial A(ξ, λ) in (3.1) be N-elliptic with parameter in [0, ∞) and assume that A degenerates regularly. Then, with a suitable numbering of the roots τ_{j}(ξ′, λ) of A(ξ′, τ, λ) with positive imaginary part, we have:

(i) Let S(ξ′) = {τ_{1}^{0}(ξ′), …, τ_{μ}^{0}(ξ′)} be the set of all zeros of A_{2μ}(ξ′, τ) with positive imaginary part. Then for all r > 0 there exists a λ_{0} > 0 such that the distance between the sets {τ_{1}(ξ′, λ), …, τ_{μ}(ξ′, λ)} and S(ξ′) is less than r for all ξ′ with |ξ′| = 1 and all λ ≥ λ_{0}. (ii) Let τ_{μ+1}^{1}, …, τ_{m}^{1} be the roots of the polynomial Q(τ) (cf. (2.4)) with positive imaginary part. Then

τ_{j}(ξ′, λ) = λτ_{j}^{1} + τ̃_{j}^{1}(ξ′, λ) (j = μ + 1, …, m) ,  (3.5)

and there exist constants K_{j} and λ_{1}, independent of ξ′ and λ, such that for λ ≥ λ_{1} the inequality

|τ̃_{j}^{1}(ξ′, λ)| ≤ K_{j}|ξ′|^{\frac{1}{k_{1}}} λ^{1− \frac{1}{k_{1}}} (|ξ′| ≤ λ)  (3.6)

holds, where k_{1} is the maximal multiplicity of the roots of Q(τ).

Theorem 3.2 Assume that the operator (A, B_{1}, …, B_{m}) is of the form (3.1)–(3.2). Assume that condition (1.4) holds and that A degenerates regularly at the boundary (cf. Definition 2.2) and (A, B_{1}, …, B_{m}) is Nelliptic with parameter in ℝ_{+}^{n} in the sense of Definition 2.4. Then for every ξ′ ∈ ℝ^{n−1}\{0} and λ ∈ [0, ∞) the ordinary differential equation (3.3)–(3.4) has a unique solution w_{j}(t, ξ′, λ), and the estimate

∥D_{t}^{l}w_{j}(·, ξ′, λ)∥_{L_{2}(ℝ_{+})} ≤

≤ C \{ \begin{matrix}|ξ′|^{l−m_{j}− \frac{1}{2}} , & j ≤ μ, & l ≤ m_{μ+1}, \\ |ξ′|^{m_{μ+1}−m_{j}}(λ + |ξ′|)^{l−m_{μ+1}− \frac{1}{2}} , & j ≤ μ, & l > m_{μ+1}, \\ |ξ′|^{l−m_{μ}− \frac{1}{2}} (λ + |ξ′|)^{m_{μ}−m_{j}}, & j > μ, & l ≤ m_{μ}, \\ (λ + |ξ′|)^{l−m_{j}− \frac{1}{2}} , & j > μ, & l > m_{μ}, \end{matrix}  (3.7)

holds with a constant C not depending on ξ′ and λ.

Proof. The existence and the uniqueness of the solution follows immediately from conditions a) and b) in Definition 2.4. From the homogeneity of the symbols and from the uniqueness of the solution we see that

w_{j}(t, ξ′, λ) = r^{−m_{j}}w_{j}(rt, \frac{ξ′}{r} , \frac{λ}{r})  (3.8)

holds for every r > 0. If we set r = |ξ′| and ω′ = \frac{ξ′}{|ξ′|} we obtain

∥D_{t}^{l}w_{j}(·, ξ′, λ)∥_{L_{2}(ℝ_{+})} = |ξ′|^{l−m_{j}− \frac{1}{2}} ∥D_{t}^{l}w_{j}(·, ω′, \frac{λ}{|ξ′|})∥_{L_{2}(ℝ_{+})} . (3.9)

The theorem will be proved if we show that for |ω′| = 1 we have

∥(D_{t}^{l}w_{j})(·, ω′, Λ)∥_{L_{2}(ℝ_{+})} ≤ \{ \begin{matrix}C , & j ≤ μ , l ≤ m_{μ+1} , \\ C Λ^{l−m_{μ+1}− \frac{1}{2}} , & j ≤ μ , l > m_{μ+1} , \\ C Λ^{m_{μ}−m_{j}}, & j > μ , l ≤ m_{μ} , \\ C Λ^{l−m_{j}− \frac{1}{2}} , & j > μ , l > m_{μ} , \end{matrix}

(3.10)

for Λ ≥ 1 and that the left-hand side is bounded by a constant for Λ ≤ 1.

The boundedness for Λ ≤ 1 follows easily from conditions a) and b) of Definition 2.4. We have to consider the case of large Λ.

To find an estimate in this case, we represent the solution in a form suggested in a paper of Frank [6]. This representation is different from the formula used for the Dirichlet boundary value problem in [5] and allows us to separate the two parts of the zeros of the polynomial A_{+}(ω′, τ, Λ) in a more adequate form.

Due to Lemma 3.1, the roots of this polynomial consist of two groups, the first group, denoted by {τ_{1}(ω′, Λ), …, τ_{μ}(ω′, Λ)}, being bounded for Λ → ∞, the other group, denoted by {τ_{μ+1}(ω′, Λ), …, τ_{m}(ω′, Λ)}, being of order Λ for Λ → ∞.

We define

A_{1}(ω′, τ, Λ) := ∏_{j=1}^{μ}(τ − τ_{j}(ω′, Λ)) .  (3.11)

Let γ^{(1)} be a contour in the upper half of the complex plane enclosing the zeros τ_{1}, …, τ_{μ}. From Lemma 3.1 we see that γ^{(1)} can be chosen independently of ω′ and Λ for all |ω′| = 1 and Λ ≥ Λ_{0}.

From the same lemma we see that A_{1}(ω′, τ, Λ) → (A_{2μ})_{+}(ω′, τ) as Λ → ∞. Therefore we obtain from condition c) in Definition 2.4 that for all |ω′| = 1 and Λ ≥ Λ_{0} the polynomials {B_{j}(ω′, τ)}_{j=1,...,μ} are independent modulo A_{1}(ω′, τ, Λ). Thus there exist polynomials (with respect to τ) N_{j}(ω′, τ, Λ), depending continuously on (ω′, Λ), such that

\frac{1}{2πi} ∫_{γ^{(1)}} \frac{B_{k}(ω′, τ)N_{j}(ω′, τ, Λ)}{A_{1}(ω′, τ, Λ)}dτ = δ_{kj} (k, j = 1, …, μ) . (3.12)

(For the construction of N_{j} cf., e.g., [2], p. 634.)

Analogously, we define

A_{2}(ω′, τ, Λ) := ∏_{j=μ+1}^{m}(τ − τ_{j}(ω′, Λ)) .  (3.13)

Let γ̃^{(2)}(ω′, Λ) be a contour in the upper half of the complex plane enclosing the zeros τ_{μ+1}(ω′, Λ), …, τ_{m}(ω′, Λ). From Lemma 3.1 we know that this contour is of order Λ for Λ → ∞. Therefore we may fix a contour γ^{(2)}, independent of ω′ and Λ such that γ^{(2)} encloses all values τ_{j}/Λ with j = μ+1, …, m. We also remark that due to the regular degeneration we may choose γ^{(2)} with a positive distance to the real axis (cf. also (3.5)).

From condition d) in 2.4 we know that {B_{j}(0, τ)}_{j=μ+1,...,m} is linearly independent modulo Q_{+}(τ). From Lemma 3.1 b) we know that

A_{2}(\frac{ω′}{Λ} , τ, 1) → Q_{+}(τ) (Λ → ∞) .

Due to continuity, the polynomials \{B_{j}(\frac{ω′}{Λ} , τ, 1)\}_{j=μ+1,...,m} are for sufficiently large Λ linearly independent modulo A_{2}(\frac{ω′}{Λ} , τ, 1). Therefore there exist polynomials (in τ) N_{j}(ω′, τ, Λ) for j = μ + 1, …, m, depending continuously on ω′ and Λ, such that

\frac{1}{2πi} ∫_{γ^{(2)}} \frac{B_{k}(\frac{ω′}{Λ} , τ)N_{j}(ω′, τ, Λ)}{A_{2}(\frac{ω′}{Λ} , τ, 1)}dτ = δ_{kj} (k, j = μ + 1, …, m) . (3.14)

Now we need a lemma which will be proved below.

Lemma 3.3 The solution w_{j}(t, ω′, Λ) of the problem (3.3)–(3.4) can be represented in the form

w_{j}(t, ω′, Λ) = \frac{1}{2πi} ∫_{γ^{(1)}} \frac{M_{j}^{(1)}(ω′, τ, Λ)}{A_{1}(ω′, τ, Λ)}e^{itτ} dτ

+ \frac{1}{2πi} ∫_{γ^{(2)}} \frac{M_{j}^{(2)}(ω′, τ, Λ)}{A_{2}(\frac{ω′}{Λ} , τ, 1)}e^{itΛτ} dτ  (3.15)

where for |τ| = O(1) and |ω′| = 1 we have

M_{j}^{(1)}(ω′, τ, Λ) ≤ \{ \begin{matrix}C , & j ≤ μ , \\ C Λ^{m_{μ}−m_{j}} , & j > μ , \end{matrix}

and

M_{j}^{(2)}(ω′, τ, Λ) ≤ \{ \begin{matrix}C Λ^{−m_{μ+1}} , & j ≤ μ , \\ C Λ^{−m_{j}} , & j > μ , \end{matrix}

As a direct corallary of the lemma we obtain

∥(D_{t}^{l}w_{j})(·, ω′, Λ)∥_{L_{2}(ℝ_{+})} ≤ \{ \begin{matrix}O(1) + O(Λ^{l−m_{μ+1}− \frac{1}{2}} ) , & j ≤ μ , \\ O(Λ^{m_{μ}−m_{j}}) + O(Λ^{l−m_{j}− \frac{1}{2}} ) , & j > μ . \end{matrix}

The estimate (3.10) trivially follows from these relations. □

Proof of Lemma 3.3. Let w(t, ω′, Λ) be a solution of the problem (3.3)–(3.4) with δ_{jk} replaced by ϕ = (ϕ_{1}, …, ϕ_{m}) ∈ ℂ^{m}. We seek the solution in the form

w(t, ω′, Λ) = ∑_{k=1}^{μ} ψ_{k}(ω′, Λ)\frac{1}{2πi} ∫_{γ^{(1)}} \frac{N_{k}(ω′, τ, Λ)}{A_{1}(ω′, τ, Λ)}e^{itτ} dτ

+ ∑_{k=μ+1}^{m} ψ_{k}(ω′, Λ)\frac{1}{2πi} ∫_{γ^{(2)}} \frac{N_{k}(ω′, τ, Λ)}{A_{2}(\frac{ω′}{Λ} , τ, 1)}e^{itΛτ} dτ (3.16)

where the functions ψ_{k} still have to be found.

Applying the boundary operator B_{l}(ξ′, D_{t}) to both sides of (3.16) and taking t = 0 we obtain the following system for the unknown functions ψ_{k}(ω′, Λ):

ψ_{l}(ω′, Λ) + Λ^{m_{l}} ∑_{k=μ+1}^{m} ψ_{k}(ω′, Λ)h_{lk}(ω′, Λ) = ϕ_{l}
(l = 1, …, μ) ,  (3.17)
∑_{k=1}^{μ} ψ_{k}(ω′, Λ)h_{lk}(ω′, Λ) + Λ^{m_{l}}ψ_{l}(ω′, Λ) = ϕ_{l}
(l = μ + 1, …, m) . (3.18)

Here we have set

h_{lk}(ω′, Λ) = \frac{1}{2πi} ∫_{γ^{(2)}} \frac{B_{l}(\frac{ω′}{Λ} , τ)N_{k}(ω′, Λ, τ)}{A_{2}(\frac{ω′}{Λ} , τ, 1)} dτ

(l = 1, …, μ; k = μ + 1, …, m) ,  (3.19)

h_{lk}(ω′, Λ) = \frac{1}{2πi} ∫_{γ^{(1)}} \frac{B_{l}(ω′, τ)N_{k}(ω′, Λ, τ)}{A_{1}(ω′, τ, Λ)} dτ

(l = μ + 1, …, m; k = 1, …, μ) . (3.20)

We remark that we have used B_{l}(ω′, Λτ) = Λ^{m_{l}}B_{l}(\frac{ω′}{Λ} , τ).

Now we write ψ = (ψ′, ψ′′), where ψ′ consists of the first μ components of the vector ψ, and ψ′′ consists of the other m − μ components. In the same way we write ϕ = (ϕ′, ϕ′′). In these notations the system (3.17)– (3.18) can be rewritten in the form

\begin{aligned}ψ′ & + & Δ_{1}H_{12}ψ′′ & = & ϕ′ , \\ H_{21}ψ′ & + & Δ_{2}ψ′′ & = & ϕ′′ , \end{aligned}

where we use the notation

Δ_{1} := ( \begin{matrix}Λ^{m_{1}} \\ & . & . & . \\ & & & & Λ^{m_{μ}} \end{matrix} ) , Δ_{2} := ( \begin{matrix}Λ^{m_{μ+1}} \\ & . & . & . \\ & & & & Λ^{m_{m}} \end{matrix} )

and

H_{12} := (h_{lk})_{{l=1,...,μ \atop k=μ+1,...,m}} , H_{21} := (h_{lk})_{{l=μ+1,...,m \atop k=1,...,μ}} .

If me multiply the second equation by the matrix Δ_{1}H_{12}Δ_{2}^{−1} from the left and subtract it from the first equation we obtain

(I − Δ_{1}H_{12}Δ_{2}^{−1}H_{12})ψ′ = ϕ′ − Δ_{1}H_{12}Δ_{2}^{−1}ϕ′′ .

In a similar way we obtain

(I − Δ_{2}^{−1}H_{21}Δ_{1}H_{12})ψ′′ = −Δ_{2}^{−1}H_{21}ϕ′ + Δ_{2}^{−1}ϕ′′ .

The matrices in brackets in the left-hand sides of above relations differ from the identity by matrices whose elements can be estimated by a constant times Λ^{m_{μ}−m_{μ+1}}. According to (1.4), their norms tend to zero as Λ → ∞. From this it follows that the matrices in brackets for large ���

Λ have inverses which we denote by G_{1} and G_{2}, respectively. Then we obtain

\begin{aligned}ψ′ & = & G_{1}ϕ′ & − & G_{1}Δ_{1}H_{12}Δ_{2}^{−1}ϕ′′ , \\ ψ′′ & = & −G_{2}Δ_{2}^{−1}H_{21}ϕ′ & + & G_{2}Δ_{2}^{−1}ϕ′′ . \end{aligned}

If we take ϕ = e_{j} (1 ≤ j ≤ μ), where e_{j} stands for the j-th unit vector, and denote by e′_{j} the first μ components of e_{j}, we obtain

ψ′_{(j)} = G_{1}e′_{j}, ψ′′_{(j)} = −G_{2}Δ_{2}^{−1}H_{21}e′_{j} .

In the same way if j > μ and e′′_{j} denotes the components μ + 1, …, m of e_{j}, we obtain

ψ′_{(j)} = −G_{1}Δ_{1}H_{12}Λ^{−m_{j}}e′′_{j}, ψ′′_{(j)} = G_{2}Λ^{−m_{j}}e′′_{j} .

The statement of the lemma directly follows from these relations. □

4. A priori estimates

Theorem 3.2 is the key result for proving a priori estimates. The norms used in these estimates are based on the Newton polygon N_{r,s} (cf. Fig.

1) defined for r > s ≥ 0 as the convex hull of the set

{(0, 0) , (0, r − s) , (s, r − s) , (r, 0) }

[figure: i; k; •; •; •; r; s; r − s; Fig. 1. The Newton polygon N_{r,s}.]

The weight function Ξ_{r,s}(ξ, λ) is defined by

Ξ_{r,s}(ξ, λ) := ∑_{(i,k)∈N_{r,s}} |ξ|^{i} |λ|^{k} ,  (4.1)

where the summation on the right-hand side is extended over all integer points of N_{r,s}. For a discussion of general Newton polygons we refer the reader to [4], [5], [7].

It is easily seen that we have the equivalence

Ξ_{r,s}(ξ, λ) ≈ (1 + |ξ|)^{s}(λ + |ξ|)^{r−s} .  (4.2)

The sign ≈ means that the quotient of the left-hand and the right-hand side is bounded from below and from above by positive constants independent of ξ and λ. Taking the right-hand side of (4.2) as a definition, we may define Ξ_{r,s} for every r, s ∈ ℝ. The Sobolev space H^{(r,s)}(ℝ^{n}) := H^{Ξ_{r,s}}(ℝ^{n}) is defined as

{u ∈ S′(ℝ^{n}) : Ξ_{r,s}(ξ, λ)F u(ξ) ∈ L_{2}(ℝ^{n})}

with the norm

∥u∥_{(r,s),ℝ^{n}} := ∥u∥_{Ξ_{r,s},ℝ^{n}} := ∥F^{−1}Ξ_{r,s}(ξ, λ)F u(ξ)∥_{L_{2}(ℝ^{n})} .  (4.3)

Here F u stands for the Fourier transform of u and S′(ℝ^{n}) denotes the space of all tempered distributions. The space H^{Ξ_{r,s}}(ℝ^{n−1}) is defined analogously with the weight function Ξ_{r,s}(ξ′, λ) := Ξ_{r,s}(ξ′, 0, λ). These spaces can be defined on the half-space ℝ_{+}^{n} in accordance with the general theory of Sobolev spaces with weight functions as it can be found, e.g., in [11]. On the manifold M and the boundary ∂M, the spaces H^{Ξ_{r,s}}(M) and H^{Ξ_{r,s}}(∂M), respectively, are defined in the usual way, using a partition of unity.

In [5], Section 2, Sobolev spaces connected with Newton polygons were investigated in detail. In particular, for r ∈ ℕ it was shown that (\frac{∂}{∂ν})^{j} : u ↦ (\frac{∂}{∂ν})^{j}u|_{∂M} acts continuously from H^{Ξ_{r,s}}(M) to H^{Ξ_{r,s}^{(−j−1/2)}}(∂M) for j = 0, …, r − 1. Here Ξ_{r,s}^{(−j−1/2)} denotes the weight function corresponding to the Newton polygon which is constructed from N_{r,s} by a shift of length j + 1/2 to the left parallel to the abscissa.

While in [5] the basic Sobolev space was H^{Ξ_{m,μ}}, we here consider more general a priori estimates. In the following, we fix integer numbers

r ≥ m_{m} + 1 and m_{μ} + 1 ≤ s ≤ m_{μ+1}  (4.4)

and consider the Newton polygon N_{r,s}, its weight function Ξ := Ξ_{r,s} and the corresponding Sobolev space. We remark that for the Dirichlet problem the values r = m and s = μ used in [5] are included as an example.

Analogously to (4.1), we define the function Φ = Φ_{r,s} by

Φ(ξ, λ) := ∑_{i,k} |ξ|^{i}λ^{k} ,  (4.5)

where the sum runs over all integer points (i, k) belonging to the side of N_{r,s} which is not parallel to one of the coordinate lines. This means that we have

Φ(ξ, λ) ≈ |ξ|^{s}(λ + |ξ|)^{r−s} .  (4.6)

By Φ^{(−l)} we again denote the corresponding function for the shifted Newton polygon. From Theorem 3.2 we obtain the following estimate for the fundamental solution w_{j} defined in (3.3)–(3.4):

Lemma 4.1 For the solution w_{j}(t, ξ′, λ) considered in Theorem 3.2 we have the estimate

∥D_{t}^{l}w_{j}(·, ξ′, λ)∥_{L_{2}(ℝ_{+})} ≤ C \frac{Φ^{(−m_{j}−1/2)}(ξ′, λ)}{Φ^{(−l)}(ξ′, λ)} .  (4.7)

Proof. To see this, we only have to remark that the right-hand side of (4.7) is equivalent to

\{ \begin{matrix}|ξ′|^{l−m_{j}− \frac{1}{2}} , & j ≤ μ, & l ≤ s, \\ |ξ′|^{s−m_{j}− \frac{1}{2}} (λ + |ξ′|)^{l−s}, & j ≤ μ, & l > s, \\ |ξ′|^{l−s}(λ + |ξ′|)^{s−m_{j}− \frac{1}{2}} , & j > μ, & l ≤ s, \\ (λ + |ξ′|)^{l−m_{j}− \frac{1}{2}} , & j > μ, & l > s. \end{matrix}

The first and fourth lines above coincide with the corresponding lines in the right-hand side of (3.7). The ratio of the second line in (3.7) and the second line above is equal to

(\frac{|ξ′|}{λ + |ξ′|})^{m_{μ+1}−s+1/2} .

Respectively, the ratio of the third line in (3.7) and the third line above is equal to

(\frac{|ξ′|}{λ + |ξ|})^{s−m_{μ}−1/2} .

Now our statement follows from (4.4). □

Theorem 4.2 Let A(x, D, λ) be an operator pencil of the form (1.1), acting on the manifold M with boundary ∂M. Let B_{j}(x, D), j = 1, …, m, be boundary operators of the form (1.3). Assume that A degenerates regularly at the boundary and that (A, B_{1}, …, B_{m}) is N-elliptic with parameter in the sense of Definition 2.4. Set Ξ = Ξ_{r,s} with r and s satisfying (4.4). For simplicity, assume that r and s are integers. Then for λ ≥ λ_{0} there exists a constant C = C(λ_{0}), independent of u and λ, such that

∥u∥_{Ξ,M} ≤ C(∥A(x, D, λ)u∥_{(r−2m,s−2μ),M}
+ ∑_{j=1}^{m} ∥B_{j}(x, D)u∥_{Ξ^{(−m_{j} −1/2)},∂M} + λ^{r−s}∥u∥_{L_{2}(M)}) .  (4.8)

Proof. The proof of this theorem is similar to the proof of Theorem 5.6 in [5]; therefore we only indicate the main steps.

By the localization method (“freezing the coefficients”), it is possible to reduce the proof to the proof of the corresponding results for model problems in ℝ^{n} and ℝ_{+}^{n}. The case of the whole space ℝ^{n} is quite elementary and needs only slight changes in comparison with [5], Proposition 5.2. The key result is the a priori estimate in the half space ℝ_{+}^{n}.

So we assume that u ∈ H^{Ξ}(ℝ_{+}^{n}) is a solution of

A(D, λ)u(x) = f in ℝ_{+}^{n} ,  (4.9)
B_{j}(D)u|_{x_{n}=0} = g_{j} (j = 1, …, m) on ℝ^{n−1} .  (4.10)

Let ψ ∈ C^{∞}(ℝ^{n}) be a cut-off function, i.e. ψ(ξ) = 1 for |ξ| ≤ 1 and ψ(ξ) = 0 for |ξ| ≥ 2. As in [5] we write u in the form

u = u_{1} + u_{2} + v = Rψ(D)Eu + R(1 − ψ(D))A^{−1}(D, λ)Ef + v . (4.11)

Here we fixed an extension operator E from ℝ_{+}^{n} to ℝ^{n}, being continuous from L_{2}(ℝ_{+}^{n}) to L_{2}(ℝ^{n}) and from H^{Ξ}(ℝ_{+}^{n}) to H^{Ξ}(ℝ^{n}); we define the distribution Ef as A(D, λ)Eu. By R we denote the operator of restriction of functions on ℝ^{n} onto ℝ_{+}^{n}. In (4.11), the pseudodifferential operator (ps.d.o.) ψ(D) in ℝ^{n} is defined as

ψ(D) := F^{−1}ψ(ξ)F .

It is easily seen that we have

∥u_{1}∥_{Ξ,ℝ_{+}^{n}} + ∥u_{2}∥_{Ξ,ℝ_{+}^{n}} ≤ C(∥f∥_{(r−2m,s−2μ),ℝ_{+}^{n}} + λ^{r−s}∥u∥_{L_{2}(ℝ_{+}^{n})}) . (4.12)

We still have to estimate v defined in (4.11). By definition, v is a solution of

A(D, λ) v = 0 in ℝ_{+}^{n} ,  (4.13)
B_{j}(D)v|_{x_{n}=0} = h_{j} (j = 1, …, m) on ℝ^{n−1} ,  (4.14)

where we set h_{j}(x′) := B_{j}(D)u(x′, 0) − B_{j}(D)u_{1}(x′, 0) − B_{j}(D)u_{2}(x′, 0). Now we use the fact that

∥v∥_{Ξ,ℝ_{+}^{n}} ≈ ∥v∥_{Φ,ℝ_{+}^{n}} + λ^{r−s}∥v∥_{L_{2}(ℝ_{+}^{n})} .

As

λ^{r−s}∥v∥_{L_{2}(ℝ_{+}^{n})} ≤ λ^{r−s}∥u∥_{L_{2}(ℝ_{+}^{n})} + ∥u_{1}∥_{Ξ,ℝ_{+}^{n}} + ∥u_{2}∥_{Ξ,ℝ_{+}^{n}} ,

it is sufficient to estimate ∥v∥_{Φ,ℝ_{+}^{n}}.

It can be seen, using the binomial formula, that

(Φ(ξ, λ))^{2} ≈ ∑_{l=0}^{r} (ξ_{n}^{l} Ξ^{(−l)}(ξ′, λ))^{2} ,

and therefore the semi-norm ∥v∥_{Φ,ℝ_{+}^{n}} is equivalent to

[∑_{l=0}^{r} ∫_{0}^{∞} ∥(D_{n}^{l}v)(·, x_{n})∥_{Φ^{(−l)},ℝ^{n−1}}^{2}dx_{n}]^{1/2} .  (4.15)

Taking the partial Fourier transform F′ with respect to x′ ∈ ℝ^{n−1}, we obtain from (4.13)–(4.14) that for ξ′ ≠ 0 the function w := F′v is a solution of

A(ξ′, D_{n}, λ)w(x_{n}) = 0 ,  (4.16)
B_{j}(ξ′, D_{n})w(x_{n})|_{x_{n}=0} = (F′h_{j})(ξ′) .  (4.17)

Due to Theorem 3.2, this solution is unique and given by

w = w(x_{n}, ξ′, λ) = ∑_{j=1}^{m} w_{j}(x_{n}, ξ′, λ)(F′h_{j})(ξ′)  (4.18)

with w_{j}(x_{n}, ξ′, λ) being the solution of (3.3)–(3.4). Now we can apply Lemma 4.1 to obtain

(Φ^{(−l)}(ξ′, λ))^{2} ∫_{0}^{∞} |D_{n}^{l}w(x_{n}, ξ′, λ)|^{2} dx_{n}
≤ C ∑ |Ξ^{(−m_{j}− \frac{1}{2} )}(ξ′, λ)(F′h_{j})(ξ′)|^{2} .

Integrating this inequality with respect to ξ′ and using the norm (4.15) we get the desired estimate for ∥v∥_{Φ,ℝ_{+}^{n}}, which finishes the proof of the theorem. □

5. The parametrix construction

In this section, we will construct a right (rough) parametrix for the operator (A, B) = (A, B_{1}, …, B_{m}), generalizing the result of [5]. We restrict ourselves to the construction of local parametrices in ℝ^{n} and ℝ_{+}^{n}; after this the definition of the parametrix on the manifold is standard (cf. also [5] for the Dirichlet problem).

For the remainder of this section, we fix integer numbers r and s satisfying (4.4) and set Ξ = Ξ_{r,s}. The following result is a slight generaliztion of [5], Proposition 6.1, which can be proved literally in the same way.

Lemma 5.1 Let A(x, D, λ) in (1.1) be N-elliptic in ℝ^{n} with coefficients of the form (2.1). Then there exists a bounded operator

P_{0} : H^{(r−2m,s−2μ)}(ℝ^{n}) → H^{(r,s)}(ℝ^{n})

such that

AP_{0} = I + T

where I denotes the identity operator in H^{(r−2m,s−2μ)}(ℝ^{n}) and

T : H^{Θ}(ℝ^{n}) → H^{(r−2m,s−2μ)}(ℝ^{n})

is bounded. Here we have set

Θ(ξ, λ) := Ξ_{r−2m+1,s−2μ+1}(ξ, λ) [ = (1 + |ξ|) Ξ_{r−2m,s−2μ}(ξ, λ)] . (5.1)

Throughout this section, by a bounded operator we understand a continuous operator with norm bounded by a constant independent of λ.

Now assume that (A, B) acts in the half space ℝ_{+}^{n} with coefficients of the form (2.1) and that (A, B) is N-elliptic in the sense of Definition 2.4. To define a parametrix, we use a cut-off function ψ′ ∈ C^{∞}(ℝ^{n−1}) with

ψ′(ξ′) = \{ \begin{matrix}0 , & |ξ′| ≤ 1 , \\ 1 , & |ξ′| ≥ 2 . \end{matrix}

For j = 1, …, m we define the ps.d.o. P_{j} in ℝ^{n−1} (with x_{n} as parameter) by

(P_{j}g)(x′, x_{n}) := ψ′(D′) w_{j}(x′, x_{n}, D′, λ) g ,  (5.2)

where w_{j}(x′, x_{n}, ξ′, λ) is the unique solution of (3.3)–(3.4) with

\begin{aligned}A(ξ′, D_{t}, λ) & = & A^{(0)}(x′, 0, ξ′, D_{t}, λ) , \\ B_{k}(ξ′, D_{t}) & = & B_{k}^{(0)}(x′, 0, ξ′, D_{t}) . \end{aligned}  (5.3)

Due to Lemma 3.3, for large λ the symbol of w_{j}(x′, x_{n}, D′, λ) can be written in the form

w_{j}(x′, x_{n}, ξ′, λ) = \frac{1}{2πi} ∫_{γ^{(1)}} \frac{M_{j}^{(1)}(x′, ξ′, τ, λ)}{A_{1}(x′, ξ′, τ, λ)} e^{ix_{n}τ} dτ

+ \frac{1}{2πi} ∫_{γ^{(2)}} \frac{M_{j}^{(2)}(x′, ξ′, τ, λ)}{A_{2}(x′, ξ′/λ, τ, 1)} e^{ix_{n}λτ} dτ . (5.4)

Lemma 5.2 The operator P_{j} defined in (5.2) is continuous from H^{Ξ^{(−m_{j} −1/2)}}(ℝ^{n−1}) to H^{Ξ}(ℝ_{+}^{n}).

Proof. Let g ∈ H^{Ξ^{(−m_{j} −1/2)}}(ℝ^{n−1}) and set u := P_{j}g. Using the equivalent norm

[∑_{l=0}^{r} ∫_{0}^{∞} ∥D_{n}^{l}u(·, x_{n})∥_{Ξ^{(−l)},ℝ^{n−1}}^{2} dx_{n}]^{1/2}  (5.5)

in H^{Ξ}(ℝ_{+}^{n}), we see that we have to show that

∥Ξ^{(−l)}(D′, λ)D_{n}^{l}P_{j} [Ξ^{(−m_{j}−1/2)}(D′, λ)]^{−1}∥_{L_{2}(ℝ^{n−1})→L_{2}(ℝ^{n−1})} ≤ C(x_{n})

for some function C = C(x_{n}) whose L_{2}(ℝ_{+})-norm is bounded by a constant independent of λ. For this it is sufficient to show that for |ξ′| ≥ 1 we have

(∫_{0}^{∞} |D_{x′}^{α′}D_{n}^{l}w_{j}(x′, x_{n}, ξ′, λ)|^{2} dx_{n})^{1/2} ≤ C \frac{Ξ^{(−m_{j}−1/2)}(ξ′, λ)}{Ξ^{(−l)}(ξ′, λ)} .

As we have for |ξ′| ≥ 1 the equivalence

Ξ_{r,s}(ξ′, λ) ≈ Φ_{r,s}(ξ′, λ)

for all r, s ∈ ℝ (with Φ_{r,s} defined by the right-hand side of (4.6)), the case α′ = 0 is already covered by Lemma 4.1. Here we take into account that, due to condition (2.1), the constant C in Lemma 4.1 applied to the symbols (5.3) may be chosen independently of x′ ∈ ℝ^{n−1}.

The case α′ > 0 follows after differentiation of (5.4) with respect to x′ along the same lines as in the proof of Lemma 3.3. □

Lemma 5.3 The operator

C_{j} := A(x, D, λ)P_{j} : H^{Ξ^{(−m_{j} −1/2)}}(ℝ^{n−1}) → H^{Θ}(ℝ_{+}^{n})

is bounded. Here Θ(ξ, λ) is defined in (5.1).

Proof. The symbol of the ps.d.o. C_{j} in ℝ^{n−1} with parameter x_{n} is given by

∑_{|α′|=1}^{2m} \frac{1}{(α′)!}∂_{ξ′}^{α′}A(x, ξ′, D_{n}, λ) D_{x′}^{α′}P_{j}(x′, x_{n}, ξ′, λ)

with

P_{j}(x, ξ′, λ) = ψ(ξ′) w_{j}(x, ξ′, λ) .

Consider the famliy F = {A(x, ξ, λ) : x ∈ ℝ_{+}^{n}} of polynomials in (ξ, λ) ∈ ℝ^{n+1}. As the degree of the polynomial A(x, ·) is equal to 2m for all x ∈ ℝ_{+}^{n}, the family F is a subset of the finite-dimensional vector space of all polynomials in (ξ, λ) of degree not greater than 2m. Therefore, there exists a finite set x^{(1)}, …, x^{(K)} ∈ ℝ_{+}^{n} such that every A ∈ F may be represented in the form

A(x, ξ, λ) = ∑_{k=1}^{K} c_{k}(x) A(x^{(k)}, ξ, λ)

with smooth coefficients c_{k}(x).

Taking into account that the operators of multiplication by c_{k}(x) are bounded in H^{Θ}(ℝ_{+}^{n}), we reduce our problem to the proof of the boundedness of operators of the form

C_{α′,l} := a_{α′,l}(D′, λ) D_{n}^{l}D_{x′}^{α′}P_{j}(x, D′, λ) : H^{Ξ^{(−m_{j} −1/2)}}(ℝ^{n−1}) → H^{Θ}(ℝ_{+}^{n}) ,

where

|a_{α′,l}(ξ′, λ)| ≤ C Ξ_{2m,2μ}^{(−l−1)}(ξ′, λ) .  (5.6)

Literally repeating the proof of Lemma 5.2 we establish the boundedness of the operator

D_{x′}^{α′}P_{j}(x, D, λ) : H^{Ξ^{(−m_{j} −1/2)}}(ℝ^{n−1}) → H^{Ξ}(ℝ_{+}^{n}) .

According to (5.6) the operator

a_{α′,l}(D′, λ) : H^{Ξ}(ℝ_{+}^{n}) → H^{Θ}(ℝ_{+}^{n})

is bounded. As C_{α′,l} is the product of of the above operators this operator is also bounded. □

Theorem 5.4 Consider in the half space ℝ_{+}^{n} the boundary value problem (A, B) = (A, B_{1}, …, B_{m}) of the form (1.1), (1.3) with coefficients of the form (2.1). Assume that A degenerates regularly at the boundary and that (A, B) is N-elliptic with parameter in [0, ∞) in the sense of Definition 2.4. Then there exists a bounded operator

P : H^{(r−2m,s−2μ)}(ℝ_{+}^{n}) × ∏_{j=1}^{m} H^{Ξ^{(−m_{j} −1/2)}}(ℝ^{n−1}) → H^{Ξ}(ℝ_{+}^{n})

such that

(A, B)P = I + T

where I stands for the identity operator in the space

H^{(r−2m,s−2μ)}(ℝ_{+}^{n}) × ∏_{j=1}^{m} H^{Ξ^{(−m_{j} −1/2)}}(ℝ^{n−1})  (5.7)

and T is a continuous operator from the space (5.7) to the space

H^{Θ}(ℝ_{+}^{n}) × ∏_{j=1}^{m} H^{Ξ^{(−m_{j} +1/2)}}(ℝ^{n−1})

with Θ(ξ, λ) being defined in (5.1).

Proof. We define

P (f, g_{1}, …, g_{m}) := P_{0}f + ∑_{j=1}^{m} P_{j}(g_{j} − B_{j}P_{0}f)

with P_{0} from Lemma 5.1 and P_{j} given by (5.2). The continuity of P follows from Lemma 5.1 and Lemma 5.2. In order to see that the operator T is continuous with respect to the spaces given in the theorem, we denote the components of T by T_{0}, T_{1}, …, T_{m}. The operator T_{0} is given by

T_{0}(f, g_{1}, …, g_{m}) = AP_{0}f − f + ∑_{j=1}^{m} AP_{j}(g_{j} − B_{j}P_{0}f) .

We see from Lemma 5.1 and Lemma 5.3 that T_{0} maps the space (5.7) continuously into H^{Θ}(ℝ_{+}^{n}).

Turning to the other components T_{1}, …, T_{m}, we remark that for j, k = 1, …, m the operator B_{k}P_{j} equals δ_{kj} I up to operators of lower order. More precisely, the operator

B_{k}(x, D)P_{j} − δ_{kj}I

is a ps.d.o. in ℝ^{n−1} which is continuous from

H^{Ξ^{(−m_{j} −1/2)}}(ℝ^{n−1}) to H^{Ξ^{(−m_{k}+1/2)}}(ℝ^{n−1}) .

This is due to the fact that w_{j}(x′, x_{n}, ξ′, λ) satisfies (3.3)–(3.4); the estimates for the lower order terms of the ps.d.o. B_{k}P_{j} can be found in the same way as it was done for AP_{j} in the proof of Lemma 5.3. From the continuity of B_{k}P_{j} − δ_{kj}I the continuity of T_{k} in the spaces given in the theorem immediately follows. □

6. Proof of the necessity

The aim of this section is to prove the following theorem.

Theorem 6.1 Let A degenerate regularly at the boundary ∂M and assume that inequality (1.4) holds. Let r and s be integers satisfying (4.4) and assume, in addition, that

r ≥ m and μ ≤ s ≤ r − m + μ .  (6.1)

If the a priori estimate (4.8) holds, then (A, B_{1}, …, B_{m}) is N-elliptic with parameter in the sense of Definition 2.4.

Note that if the orders m_{j} of the boundary operators B_{j} are all different (e.g., if the boundary operators are normal), then (1.4) is satisfied and (6.1) is a consequence of (4.4). The proof of this theorem is divided into several steps.

Necessity of condition 2.4 a). First of all note that applying estimate (4.8) to functions whose support does not intersect with the boundary, we obtain the estimate in ℝ^{n}

||u||_{r,s} ≤ C(||A(x, D, λ)u||_{r−2m,s−2μ} + λ^{r−s}||u||_{L_{2}})  (6.2)

Proposition 6.2 Suppose that (6.2) takes place and x^{0} is an arbitrary point of ℝ^{n}. Denote A(D, λ) = A^{0}(x^{0}, D, λ). Then the estimate

∥|D|^{s}(|D| + λ)^{r−s}u∥_{L_{2}}
≤ C∥|D|^{s−2μ}(|D| + λ)^{r−s−2m+2μ}A(D, λ)u∥_{L_{2}}  (6.3)

holds, where the constant C does not depend on x^{0} or λ.

The necessity of a) easily follows from (6.3). Indeed, applying the Fourier transform, we can rewrite (6.3) in the form

∫_{ℝ^{n}} [|ξ|^{2s}(|ξ| + λ)^{2(r−s)}

−C^{2}|ξ|^{2(s−2μ)}(|ξ| + λ)^{2(r−s−2m+2μ)}|A(ξ, λ)|^{2}]|(F u)(ξ)|^{2}dξ ≤ 0 .

Since u ∈ D is arbitrary, the expression in the square brackets is nonpositive. From this part a) follows.

To prove (6.3) we replace in (6.2) λ by ρλ with ρ > 0 and u(x) by

u_{ρ}(x) = ρ^{−r+n/2}u(ρ(x − x^{0}))  (6.4)

and tend ρ to +∞. To carry out the calculations we need the following

Lemma 6.3 Denote

(S_{ρ,x^{0}}u)(x) = u(ρ(x − x^{0})).  (6.5)

Then for an arbitrary ps.d.o. a(x, D) we have

[a(x, D)S_{ρ,x^{0}}u](x) = [S_{ρ,x^{0}}a(x^{0} + ρ^{−1}x, ρD)u](x) .  (6.6)

Proof. Direct calculation shows that

(F S_{ρ,x^{0}}u)(ξ) = ρ^{−n} exp(−ix^{0}ξ)(F u)(\frac{ξ}{ρ}) .

If we substitute the last expression in the left-hand side of (6.6) and change ξ to ρξ we obtain the right-hand side of (6.6). □

Proof of Proposition 6.2. Applying the a priori estimate (6.2) to the function u_{ρ} (cf. (6.4)), we obtain, according to the lemma,

((1 + |D|)^{s}(ρλ + |D|)^{r−s}u_{ρ})(x)
= ρ^{n/2}S_{ρ,x^{0}}[(ρ^{−1} + |D|)^{s}(λ + |D|)^{r−s}u](x) .

The L_{2}(ℝ^{n}) norm of this expression tends to the left-hand side of (6.3), as ρ tends to +∞.

Now we turn to the right-hand side of (6.2). We have

(1 + |D|)^{s−2μ}(ρλ + |D|)^{r−s−2m+2μ}A(x, D, ρλ)u_{ρ}(x)

= ρ^{n/2}S_{ρ,x^{0}}[(ρ^{−1} + |D|)^{s−2μ}(λ + |D|)^{r−s−2m+2μ}h_{ρ}](x)

where

h_{ρ}(x) = ρ^{−2m}A(x^{0} + ρ^{−1}x, ρD, ρλ)u(x) → A(D, λ)u

as ρ → +∞.

It is easy to check that the limit of the second term of the right-hand side of (6.2) is equal to zero. □

To prove the necessity of 2.4 b), c) and d) we consider (4.8) for functions with supports belonging to a small neighbourhood of a point x^{0} ∈ ∂M. In this case the norms in (4.8) can be taken in ℝ_{+}^{n} and ℝ^{n−1}, respectively. Now we use the fact that we have the norm equivalence (5.5) and the equivalence

Ξ^{(−l)}(ξ, λ) ≈ \{ \begin{matrix}(1 + |ξ|)^{s−l} (λ + |ξ|)^{r−s} , & l ≤ s , \\ (λ + |ξ|)^{r−l} , & l > s . \end{matrix}

According to [5], Section 2, the norm

∥(iD_{n} + \sqrt{1 + |D′|^{2}})^{q} (iD_{n} + \sqrt{λ^{2} + |D′|^{2}})^{p−q}u∥_{L_{2}(ℝ_{+}^{n})}

is defined for any p, q ∈ ℝ and is equivalent to ∥u∥_{(p,q),ℝ_{+}^{n}} (cf. (4.3)). Substituting these expressions into the a priori estimate (4.8) for the half space, we obtain in explicit form

∑_{l=0}^{s} ∥(1 + |D′|^{2})^{(s−l)/2}(λ^{2} + |D′|^{2})^{(r−s)/2}D_{n}^{l}u∥_{L_{2}(ℝ_{+}^{n})}

+ ∑_{l=s+1}^{r} ∥(λ^{2} + |D′|^{2})^{(r−l)/2}D_{n}^{l}u∥_{L_{2}(ℝ_{+}^{n})}

≤ C(∥σ(D, λ)A(x, D, λ)u∥_{L_{2}(ℝ_{+}^{n})}
+ λ^{r−s}||u||_{L_{2}(ℝ_{+}^{n})}
+ ∑_{j=1}^{μ} ∥(1 + |D′|^{2})^{(s−m_{j}−1/2)/2}(λ^{2} + |D′|^{2})^{(r−s)/2}B_{j}(x′, D)u∥_{L_{2}(ℝ^{n−1})}

+ ∑_{j=μ+1}^{m} ∥(λ^{2} + |D′|^{2})^{(r−m_{j}−1/2)/2}B_{j}(x′, D)u∥_{L_{2}(ℝ^{n−1})}) ,  (6.7)

where we used the abbreviation

σ(D, λ) := (iD_{n} + \sqrt{1 + |D′|^{2}})^{s−2μ}(iD_{n} + \sqrt{λ^{2} + |D′|^{2}})^{r−s−2m+2μ} .

Proposition 6.4 Suppose estimate (6.7) holds. Let x^{0} be an arbitrary point in ℝ^{n−1} and set A(D, λ) = A^{(0)}(x^{0}, D, λ), B_{j}(D, λ) = B_{j}^{(0)}(x^{0}, D, λ), j = 1, …, m. Then the following estimate holds

∑_{l=0}^{s} ∥|D′|^{s−l}(λ^{2} + |D′|^{2})^{(r−s)/2}D_{n}^{l}u∥_{L_{2}(ℝ_{+}^{n})}

+ ∑_{l=s+1}^{r} ∥(λ^{2} + |D′|^{2})^{(r−l)/2}D_{n}^{l}u∥_{L_{2}(ℝ_{+}^{n})}

≤ C(∥σ̃(D, λ)A(D, λ)u∥_{L_{2}(ℝ_{+}^{n})}

+ ∑_{j=1}^{μ} ∥|D′|^{s−m_{j}−1/2}(λ^{2} + |D′|^{2})^{(r−s)/2}B_{j}(D)u∥_{L_{2}(ℝ^{n−1})}

+ ∑_{j=μ+1}^{m} ∥(λ^{2} + |D′|^{2})^{(r−m_{j}−1/2)/2}B_{j}(D)u∥_{L_{2}(ℝ^{n−1})}) , (6.8)

where we have set

σ̃(D, λ) := (iD_{n} + |D′|)^{s−2μ}(iD_{n} + \sqrt{λ^{2} + |D′|^{2}})^{r−s−2m+2μ} .

Proof. We apply (6.7) with λ replaced by ρλ to the function u_{ρ} defined in (6.4), noting that S_{ρ,x^{0}}u is again defined in ℝ_{+}^{n} because of x^{0} ∈ ℝ^{n−1} and ρ > 0. From Lemma 6.3 and the fact that for any function v ∈ L_{2}(ℝ_{+}^{n}) we have

ρ^{n/2}∥S_{ρ,x^{0}}v∥_{L_{2}(ℝ_{+}^{n})} = ∥v∥_{L_{2}(ℝ_{+}^{n})} ,

we see that the l-th term in the first sum in (6.7) is equal to

∥(\frac{1}{ρ^{2}} + |D′|^{2})^{(s−l)/2} (λ^{2} + |D′|^{2})^{(r−s)/2}D_{n}^{l}u∥_{L_{2}(ℝ_{+}^{n})}

which tends to the corresponding term in (6.8) for ρ → ∞. The remaining expressions in (6.7) can be treated analogously; the term (ρλ)^{r−s}∥u_{ρ}∥_{L_{2}(ℝ_{+}^{n})} tends to zero for ρ → ∞.

For the terms involving the boundary operators we remark that γ_{0}S_{ρ,x^{0}} = S_{ρ,x^{0}}γ_{0} where γ_{0} : u ↦ u(·, 0) stands for the trace operator. Therefore we may apply Lemma 6.3 to the function B_{j}(x′, D)u_{ρ} defined in ℝ^{n−1}. □

If we apply (6.8) to a function of the form

u(x) = ϕ(x′)V (x_{n}), ϕ(x′) ∈ D(ℝ^{n−1})

we obtain an estimate on the half-line (cf. [8], Chapter 3, Proposition 2 in Subsection 2.3):

∑_{l=0}^{s} |ξ′|^{s−l}(λ^{2} + |ξ′|^{2})^{(r−s)/2}||D_{n}^{l}V ||_{L_{2}(ℝ_{+})}

+ ∑_{l=s+1}^{r}(λ^{2} + |ξ′|^{2})^{(r−l)/2}||D_{n}^{l}V ||_{L_{2}(ℝ_{+})}

≤ C(∥σ̃(ξ′, D_{n}, λ)A(ξ′, D_{n}, λ)V ∥_{L_{2}(ℝ_{+})}

+ ∑_{j=1}^{μ} |ξ′|^{s−m_{j}−1/2}(λ^{2} + |ξ′|^{2})^{(r−s)/2}|B_{j}(ξ′, D_{n})V (0)|

+ ∑_{j=μ+1}^{m}(λ^{2} + |ξ′|^{2})^{(r−m_{j}−1/2)/2}|B_{j}(ξ′, D_{n})V (0)|) .  (6.9)

Necessity of condition 2.4 b). Suppose V (x_{n}) ∈ L_{2}(ℝ_{+}) is a solution of the homogeneous equation

A(ξ′, D_{n}, λ)V (x_{n}) = 0, x_{n} > 0 .

Then this function satisfies the equation

A_{+}(ξ′, D_{n}, λ)V (x_{n}) = 0, x_{n} > 0.  (6.10)

Now from (6.9) we deduce the estimate

c(ξ′, λ) ∑_{l=0}^{r} ||D_{n}^{l}V ||_{L_{2}(ℝ_{+})} ≤ ∑_{j=1}^{m} |B′_{j}(ξ′, λ, D_{n})V (0)|  (6.11)

Here B′_{j} are remainders of B_{j} after the division by A_{+} and c(ξ, λ) > 0 for ξ′ ≠ 0 and λ ≥ 0. From a standard trace result for Sobolev spaces on ℝ_{+} we know that

∑_{j=1}^{r} |D_{n}^{j−1}V (0)| ≤ C ∑_{j=1}^{r+1} ∥D_{n}^{j−1}V ∥_{L_{2}(ℝ_{+})} .  (6.12)

From this and (6.11) we obtain, using r ≥ m (see (6.1)),

c̃(ξ′, λ) ∑_{j=1}^{m} |D_{n}^{j−1}V (0)| ≤ ∑_{j=1}^{m} | ∑_{k=1}^{m} b_{jk}(ξ′, λ)D_{n}^{k−1}V (0)|,  (6.13)

where

B′_{j}(ξ′, λ, z) = ∑_{k=1}^{m} b_{jk}(ξ′, λ)z^{k−1} .  (6.14)

The constant c̃(ξ′, λ) in (6.13) is positive for ξ′ ≠ 0 and λ ≥ 0. Note that the Cauchy problem

D_{n}^{k−1}V (0) = ζ_{k}, k = 1, …, m

for ODE (6.10) has a unique solution for arbitrary ζ = (ζ_{1}, …, ζ_{m}) ∈ ℂ^{m}. This means that for an arbitrary complex vector ζ we have the estimate

c̃(ξ′, λ)|ζ| ≤ |B(ξ′, λ)ζ|

where B(ξ′, λ) := (b_{jk}(ξ′, λ))_{j,k=1,...,m}. The last inequality means that the matrix B(ξ′, λ) is nonsingular as |ξ′| ≠ 0, λ ≥ 0, i.e. the necessity of b) is proved.

Proposition 6.5 Suppose the estimate (6.7) holds. Let x^{0} be an arbitrary point of ℝ^{n−1}. Then the inequality

∑_{l=0}^{s} ∥|D′|^{s−l}D_{n}^{l}u∥_{L_{2}(ℝ_{+}^{n})}

≤ C(∥(iD_{n} + |D′|)^{s−2μ}A_{2μ}(D)u∥_{L_{2}(ℝ_{+}^{n})}

+ ∑_{j=1}^{μ} ∥|D′|^{s−m_{j}−1/2}B_{j}(D)u∥_{L_{2}(ℝ^{n−1})})  (6.15)

holds, where A_{2μ}(D) = A_{2μ}^{(0)}(x^{0}, D), B_{j}(D) = B_{j}^{(0)}(x^{0}, D), j = 1, …, μ.

Proof. This can be seen in exactly the same way as Proposition 6.4, now applying the a priori estimate (6.7) with ρ^{t}λ instead of λ to the function

u_{ρ}(x) := ρ^{−t(r−s)−s+n/2}u(ρ(x − x^{0}))

where t > 1 is fixed and ρ > 0 tends to infinity. □

Necessity of condition 2.4 c). From Proposition 6.5 the estimate on the half-line can be obtained

∑_{l=0}^{s} |ξ′|^{s−l}||D_{n}^{l}V ||_{L_{2}(ℝ_{+})}

≤ C(∥(iD_{n} + |ξ′|)^{s−2μ}A_{2μ}(ξ′, D_{n})V ∥_{L_{2}(ℝ_{+})}

+ ∑_{j=1}^{μ} |ξ′|^{s−m_{j}−1/2}|B_{j}(ξ′, D_{n})V (0)|) .

As above we see that for solutions V (x_{n}) ∈ L_{2}(ℝ_{+}) of

A_{2μ}(ξ′, D_{n})V (x_{n}) = 0, x_{n} > 0

we obtain the inequality

∑_{l=1}^{s} |D_{n}^{l−1}V (0)| ≤ C ∑_{j=1}^{μ} |B′_{j}(ξ′, λ, D_{n})V (0)|  (6.16)

with a constant C independent of ξ′, |ξ′| = 1, and λ, where now B′_{j} denotes the remainder of B_{j} after division by (A_{2μ})_{+}. Replacing in (6.16) the germ of V in 0 by an arbitrary vector ζ ∈ ℂ^{μ} and using s ≥ μ (see (6.1)), we obtain the necessity of c).

Proposition 6.6 Suppose the estimate (6.7) holds and x^{0} is an arbitrary point of ℝ^{n−1}. Then the estimate

∑_{l=s}^{r} ||D_{n}^{l}u||_{L_{2}(ℝ_{+}^{n})} ≤ C(||(D_{n} − i)^{r−s−2m+2μ}D_{n}^{s}Q(x^{0}, D_{n})u||_{L_{2}(ℝ_{+}^{n})}

+ ∑_{j=μ+1}^{m} ||B_{j}^{(0)}(x^{0}, 0, D_{n})u||_{L_{2}(ℝ^{n−1})})  (6.17)

holds.

Proof. We apply (6.7) with λ replaced by ρ to the function

u_{ρ}(x) := ρ^{1/2+ε(n−1)/2−r}u(ρ^{ε}(x′ − x^{0}), ρx_{n})

with 0 < ε < 1 fixed. Now we use

u(ρ^{ε}(x′ − x^{0}), ρx_{n}) = [S_{ρ^{ε},x^{0}}^{(x′)} S_{ρ,0}^{(x_{n})}u](x) ,

where S_{···}^{(x′)} indicates that the operator S_{···} acts on the first n−1 variables (and analogously that S_{···}^{(x_{n})} acts on the last variable), and apply Lemma 6.3 twice. For the l-th term in the first sum of (6.7) we obtain the expression

ρ^{(1−ε)(l−s)}∥(ρ^{−2ε} + |D′|^{2})^{(s−l)/2}(1 + ρ^{2(ε−1)}|D′|^{2})^{(r−s)/2}D_{n}^{l}u∥_{L_{2}(ℝ_{+}^{n})}.

For l ≤ s − 1 this expression tends to zero for ρ → ∞, for l = s its limit equals ∥D_{n}^{s}u∥_{L_{2}(ℝ_{+}^{n})}.

The remaining terms can be treated analogously; to finish the proof we use

ρ^{−2m} A(x^{0} + \frac{x′}{ρ^{ε}} , \frac{x_{n}}{ρ} , ρ^{ε}D′, ρD_{n}, ρ) → D_{n}^{2μ}Q(x^{0}, D_{n})

and

ρ^{−m_{j}}B_{j}(x^{0} + \frac{x′}{ρ^{ε}} , ρ^{ε}D′, ρD_{n}) → B_{j}^{(0)}(x^{0}, 0, D_{n})

as ρ → ∞. □

Necessity of condition 2.4 d). From Proposition 6.6 we obtain the estimate on the half-line

∑_{l=s}^{r} ||D_{n}^{l}V ||_{L_{2}(ℝ_{+})} ≤ C(∥(D_{n} − i)^{r−s−2m+2μ}D_{n}^{s}Q(x^{0}, D_{n})V ∥_{L_{2}(ℝ_{+})}

+ ∑_{j=μ+1}^{m} |B_{j}^{(0)}(x^{0}, 0, D_{n})V (0)|) .  (6.18)

Since m_{j} ≥ s for j ≥ μ + 1, each B_{j}^{(0)}(x^{0}, 0, τ) contains the factor τ^{s}, and it is easily seen that condition d) follows from the analogous condition for

B̃_{j}^{(0)}(x^{0}, 0, τ) := τ^{−s}B_{j}^{(0)}(x^{0}, 0, τ) .

Now we apply (6.18) to a solution V ∈ L_{2}(ℝ_{+}) of

Q(x^{0}, D_{n})V (x_{n}) = 0, x_{n} > 0

and substitute W (x_{n}) := D_{n}^{s}V (x_{n}). We obtain

∑_{l=1}^{r−s+1} ∥D_{n}^{l−1}W ∥_{L_{2}(ℝ_{+})} ≤ C ∑_{j=μ+1}^{m} |B′_{j}(x^{0}, 0, D_{n})W (0)|,  (6.19)

where now B′_{j} stands for the remainder of B̃_{j}^{(0)} after division by Q_{+}. Using r − s ≥ m − μ (cf. (6.1)) and the trace result (6.12), we obtain the linear independence of B′_{j} modulo Q_{+} from (6.19) and therefore condition d).

Appendix. Singularly perturbed problems

One of the most important features of the Newton polygon approach is to provide an easy formulation and proof of a priori estimates in the theory of singularly perturbed problems. All results of the previous sections can be rewritten for boundary value problems with small parameter as treated by Vishik–Lyusternik [10], Nazarov [9], Frank [6] and others. (Cf. also [5], Appendix, for the Dirichlet problem.) As an example, we formulate an a priori estimate for such problems.

Consider for ε > 0 the operator

A_{ε}(x, D) := ε^{2m−2μ}A_{2m}(x, D) + ε^{2m−2μ−1}A_{2m−1}(x, D) + … + A_{2μ}(x, D)

with A_{j} of the form (1.2). Let A_{ε} act on a smooth compact manifold M with boundary ∂M and assume that we have boundary conditions B_{1}(x, D), …, B_{m}(x, D) of the form (1.3) satisfying (1.4).

We fix integer numbers r and s satisfying (4.4) and consider the weight function

Ξ_{ε}(ξ) := Ξ_{ε,(r,s)}(ξ) := (1 + |ξ|)^{s} (1 + ε|ξ|)^{r−s} .

The norms corresponding to this weight function will be denoted by ∥ · ∥_{Ξ_{ε},M} = ∥ · ∥_{ε,(r,s),M}.

Definition A.1 a) The operator A_{ε}(x, D) is called N-elliptic if

|A_{ε}^{(0)}(x, ξ)| ≥ C|ξ|^{2μ} (1 + ε|ξ|)^{2m−2μ} (ξ ∈ ℝ^{n} , ε > 0 , x ∈ \overline{M})

holds where C does not depend on x, ξ or ε.

b) The operator A_{ε} is said to degenerate regularly at the boundary if the polynomial

Q(x^{0}, τ) := τ^{−2μ}A_{1}^{(0)}(x^{0}, 0, τ)

has exactly m − μ roots in the upper half plane.

Definition A.2 The boundary problem (A_{ε}, B_{1}, …, B_{m}) is called Nelliptic if the following conditions hold:

a) The operator A_{ε}(x, D) is N-elliptic in the sense of Definition A.1. b) For every fixed x^{0} ∈ ∂M the boundary problem

(A_{ε}^{(0)}(x^{0}, ξ′, D_{n}), B_{1}^{(0)}(x^{0}, ξ′, D_{n}), …, B_{m}^{(0)}(x^{0}, ξ′, D_{n}) )

for each ε > 0 and ξ′ ≠ 0 is uniquelly solvable on the half-line x_{n} ≥ 0 in the space of functions tending to zero as x_{n} → ∞. Moreover we suppose that the problem

(A_{2m}^{(0)}(x^{0}, ξ′, D_{n}), B_{1}^{(0)}(x^{0}, ξ′, D_{n}), …, B_{m}^{(0)}(x^{0}, ξ′, D_{n}) )

(corresponding to ε = ∞) has the same property.

c) For every x^{0} ∈ ∂M the boundary problem

(A_{2μ}(x^{0}, D), B_{1}(x^{0}, D), …, B_{μ}(x^{0}, D))

fulfills the Shapiro–Lopatinskii condition.

d) For every x^{0} ∈ ∂M the polynomials (B_{j}^{(0)}(x^{0}, 0, τ))_{j=μ+1,...,m} are linearly independent modulo Q_{+}(x^{0}, τ) with Q_{+} defined in Definition 2.4 d).

If the conditions of Definition A.1 and A.2 hold, we can apply Theorem 4.2 to the operator

A(x, D, λ) := λ^{2m−2μ} A_{1/λ}(x, D) .

The connection between Ξ_{ε}(ξ) and Ξ(ξ, ε^{−1}) (defined in (4.2)) is given by

Ξ_{ε}(ξ) = ε^{r−s} Ξ(ξ, ε^{−1})

and

Ξ_{ε}^{(−m_{j}−1/2)}(ξ) = \{ \begin{matrix}ε^{r−s} Ξ^{(−m_{j}−1/2)}(ξ, ε^{−1}) & if j ≤ μ , \\ ε^{r−m_{j}−1/2} Ξ^{(−m_{j}−1/2)}(ξ, ε^{−1}) & if j > μ . \end{matrix}

Using these relations, we obtain from Theorem 4.2 the following result which can be found (without the notation of the Newton polygon) in [6]:

Theorem A.3 Assume that A_{ε} degenerates regularly and that (A_{ε}, B_{1}, …, B_{m}) is N-elliptic in the sense of Definition A.2. Then the following a priori estimate holds with a constant C independent of ε > 0:

∥u∥_{Ξ_{ε},M} ≤ C(∥A_{ε}u∥_{ε,(r−2m,s−2μ),M} + ∑_{j=1}^{μ} ∥B_{j}u∥_{Ξ_{ε}^{(−m_{j} −1/2)},∂M}

+ ∑_{j=μ+1}^{m} ε^{m_{j}+1/2−s} ∥B_{j}u∥_{Ξ_{ε}^{(−m_{j} −1/2)},∂M} + ∥u∥_{L_{2}(M)}) .

References

[1] Agmon, S.: On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Comm. Pure Appl. Math. 15 (1962), 119-147.

[2] Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 22 (1959), 623-727.

[3] Agranovich, M. S., Vishik, M. I.: Elliptic problems with parameter and parabolic problems of general form (Russian). Uspekhi Mat. Nauk 19 (1964), No. 3, 53-161. English transl. in Russian Math. Surv. 19 (1964), No. 3, 53-157.

[4] Denk, R., Mennicken, R., Volevich, L.: The Newton polygon and elliptic problems with parameter. Math. Nachr. 192 (1998), 125-157.

[5] Denk, R., Mennicken, R., Volevich, L.: Boundary value problems for a class of elliptic operator pencils. Keldysh Inst. Appl. Math. Preprint 58 (1998).

[6] Frank, L.: Coercive singular perturbations. I. A priori estimates. Ann. Mat. Pura Appl. (4) 119 (1979), 41–113.

[7] Gindikin, S. G., Volevich, L. R.: The Method of Newton’s Polyhedron in the Theory of Partial Differential Equations, Math. Appl. (Soviet Ser.) 86, Kluwer Academic, Dordrecht, 1992.

[8] Gindikin, S. G., Volevich, L. R.: Mixed problem for partial differential equations with quasihomogeneous principal part. Transl. Math. Monogr. 147, Amer. Math. Soc., Providence, RI, 1996.

[9] Nazarov, S. A.: The Vishik–Lyusternik method for elliptic boundary value problems in regions with conic points. I. The problem in a cone (Russian). Sibirsk. Mat. Zh. 22 (1981), No. 4, 142-163. English transl. in Siberian Math. J. 22 (1982), 594-611.

[10] Vishik, M. I., Lyusternik, L. A.: Regular degeneration and boundary layer for linear differential equations with small parameter (Russian). Uspehi Mat. Nauk (N.S.) 12 (1957), No. 5 (77), 3-122. English transl. in Amer. Math. Soc. Transl. (2) 20 (1962), 239-364.

[11] Volevich, L. R., Paneah, B. P.: Some spaces of generalized functions and embedding theorems (Russian). Uspehi Mat. Nauk 20 (1965), No. 1 (121), 3-74. English transl. in Russian Math. Surv. 20 (1964), No. 1, 1-73.