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

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

BOUNDARY VALUE PROBLEMS FOR A CLASS OF

ELLIPTIC OPERATOR PENCILS

Preprint 58 (1998)

Moscow 1998

Abstract

R. Denk, R. Mennicken, and L. R. Volevich^{1}. Boundary value problems for a class of elliptic operator pencils.

In this paper operator pencils A(x, D, λ) are studied which act on a manifold with boundary and satisfy the condition of N-ellipticity with parameter, a generalization of the notion of ellipticity with parameter as introduced by Agmon and Agranovich–Vishik. Sobolev spaces corresponding to a Newton polygon are defined and investigated; in particular it is possible to describe their trace spaces. With respect to these spaces, an a priori estimate holds for the Dirichlet boundary value problem connected with an N-elliptic pencil, and a right parametrix is constructed.

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 operator pencils 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)

acting on a smooth manifold M with smooth boundary Γ. Here m and μ are integer numbers with m > μ ≥ 0, A_{2μ}, …, A_{2m} are partial differential operators in M with infinitely smooth coefficients and λ is a complex parameter. We assume that

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

is a differential operator of order j with scalar coefficients a_{αj}(x) ∈ C^{∞}(\overline{M}). As usual, we use for multi-indices α = (α_{1}, …, α_{n}) the notation

D^{α} = D_{1}^{α_{1}} ⋯ D_{n}^{α_{n}} , D_{j} = −i\frac{∂}{∂x_{j}} , |α| = α_{1} + … + α_{n} . (1.3)

There is a close connection between pencils of the form (1.1) and general parabolic problems which we want to describe briefly. An important tool in the field of elliptic and parabolic problems is the concept of the Newton polygon. For a given polynomial

P (ξ, λ) = ∑_{α,k} a_{αk}ξ^{α}λ^{k} ,  (1.4)

where ξ = (ξ_{1}, …, ξ_{n}) ∈ ℝ^{n} and λ ∈ ℂ, let ν(P) be the set of all integer points (i, k) such that an α exists with |α| = i and p_{αk} ≠ 0. Then the Newton polygon N(P) is defined as the convex hull of all points in ν(P), their projections on the coordinate axes and the origin. The polynomial P (ξ, λ) is called N-parabolic (see [8], Chapter 2) if N(P) has no sides parallel to the coordinate axes and if the inequality

|P (ξ, λ)| > δ ∑_{(i,k)∈N(P)} |ξ|^{i} |λ|^{k}  (1.5)

holds for all λ ∈ ℂ with Im λ < λ_{0} where δ > 0 and λ_{0} are constants. An important example of such polynomials is the product of polynomials P_{1}(ξ, λ), …, P_{N}(ξ, λ) which are quasi-homogeneous and 2b_{j}-parabolic in the sense of Petrovskii with different weights 2b_{j} (j = 1, …, N). Note that in this case P (ξ, λ) is no quasi-homogeneous function in (ξ, λ). Similarly (see [5]), the polynomial P (ξ, λ) is called N-elliptic with parameter along some ray L of the complex plane if (1.5) holds for all ξ ∈ ℝ^{n} and all λ ∈ L , |λ| ≥ R , with large enough R. This type of polynomials appears, for instance, if one considers Douglis–Nirenberg systems A(x, D) − λI which are elliptic with parameter. On manifolds without boundary Douglis–Nirenberg systems were investigated by Kozhevnikov [9] and by the authors in [5]. It turned out that an equivalent condition for unique solvability of such a system and sharp a priori estimate is the condition that for every x the determinant

P (x, ξ, λ) = det(A(x, D) − λI)

satisfies inequality (1.5).

The basic idea of the Newton polygon method for the problems mentioned above is to assign to λ various weights r_{j} which are defined by the Newton polygon. For each of these weights we obtain a different principal part of P (ξ, λ) which we denote by P_{r_{j}}(ξ, λ). On a manifold without boundary there is a finite open covering {U_{j}}_{j} of the set of all (ξ, λ) and a corresponding partition of unity ∑_{j} ψ_{j}(ξ, λ) ≡ 1 such that P (D, λ)ψ_{j}(D, λ) differs from the corresponding principal part P_{r_{j}}(D, λ)ψ_{j}(D, λ) only by a small regular perturbation. This allows estimates and existence results for the operators P (D, λ), cf. [5] for N-elliptic systems and [8] for parabolic problems.

Now let us consider the same problems on a manifold with boundary. For instance, let P (D, λ) be the product of two operators which are parabolic in the sense of Petrovskii, i.e.

P (D, λ) = (λ + A_{2p}(D)) (λ + A_{2q}(D)) ,

where λ + A_{2p}(D) and λ + A_{2q}(D) are 2p- and 2q-parabolic operators, respectively, with p > q. If, for instance, we assign to λ the weight r_{1} = 2q, we obtain the principal part P_{r_{1}}(D, λ) = A_{2p}(D)A_{2q}(D) + λA_{2p}(D) which is of the form (1.1). If we take the weight r_{2} with 2q < r_{2} < 2p the corresponding principal part is P_{r_{2}}(D, λ) = λA_{2p}(D). The operator P_{r_{2}}(D, λ) has to be supplied with p boundary conditions while the operator P (D, λ) needs p + q boundary conditions. Thus we can see here that P (D, λ) is (after dividing by λ) a singular perturbation of the principal part P_{r_{2}}(D, λ). A similar situation occurs if the weight of λ is larger than 2p + 2q.

So we see from this example that operator pencils of the form (1.1) and singular perturbations naturally arise when we deal with N-parabolic problems on manifolds with boundary. If we consider boundary value problems elliptic in the sense of Douglis–Nirenberg, the situation is the same or even more complicated.

As a first step to handle these problems we consider as a model problem operator pencils of the form (1.1) and the corresponding Dirichlet problem on manifolds with boundary. The aim of this paper is to show that the Newton polygon provides an easy and direct approach to the Sobolev spaces where the pencil acts and to the proof of a priori estimate. In particular, we obtain a description of the trace spaces which is formulated in the general context of Sobolev spaces corresponding to Newton polygons. We hope to study in a subsequent paper boundary value problems for general N-parabolic operators on manifolds with boundary.

For r, s ∈ ℝ let the Sobolev space H^{(r,s)}(ℝ^{n}) be defined by

H^{(r,s)}(ℝ^{n}) =  (1.6)

{u ∈ S′(ℝ^{n}) : (|ξ|^{2} + 1)^{s/2}(|ξ|^{2} + |λ|^{2})^{(r−s)/2}F u(ξ) ∈ L_{2}(ℝ^{n})}

where F u denotes the Fourier transform of u. The norm in H^{(r,s)}(ℝ^{n}) is given by

∥u∥_{r,s} := (∫_{ℝ^{n}}(|ξ|^{2} + 1)^{s}(|ξ|^{2} + |λ|^{2})^{r−s}|F u(ξ)|^{2} dξ)^{\frac{1}{2}} .  (1.7)

Restricting the distributions belonging to H^{(r,s)}(ℝ^{n}) to the right half space ℝ_{+}^{n} := {x = (x′, x_{n}) ∈ ℝ^{n} : x_{n} > 0}, we obtain the Sobolev space H^{(r,s)}(ℝ_{+}^{n}). See Section 2 for the description of the norm in this Sobolev space. In the standard way we can also define H^{(r,s)}(M) using local coordinates.

For every r and s the operator pencil (1.1) acts continuously from H^{(r,s)} to H^{(r−2m,s−2μ)}. In what follows in connection with the Dirichlet problem for (1.1) we will restrict ourselves to the case r = m, s = μ, i.e. we will realize (1.1) as an operator from H^{(m,μ)} onto H^{(−m,−μ)}. We will assume this pencil to be elliptic with parameter along the ray [0, ∞) in the following sense: Denote by

A_{j}^{(0)}(x, ξ) := ∑_{|α|=j} a_{αj}(x)ξ^{α} (j = 2μ, …, 2m)  (1.8)

the principal symbol of A_{j}, where ξ^{α} = ξ_{1}^{α_{1}} ⋯ ξ_{n}^{α_{n}} for ξ = (ξ_{1}, …, ξ_{n}) = (ξ′, ξ_{n}), and by

A^{(0)}(x, ξ, λ) := A_{2m}^{(0)}(x, ξ) + λA_{2m−1}^{(0)}(x, ξ) + … + λ^{2m−2μ}A_{2μ}^{(0)}(x, ξ) (1.9)

the principal symbol of A(x, D, λ). Then our main assumption is that the estimate

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

(1.10)

holds where the constant C does not depend on x, ξ or λ. In the case μ = 0 this is the usual definition of ellipticity with parameter which was introduced by Agmon [1] and Agranovich–Vishik [3]. Therefore we may assume in the following that μ > 0. In this case even for λ ≠ 0 the principal symbol A^{(0)}(x, ξ, λ) vanishes for ξ = 0 which causes the main difficulties in proving existence results and estimates. Note that the symbol A^{(0)}(x, ξ, λ) is homogeneous in ξ and λ of degree 2m, as it is the case for the problems treated in [3].

We will consider boundary value problems in ℝ^{n}, ℝ_{+}^{n} and M. For this we will describe the space of traces of functions u ∈ H^{(m,μ)}(ℝ_{+}^{n}), i.e. the space

{D_{n}^{j−1}u(x′, 0) : u ∈ H^{(m,μ)}(ℝ_{+}^{n})} for j = 1, …, m .  (1.11)

This will be done in a more general context in Section 2 where Sobolev spaces corresponding to Newton polygons are considered. The space H^{(m,μ)}(ℝ_{+}^{n}) appears to be a special case of the space H^{Ξ}(ℝ_{+}^{n}) where Ξ(ξ, λ) is the weight function corresponding to the Newton polygon N(P) of a polynomial P (ξ, λ) in ξ ∈ ℝ^{n} and λ ∈ ℂ. It turns out that the trace

space \{D_{n}^{j−1}u(x′, 0) : u ∈ H^{Ξ}(ℝ_{+}^{n})\} is given by H^{Ξ^{(−j+ \frac{1}{2} )}}(ℝ^{n−1}) where Ξ^{(−j+ \frac{1}{2} )}(ξ′, λ) denotes the weight function corresponding to the Newton polygon which is constructed from N(P) by a shift of length j − \frac{1}{2} to the left. Cf. Section 2 for details. In particular, in the case of the operator pencil (1.1) the trace spaces have the form H^{(m_{j},μ_{j})}(ℝ^{n−1}) where the parameters m_{j} and μ_{j} can be seen directly from the corresponding Newton polygon.

In Section 5 we consider the Dirichlet boundary problem

A(x, D, λ) u(x) = f(x) in M ,  (1.12)
(\frac{∂}{∂ν} )^{j−1} u(x) = g_{j}(x) (j = 1, …, m) on Γ .  (1.13)

where \frac{∂}{∂ν} denotes the derivative in the direction of the inner normal to the boundary. The main theorem states that for every solution u ∈ H^{(m,μ)}(M) of the boundary value problem (1.12)–(1.13) the a priori estimate

∥u∥_{m,μ} ≤ C(∥f∥_{−m,−μ} + ∑_{j=1}^{m} ∥g_{j}∥_{m_{j},μ_{j}} + λ^{m−μ}∥u∥_{L_{2}(M)}) (1.14)

holds for λ ≥ λ_{0} with a constant C not depending on λ or u. The proof of this theorem is essentially based on estimates of the solution of an ordinary differential equation which arises from (1.12)–(1.13) by fixing x ∈ Γ, rewriting the boundary value problem in coordinates corresponding to x and taking the partial Fourier transform with respect to the first n − 1 variables. Estimates for the fundamental solution of the resulting ordinary differential equation can be found in Section 4 and use the precise knowledge about the zeros of the principal symbol A^{(0)}(x, ξ, λ) considered as a polynomial in ξ_{n}.

These zeros can (for large λ) be arranged in two groups, one group remaining bounded for λ → ∞, the other group of zeros being exactly of order O(λ) for λ → ∞. To obtain this result we have to impose an additional condition on the principal symbol A^{(0)}(x, ξ, λ) which is the analogue of the condition of regular degeneration which is known from the theory of singular perturbations (cf. Vishik-Lyusternik [13]). The details can be found in Section 3.

As mentioned above, there is a close connection between pencils of the form (1.1) and elliptic boundary value problems with small parameter. Nazarov obtained in [12] a priori estimates under the assumption that the fundamental solutions fulfill some estimates which are similar to those proved in Section 4 below. (The norms used in [12] differ slightly from the norms used in the present paper.) In several papers Frank and other authors investigated singular perturbed problems and corresponding a priori estimates, cf. [6] and the references therein. The use of the Newton polygon method which gives the connection to general parabolic problems as described above, seems to be new even for singular perturbed problems.

2. Newton’s polygon and functional spaces corresponding to it

In this section we consider a polynomial P (ξ, λ) of the form (1.4) and its Newton polygon N(P) which was defined in the Introduction. For a detailed discussion of the Newton polygon, we refer the reader to Gindikin-Volevich [8], Chapters 1 and 2.

To construct function spaces corresponding to the Newton polygon, we consider the weight function

Ξ_{P} (ξ, λ) := ∑_{(i,k)∈N(P)} |ξ|^{i} |λ|^{k} ,  (2.1)

where the summation on the right-hand side is extended over all integer points of N(P). The Sobolev space H^{Ξ} will arise as a special case of the following more general definition which is taken from Volevich-Paneah [15]. It can be seen directly that the function σ(ξ) := Ξ_{P} (ξ, λ) satisfies the condition which appears in this definition (cf. also Remark 2.4 below). In the following, the Fourier transform F is defined by

F u(ξ) = \frac{1}{(2π)^{\frac{n}{2}}} ∫_{ℝ^{n}} e^{−ix·ξ}u(x) dx  (2.2)

for u ∈ S(ℝ^{n}), the definition is extended in the usual way to distributions u ∈ S′(ℝ^{n}).

Definition 2.1 Let σ(ξ) be a continuous function on ℝ^{n} with values in ℝ_{+} and assume that σ(ξ)σ^{−1}(η) ≤ C(1 + |ξ − η|^{N}) holds for all ξ, η ∈ ℝ^{n} with constants C and N not depending on ξ or η. Then H^{σ} is defined as the space of all distributions u in S′(ℝ^{n}) such that σ(ξ)F u(ξ) ∈ L_{2}(ℝ^{n}). The space H^{σ} is endowed with the norm

∥u∥_{σ,ℝ^{n}} := ( ∫_{ℝ^{n}} σ^{2}(ξ)|F u(ξ)|^{2} dξ)^{1/2} .  (2.3)

Proposition 2.2 (See Volevich-Paneah [15].) Let σ(ξ, λ) be a continuous function of ξ and assume that

σ(ξ, λ)σ^{−1}(η, λ) ≤ C_{1}(1 + |ξ − η|^{N})

holds with a constant C_{1} not depeding on ξ, η or λ. Let

σ′_{l}(ξ′, λ) := (∫_{−∞}^{∞} \frac{ξ_{n}^{2l}}{σ^{2}(ξ, λ)} dξ_{n})^{−1/2} < ∞ .

Then D_{n}^{l}u(x′, 0) is well-defined as an element of H^{σ′_{l}}(ℝ^{n−1}) for every u ∈ H^{σ}(ℝ^{n}), and there exists a constant C, independent of u and λ, such that

∥D_{n}^{l}u(x′, 0)∥_{σ′_{l},ℝ^{n−1}} ≤ C∥u∥_{σ,ℝ^{n}} .  (2.4)

We will apply Proposition 2.2 to the case where σ(ξ, λ) is given by Ξ_{P} (ξ, λ) (see (2.1)).

Let one of the functions σ(ξ, λ) or σ_{1}(ξ, λ) for each λ satisfy the condition of Definition 2.1 and σ(ξ, λ) ≈ σ_{1}(ξ, λ). The symbol ≈ means that there exist positive constants C_{1} and C_{2}, independent of ξ and λ, such that

C_{1}σ(ξ, λ) ≤ σ_{1}(ξ, λ) ≤ C_{2}σ(ξ, λ) .

Then the other function also satisfies the condition of Definition 2.1 and, evidently, the statement of Proposition 2.2 remains valid, if we replace σ by the equivalent function σ_{1}. In the following we will construct an equivalent function for Ξ_{P} (ξ, λ) (cf. [5], Section 2). For this purpose we introduce some simple geometric notions connected with the Newton polygon (see, e.g., [8], Chapter 1).

[figure: i; k; •; •; •; •; •; •; Γ_{1}; Γ_{2}; ^{...}; Γ_{S}; Fig. 1]

Let Γ_{1}, …, Γ_{S} be the sides of the Newton polygon not lying on the coordinate axes and indexed in the clockwise direction (cf. Fig. 1). Suppose that

(0, 0), (a_{1}, b_{1}), …, (a_{S+1}, b_{S+1}) , a_{1} = 0, b_{S+1} = 0 ,

are the vertices of the polygon N(P). Then the side Γ_{s} is given by

Γ_{s} = {(a, b) ∈ ℝ^{2} : 1 · a + r_{s} · b = d_{s}} (s = 1, …, S)  (2.5)

where r_{s} = (a_{s+1}−a_{s})/(b_{s}−b_{s+1}). The vector (1, r_{s}) is an exterior normal to the side Γ_{s}, where we admit r_{1} = ∞ if Γ_{1} is horizontal. Further we have r_{S} = 0 in the case that Γ_{S} is vertical. In what follows we will suppose that Γ_{S} is not vertical. Since N(P) is convex, we have

∞ ≥ r_{1} > … > r_{S} > 0 .

The r_{s}-principal part of P is defined by

P_{r_{s}}(ξ, λ) := ∑_{|α|+r_{s}k=d_{s}} a_{αk}ξ^{α}λ^{k} .  (2.6)

Here d_{s} is the so-called r_{s}-degree of P which may be defined by

d_{s} := max_{(a,b)∈N(P)}(1 · a + r_{s} · b) .  (2.7)

Now we set

Ξ_{(s)}(ξ, λ) = |ξ|^{−a_{s}} |λ|^{−b_{s+1}} ∑_{i+r_{s}k=d_{s}} |ξ|^{i}|λ|^{k} .

This function will be a polynomial of |ξ| and |λ|.

Repeating the argument in [8], Theorem 1.1.3, we can prove that

∏_{s=1}^{S} Ξ_{(s)}(ξ, λ) = ∑_{s=1}^{S} |ξ|^{a_{s}}|λ|^{b_{s}} + …,  (2.8)

where the dots denote the sum of monomials |ξ|^{i}|λ|^{k} with (i, k) ∈ N(P). For |λ| ≥ 1 the right-hand side can be estimated from below by

1 + ∑_{s=1}^{S} |ξ|^{a_{s}}|λ|^{b_{s}} .

This function can be estimated from below by Ξ_{P} (ξ, λ) (see [5], Subsection 3.2). From this it follows that the left-hand side of (2.8) is equivalent to Ξ_{P}. Denote by 2m_{s} the largest degree of |ξ| in Ξ_{(s)}. It is obvious that Ξ_{(s)} is equivalent to (|ξ| + |λ|^{\frac{1}{rs}})^{2m_{s}}, and consequently

Ξ_{P} (ξ, λ) ≈ ∏_{s=1}^{S} (|ξ|^{2} + |λ|^{\frac{2}{rs}} )^{m_{s}} .  (2.9)

We will suppose further, as in the case of parabolic polynomials (cf. [8], Chapter 2), that m_{1}, …, m_{S} are integers.

Remark 2.3 In the case r_{1} = ∞ (i.e. Γ_{1} is horizontal) (2.6) and (2.7) have no sense and (2.6) should be replaced by

P_{r_{1}} := ∑_{|α|=a_{2}} a_{αb_{1}}ξ^{α}λ^{b_{1}} .

As for the equivalence (2.9), it will be valid for |λ| > λ_{0} with arbitrary λ_{0} > 0 and the equivalence constants, of course, depend on λ_{0}.

Remark 2.4 The fact that Ξ(ξ, λ) satisfies the condition of Definition 2.1 is an immediate consequence of (2.9) as this condition is fulfilled for each factor on the right-hand side.

Remark 2.5 From (2.9) it follows that the r_{s}-degree d_{s} (cf. (2.7)) is given by

d_{s} = 2 (∑_{j=1}^{s} m_{j} + ∑_{j=s+1}^{S} \frac{r_{s}}{r_{j}} m_{s}) .  (2.10)

To see this, we use the relation

Ξ_{P} (tξ, t^{r_{s}}λ) = t^{d_{s}}Ξ_{P_{rs}} (ξ, λ) + o(t^{d_{s}}), t → +∞ ,  (2.11)

cf. [8], Section 1.1.2. In our case we obtain, denoting the right-hand side of (2.10) by d′_{s},

Ξ_{P} (tξ, t^{r_{s}}λ) = ∏_{j=1}^{S} (t^{2}|ξ|^{2} + t^{2 \frac{rs}{r_{j}}} |λ|^{\frac{2}{r_{j}}} )^{m_{j}}

= t^{d′_{s}} ∏_{j=1}^{s} (|ξ|^{2} + t^{2( \frac{rs}{r_{j}} −1)}|λ|^{\frac{2}{r_{j}}} )^{m_{j}} ∏_{j=s+1}^{S}(t^{2(1− \frac{rs}{r_{j}})}|ξ|^{2} + |λ|^{\frac{2}{r_{j}}} )^{m_{j}}
= t^{d′_{s}}Ξ_{P_{rs}} (ξ, λ) + o(t^{d′_{s}}) ,

which shows d_{s} = d′_{s}.

Now we will describe the trace spaces of the spaces H^{Ξ}. For this we use the following lemma:

Lemma 2.6 Let 1 ≤ a_{1} < a_{2} < … < a_{S} < ∞ and m_{1}, …, m_{S} ∈ ℕ. For l ∈ ℕ with 0 ≤ l < 2(m_{1} + … + m_{S}) define the index κ by

2m_{1} + … + 2m_{κ−1} ≤ l < 2m_{1} + … + 2m_{κ} .  (2.12)

Then there exists a constant C > 0, independent of a_{1}, …, a_{S}, such that

\begin{matrix}C^{−1}a_{κ}^{2l+1−4m_{1}−...−4m_{κ}} ∏_{s=κ+1}^{S} a_{s}^{−4m_{s}} ≤ ∫_{−∞}^{∞} \frac{t^{2l}}{∏_{s=1}^{S}(t^{2} + a_{s}^{2})^{2m_{s}}} dt \\ ≤ Ca_{κ}^{2l+1−4m_{1}−...−4m_{κ}} ∏_{s=κ+1}^{S} a_{s}^{−4m_{s}} . \end{matrix}

(2.13)

In the case 0 ≤ l < 2m_{1}, we set m_{0} = 0 in (2.12).

Proof. Substituting in the integral t = a_{S}τ, we obtain

I := ∫_{−∞}^{∞} t^{2l} ∏_{s=1}^{S}(t^{2} + a_{s}^{2})^{−2m_{s}}dt

= 2a_{S}^{2l+1−4m_{1}−...−4m_{S}} ∫_{0}^{∞} t^{2l} ∏_{s=1}^{S} (t^{2} + (\frac{a_{s}}{a_{S}} )^{2})^{−2m_{s}} dt.

For t ≥ 1 we use

t^{2l}(1+t^{2})^{−2m_{1}−...−2m_{S}} ≤ t^{2l} ∏_{s=1}^{S} (t^{2} + (\frac{a_{s}}{a_{S}} )^{2})^{−2m_{s}} ≤ t^{2l−4m_{1}−...−4m_{S}} .

As l < 2 ∑_{s=1}^{S} m_{s}, the left-hand and right-hand side of this inequality are integrable functions over [1, ∞), and we obtain

C_{1}^{−1} ≤ ∫_{1}^{∞} t^{2l} ∏_{s=1}^{S} (t^{2} + (\frac{a_{s}}{a_{S}} )^{2})^{−2m_{s}}dt ≤ C_{1}

for some C_{1} > 0.

For 0 ≤ t ≤ 1 we have 1 ≤ 1 + t^{2} ≤ 2, and therefore

∫_{0}^{1} … dt ≈ ∫_{0}^{1} t^{2l} ∏_{s=1}^{S−1} (t^{2} + \frac{a_{s}^{2}}{a_{S}^{2}} )^{−2m_{s}} dt .

Now we substitute t = \frac{a_{S−1}}{a_{S}} τ and get

∫_{0}^{1}… dt ≈ (\frac{a_{S−1}}{a_{S}} )^{2l+1−4m_{1}−...−4m_{S−1}}∫0^{\frac{a_{S}}{a_{S−1}}}t^{2l} ∏_{s=1}^{S−1} (t^{2} + \frac{a_{s}^{2}}{a_{S−1}^{2}} )^{−2m_{s}}dt .

Again we split up ∫0^{\frac{a_{S}}{a_{S−1}}} … = ∫_{0}^{1} … + ∫1^{\frac{a_{S}}{a_{S−1}}} … and use an estimate of

the form C_{2}^{−1} ≤ ∫1^{\frac{a_{S}}{a_{S−1}}} … ≤ C_{2} for the second integral.

Proceding in this way, we receive

I ≈ a_{S}^{2l+1−4m_{1}−...−4m_{S}} (\frac{a_{S−1}}{a_{S}} )^{2l+1−4m_{1}−...−4m_{S−1}} · . . .

· (\frac{a_{κ}}{a_{κ+1}} )^{2l+1−4m_{1}−...−4m_{κ}} ∫0^{\frac{a_{κ+1}}{aκ}} t^{2l} ∏_{s=1}^{κ} (t^{2} + \frac{a_{s}^{2}}{a_{κ}^{2}} )^{−2m_{s}} dt .

For the last integral we use

t^{2l}(t^{2} + 1)^{−2m_{1}−...−2m_{κ}} ≤ t^{2l} ∏_{s=1}^{κ} (t^{2} + \frac{a_{s}^{2}}{a_{κ}^{2}} )^{−2m_{s}}
≤ t^{2l−4m_{1}−...−4m_{κ−1}}(t^{2} + 1)^{−2m_{κ}} .

As 2m_{1} +…+2m_{κ−1} ≤ l < 2m_{1} +…+2m_{κ}, the left-hand and the righthand side of this inequality are integrable functions on [0, ∞). Therefore

I ≈ a_{κ}^{2l+1−4m_{1}−...−4m_{κ}}a_{κ+1}^{−4m_{κ+1}} · . . . · a_{S}^{−4m_{S}} .

□

Remark 2.7 Using the substitution t = a_{1}τ, it is easily seen that the condition a_{1} ≥ 1 in Lemma 2.6 may be replaced by a_{1} > 0.

As in the Introduction, we denote by Ξ_{P}^{(−l)}(ξ, λ) the function corresponding to the Newton polygon which is constructed from N(P) by a shift of length l to the left parallel to the abscissa. We preserve the notation H^{Ξ_{P}^{(−l)}}(ℝ^{n−1}) for the spaces in ℝ^{n−1} corresponding to the weight functions Ξ_{P}^{(−l)}(ξ′, λ) := Ξ_{P}^{(−l)}(ξ′, 0, λ).

Lemma 2.8 Let λ_{0} > 0. Then for |λ| ≥ λ_{0} we have

σ′_{l}(ξ′, λ) ≈ Ξ^{(−l− \frac{1}{2} )}(ξ′, λ) ,  (2.14)

where σ′_{l} is defined by

σ′_{l}(ξ′, λ) := (∫_{−∞}^{∞} \frac{ξ_{n}^{2l}}{Ξ_{P}^{2} (ξ, λ)} dξ_{n})^{− \frac{1}{2}} .  (2.15)

Proof. Instead of Ξ_{P} we use the right-hand side of (2.9). From Lemma 2.6 with a_{s}^{2} = |ξ′|^{2} + |λ|^{\frac{2}{rs}} we obtain (see Remark 2.7) that

σ′_{l}(ξ′, λ) ≈ (|ξ′|^{2} + |λ|^{\frac{2}{rκ}} )^{m_{1}+...+m_{κ}− \frac{l}{2} − \frac{1}{4}} ∏_{s=κ+1}^{S} (|ξ′|^{2} + |λ|^{\frac{2}{rs}} )^{m_{s}} ,

(2.16)

where κ is chosen according to Lemma 2.6. From Remark 2.5 applied to σ′_{l}(ξ′, λ) we see that the sides of the Newton polygon corresponding to the weight function (2.16) are given by

Γ_{j} = {(a, b) ∈ ℝ^{2} : a + r_{j}b = d′_{j}}

with d′_{j} = d_{j} − l − \frac{l}{2} (j = κ, …, S). But this means that the Newton polygon for σ′_{l} is constructed from N(P) by a shift of l + \frac{1}{2} to the left,

i.e. we have σ′_{l}(ξ′, λ) ≈ Ξ_{P}^{(−l− \frac{1}{2} )}(ξ′, λ). □

The following theorem is an immediate consequence of Proposition 2.2 and Lemma 2.8.

Theorem 2.9 For every λ_{0} > 0 there exists a constant C > 0, independent of u and λ, such that

∥D_{n}^{l}u(x′, 0)∥_{Ξ_{P}^{(−l− \frac{1}{2} )},ℝ^{n−1}} ≤ C∥u∥_{Ξ_{P} ,ℝ^{n}} (l = 0, …, 2m_{1} + … + 2m_{S} − 1)

(2.17)

holds for u ∈ H^{Ξ_{P}} (ℝ^{n}) and λ ∈ ℂ with |λ| ≥ λ_{0}.

In the following, we will also consider the function spaces in the half space ℝ_{+}^{n} which correspond to Newton polygons. Using the binomial formula, it is easily seen that

Ξ_{P}^{2} (ξ, λ) ≈ ∑_{l=0}^{M} ξ_{n}^{2l} (Ξ_{P}^{(−l)}(ξ′, λ))^{2}  (2.18)

where M = 2m_{1} + … + 2m_{S}. From this it follows that we can take

( ∑_{l=0}^{M} ∫_{−∞}^{∞} ∥(D_{n}^{l}u)(·, x_{n})∥_{Ξ_{P}^{(−l)},ℝ^{n−1}}^{2} dx_{n})^{1/2}  (2.19)

as an equivalent norm in H^{Ξ_{P}} (ℝ^{n}). Replacing the integral over ℝ by the integral over x_{n} ≥ 0 we define a norm in H^{Ξ_{P}} (ℝ_{+}^{n}).

To define the space H^{\frac{1}{Ξ_{P}}} (ℝ_{+}^{n}), we use the more general approach which can be found, e.g., in [15]. Let σ(ξ) be a weight function fulfilling the condition in Definition 2.1. Denote by H^{σ}(ℝ^{n})_{±} the subspace of H^{σ}(ℝ^{n}) consisting of elements with supports in the closure of ℝ_{±}^{n}. Then we define

H^{σ}(ℝ_{+}^{n}) = H^{σ}(ℝ^{n})/H^{σ}(ℝ^{n})_{−}  (2.20)

endowed with the natural quotient norm

∥f∥_{σ,ℝ_{+}^{n}} = inf_{f_{−}∈H^{σ}(ℝ^{n})_{−}} ∥f_{0} + f_{−}∥_{σ,ℝ^{n}} ,  (2.21)

where f_{0} is an arbitrary representative of the conjugacy class of f.

Suppose that σ(ξ′, ξ_{n}) for a fixed ξ′ ∈ ℝ^{n−1} can be extended as a holomorphic function in ξ_{n} of polynomial growth in the lower half-plane Im ξ_{n} < 0. In this case the quotient norm of f ∈ H^{σ}(ℝ_{+}^{n}) coincides with the norm

∥σ(D′, D_{n})f_{0}∥_{L_{2}(ℝ_{+}^{n})}  (2.22)

which does not depend on the choice of the element f_{0} in the conjugacy class. In (2.22) the pseudo-differential operator (ps.d.o.) σ(D′, D_{n}) = σ(D) is defined by

σ(D)f := F^{−1}σ(ξ)(F f)(ξ)

In the case when

σ ≈ ∏_{j=1}^{S}(|ξ|^{2} + |λ|^{2/r_{j}})^{m_{j}}

we replace σ in the definition of H^{σ}(ℝ_{+}^{n}) by

∏_{j=1}^{S} (iξ_{n} + (|ξ|^{2} + |λ|^{2/r_{j}})^{1/2})^{2m_{j}} .

3. The zeros of the symbol

Now we come back to the operator pencil (1.1) and consider the corresponding model problem with constant coefficients and without lower order terms. Let A(ξ, λ) be a polynomial in ξ ∈ ℝ^{n} and λ ∈ ℂ of 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.

Definition 3.1 The polynomial A(ξ, λ) is called N-elliptic with parameter in [0, ∞) if the estimate

|A(ξ, λ)| ≥ C |ξ|^{2μ} (λ + |ξ|)^{2m−2μ} (ξ ∈ ℝ^{n}, λ ∈ [0, ∞))  (3.2)

holds with a constant C independent of ξ and λ.

Lemma 3.2 The polynomial A(ξ, λ) is N-elliptic with parameter in [0, ∞) if and only if the following conditions are satisfied:

(i) A_{2m}(ξ) is elliptic, i.e. A_{2m}(ξ) ≠ 0 for ξ ∈ ℝ^{n}\{0}.

(ii) A_{2μ}(ξ) is elliptic.

(iii) A(ξ, λ) ≠ 0 for ξ ∈ ℝ^{n}\{0} and λ ∈ [0, ∞).

Proof. From (3.2) we trivially obtain condition (iii) and, setting λ = 0, condition (i). Taking ε = \frac{1}{λ} and dividing (3.2) by ε^{2μ−2m}, we receive

|A_{2μ}(ξ)+εA_{2μ+1}(ξ)+…+ε^{2m−2μ}A_{2m}(ξ)| ≥ C|ξ|^{2μ}(1+ε|ξ|)^{2m−2μ} . (3.3)

Taking the limit for ε → 0, we obtain (ii).

Now let conditions (i)–(iii) be fulfilled. For ξ ∈ ℝ^{n}\{0} we write A(ξ, λ) in the form

A(ξ, λ) = A_{2μ}(ξ)B_{2m−2μ}(ξ, λ)  (3.4)

with

B_{2m−2μ}(ξ, λ) = \frac{A_{2m}(ξ)}{A_{2μ}(ξ)} + λ\frac{A_{2m−1}(ξ)}{A_{2μ}(ξ)} + … + λ^{2m−2μ} .  (3.5)

The coefficients of B_{2m−2μ}(ξ, λ) (considered as a polynomial in λ) are homogeneous functions in ξ ∈ ℝ^{n}\{0}, and therefore B(ξ, λ) is a homogeneous function in (ξ, λ) of degree 2m − 2μ. From this and from conditions (ii) and (iii) it follows that

|A_{2μ}(ξ)| ≥ C|ξ|^{2μ} , |B_{2m−2μ}(ξ, λ)| ≥ C(λ + |ξ|)^{2m−2μ} .  (3.6)

Multiplying these estimates, we see that A is N-elliptic with parameter in [0, ∞). □

Denote by τ_{j}(ξ′, λ) (j = 1, …, 2m) the zeros of the algebraic equation

A(ξ′, τ, λ) = 0 (ξ′ ∈ ℝ^{n−1}\{0}, λ ∈ [0, ∞)) .  (3.7)

Due to Lemma 3.2 (iii), this equation has no real roots. The number m_{+} of roots with positive imaginary part is independent of (ξ′, λ) and therefore coincides with the corresponding number for λ = 0. It is easily seen (cf. [4], Section 1.2) that in the case n > 2 the set {(ξ′, λ) : ξ′ ∈ ℝ^{n−1}\{0}, λ ∈ [0, ∞)} is connected, and therefore we have m_{+} = m. In the case n ≤ 2 the relation m_{+} = m is an additional condition which will be assumed to hold in the following. We denote the roots of A(ξ′, τ, λ) with positive imaginary part by τ_{1}(ξ′, λ), …, τ_{m}(ξ′, λ).

To investigate the elliptic pencil corresponding to A(ξ′, τ, λ) we will need an additional assumption which is closely related to the condition of regularity of degeneration in the theory of singular perturbations (cf. Vishik-Lyusternik [13], Section 1.1). To formulate this assumption we consider the auxiliary polynomial of degree 2m − 2μ given by

Q(τ) := τ^{−2μ}A(0, τ, 1) .  (3.8)

From inequality (3.2) with ξ′ = 0 and λ = 1 we obtain for τ ≠ 0 the estimate

|Q(τ)| ≥ C(|τ| + 1)^{2m−2μ}  (3.9)

with a constant independent of τ. By continuity we obtain that Q(0) ≠ 0, and thus Q(τ) has no real roots.

Definition 3.3 The polynomial A(ξ′, τ, λ) is said to degenerate regularly for λ → ∞ if the polynomial Q(τ) defined in (3.9) has exactly m−μ roots with positive imaginary part (counted according to their multiplicities).

Remark 3.4 a) Suppose that the polynomial A(ξ, λ) contains only terms of even order, i.e.

A(ξ, λ) = A_{2m}(ξ) + λ^{2}A_{2m−2}(ξ) + …
+ λ^{2m−2μ−2}A_{2μ+2}(ξ) + λ^{2m−2μ}A_{2μ}(ξ) . (3.10)

Then the polynomial Q(τ) is a polynomial of degree m−μ in the variable τ^{2} and A(ξ, λ) degenerates regularly for λ → ∞.

b) (Cf. [13], Lemma 3.4.) Assume that A(ξ, λ) is the symbol of a differential operator Ã(\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} , λ) with real coefficients. Then the polynomials of even order A_{2m−2j}(ξ) (j = 0, …, m − μ) are real and the polynomials of odd order A_{2m−2j−1}(ξ) (j = 0, …, m − μ − 1) are purely imaginary. Assume that Ã is strongly elliptic, i.e. we have

Re A(ξ, λ) ≥ C|ξ|^{2μ}(λ + |ξ|)^{2m−2μ} .  (3.11)

Then we obtain that Re A = A_{2m} + λ^{2}A_{2m−2} + … + λ^{2m−2μ}A_{2μ} satisfies (3.2), and due to part a) the polynomial Re Q(τ) has m − μ roots with positive imaginary part and m − μ roots with negative imaginary part. Since the polynomial

Q_{δ}(τ) := Re Q(τ) + δi Im Q(τ) (0 ≤ δ ≤ 1)  (3.12)

satisfies

Re Q_{δ}(τ) ≥ C(|τ| + 1)^{2m−2μ} (0 ≤ δ ≤ 1) ,  (3.13)

the number of roots of Q_{δ} in the upper half complex plane does not depend on δ ∈ [0, 1], and A(ξ, λ) degenerates regularly for λ → ∞.

Lemma 3.5 Let the polynomial A(ξ, λ) in (3.1) be N-elliptic with parameter in [0, ∞) and assume that A degenerates regularly for λ → ∞. 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. (3.8)) with positive imaginary part. Then

τ_{j}(ξ′, λ) = λτ_{j}^{1} + τ̃_{j}^{1}(ξ′, λ) (j = μ + 1, …, m) ,  (3.14)

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)  (3.15)

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

Proof. (i) We write ξ′ = ρω with |ω| = 1 and set τ̃ = \frac{τ}{ρ} , ε = \frac{ρ}{λ}. After division of A(ξ′, τ, λ) by λ^{2m−2μ}ρ^{2μ} we obtain the equation

B(ω, τ̃, ε) := A_{2μ}(ω, τ̃) + εA_{2μ+1}(ω, τ̃) + … + ε^{2m−2μ}A_{2m}(ω, τ̃) = 0 .

(3.16)

First we fix ω with |ω| = 1. Let τ̃_{j} = … = τ̃_{j+p−1} be a zero of B(ω, τ̃, 0) = A_{2μ}(ω, τ̃) of multiplicity p. Then there exists an α = α(ω) > 0 such that

\frac{1}{2πi} ∫_{|z−τ̃_{j}|=α} \frac{\frac{d}{dz} B(ω, z, ε)}{B(ω, z, ε)} dz = \frac{1}{2πi} ∫_{|z−τ̃_{j}|=α} \frac{\frac{d}{dz} B(ω, z, 0)}{B(ω, z, 0)} dz = p

(3.17) holds for all ε < ε_{0} = ε_{0}(ω). Therefore, for every ε < ε_{0} the equation (3.16) has exactly p roots in {z ∈ ℂ : |z − τ̃_{j}| < α} which we denote by τ̃_{j}(ω, ε), …, τ̃_{j+p−1}(ω, ε). Proceding in this way for all zeros of A_{2μ}(ω, τ̃), we obtain the set S(ω, ε) := {τ̃_{1}(ω, ε), …, τ̃_{μ}(ω, ε)} of zeros of B(ω, τ̃, ε).

Now we assume that the statement in (i) is false. Then there exists a sequence (ω_{n})_{n≥1} with |ω_{n}| = 1 and a constant C > 0 such that dist(S(ω_{n}), S(ω_{n}, ε_{n})) ≥ C for all n ≥ 1 where we have set ε_{n} = \frac{1}{n}. Due to compactness, we may assume that ω_{n} converges to ω_{0}. As the zeros of A_{2μ}(ω, τ̃) depend continuously on ω, we obtain for large n that

dist(S(ω_{0}), S(ω_{n}, ε_{n})) ≥ \frac{C}{2} .  (3.18)

But from the same considerations as above we see that for every sufficiently small α > 0 there exists an ε_{0} = ε_{0}(ω_{0}) and an s > 0 such that B(ω, τ̃, ε) has exactly μ roots in ⋃_{j}{z ∈ ℂ : |z − τ̃_{j}(ω_{0})| < α} for all |ω − ω_{0}| < s and 0 < ε < ε_{0}. Taking α < \frac{C}{2} , we obtain a contradiction to (3.18).

(ii) We set τ̃ = \frac{τ}{λ} and ε = \frac{|ξ′|}{λ} and obtain the equation B(ω, τ̃, ε) := A(εω, τ̃, 1) = 0 with ω := \frac{ξ′}{ε}. First we fix ω with |ω| = 1. We write

0 = B(ω, τ̃, ε) = A(0, τ̃, 1) + ∑_{k=1}^{2m} (\frac{∂}{∂ε})^{k}B(ω, τ̃, 0) \frac{ε^{k}}{k!} .  (3.19)

Let τ_{j}^{1} = … = τ_{j+p−1}^{1} be a zero of Q(τ) of multiplicity p. Then we know from the theory of algebraic functions that there exist p roots τ̃_{j}(ω, ε), …, τ̃_{j+p−1}(ω, ε) of B(ω, τ̃, ε) for which we have an expansion (Puiseux series) of the form

τ̃_{s}(ω, ε) = τ_{j}^{1} + ∑_{k=1}^{∞} c_{jk}(ω)ε^{k/p} (s = j, …, j + p − 1)  (3.20)

(cf., e.g., [7], Section 7). In formula (3.20) we have to take the p different branches of the function ε^{\frac{1}{p}} to obtain the zeros τ̃_{j}(ε), …, τ̃_{j+p−1}(ε). The series on the right-hand side is a holomorphic function in ε^{\frac{1}{p}} for |ε| ≤ ε_{1}(ω) for some ε_{1}(ω) > 0.

From the construction of the Puiseux series (cf. [7], Section 8) we know that the coefficients c_{jk}(ω) in the series (3.20) depend continuously on the coefficients of the polynomial B(ω, τ̃, ε) and therefore on ω. Thus there exists an ε_{1} > 0, independent of ω, such that the right-hand side of (3.20) is a holomorphic function in ε^{\frac{1}{p}} for |ε| ≤ ε_{1}. As the function

(τ̃_{j}(ω, ε) − τ_{j}^{1})ε^{− \frac{1}{p}} = ∑_{k=1}^{∞} c_{jk}(ω)ε^{\frac{k−1}{p}}

is continuous in ω and ε for |ω| = 1 and 0 ≤ ε ≤ ε_{0}, it is bounded by some constant K_{1}, independent of ω and ε, which finishes the proof of part (ii). □

4. Estimates for ordinary differential equations

In this section we consider the polynomial A(ξ, λ) given by (3.1) and assume that this polynomial is N-elliptic with parameter in [0, ∞) and degenerates regularly for λ → ∞. The Newton polygon corresponding to A has the shape indicated in Figure 2 with r = 2m and s = 2μ.

[figure: i; k; •; •; •; r; s; r − s; Fig. 2. The Newton polygon N_{r,s}.]

For fixed ξ′ ∈ ℝ^{n−1}, λ ∈ [0, ∞) and j = 1, …, m we consider the ordinary differential equation on the half-line

A(ξ′, D_{t}, λ) w_{j}(t) = 0  (t > 0) ,  (4.1)
D_{t}^{k−1} w_{j}(t)|_{t=0} = δ_{jk} (k = 1, …, m) ,  (4.2)
w_{j}(t) → 0  (t → +∞) .

Here D_{t} stands for −i\frac{∂}{∂t}.

Theorem 4.1 For every ξ′ ∈ ℝ^{n−1}\{0} and λ ∈ [0, ∞) the ordinary differential equation (4.1)–(4.2) has a unique solution w_{j}(ξ′, t, λ), and the estimate

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

(4.3)

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

Proof. The existence and the uniqueness of the solution follows immediately from the fact that A(ξ′, τ, λ) (considered as a polynomial in τ) has exactly m roots with positive imaginary part. Let γ(ξ′, λ) be a closed contour in the upper half of the complex plane enclosing all roots τ_{1}(ξ′, λ), …, τ_{m}(ξ′, λ) with positive imaginary part. Then w_{j}(ξ′, t, λ) is given by

w_{j}(ξ′, t, λ) = \frac{1}{2πi} ∫_{γ(ξ′,λ)} \frac{M_{j}(ξ′, τ, λ)}{A_{+}(ξ′, τ, λ)}e^{itτ} dτ  (4.4)

where

A_{+}(ξ′, τ, λ) = ∏_{k=1}^{m} (τ − τ_{k}(ξ′, λ)) =: ∑_{k=0}^{m} a_{k}(ξ′, λ)τ^{k}  (4.5)

and

M_{j}(ξ′, τ, λ) = ∑_{k=0}^{m−j} a_{k}(ξ′, λ)τ^{m−j−k} .  (4.6)

(Cf., e.g., [2], Section 1.) The coefficients are given by the formula of Vieta,

a_{k}(ξ′, λ) = ∑_{1≤l_{1}<...<l_{k}≤m}(−1)^{k}τ_{l_{1}}(ξ′, λ) · . . . · τ_{l_{k}}(ξ′, λ) .  (4.7)

From (4.4) we see, substituting τ = rτ̃, that

r^{1−j+l}(D_{t}^{l}w_{j})(rξ′, \frac{t}{r} , rλ) = D_{t}^{l}w_{j}(ξ′, t, λ) ,  (4.8)

and therefore

∥D_{t}^{l}w_{j}(ξ′, ·, λ)∥_{L_{2}(ℝ_{+})} = r^{\frac{1}{2} −j+l}∥D_{t}^{l}w_{j}(\frac{ξ′}{r} , ·, \frac{λ}{r})∥_{L_{2}(ℝ_{+})} .  (4.9)

If we set r = |ξ′| and ω′ = \frac{ξ′}{|ξ′|} we obtain

∥D_{t}^{l}w_{j}(ξ′, ·, λ)∥_{L_{2}(ℝ_{+})} = |ξ′|^{\frac{1}{2} −j+l}∥D_{t}^{l}w_{j}(ω′, ·, \frac{λ}{|ξ′|})∥_{L_{2}(ℝ_{+})} . (4.10)

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 ≤ μ , \\ C Λ^{l−μ− \frac{1}{2}} , & j ≤ μ , l > μ \\ C Λ^{μ−j+ \frac{1}{2}} , & j > μ , l ≤ μ , \\ C Λ^{l−j+ \frac{1}{2}} , & j > μ , l > μ , \end{matrix}  (4.11)

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

The boundedness for Λ ≤ 1 easily follows from the ellipticity of A(ω′, τ, Λ) and inequality (1.10).

For large Λ we write

γ(ω′, Λ) = γ^{(1)}(ω′, Λ) ∪ γ^{(2)}(ω′, Λ)

where γ^{(1)}(ω′, Λ) encloses the zeros τ_{1}(ω′, Λ), …, τ_{μ}(ω′, Λ) and γ^{(2)}(ω′, Λ) encloses the zeros τ_{μ+1}(ω′, Λ), …, τ_{m}(ω′, Λ). Here we assume that the zeros are numbered according to Lemma 3.5. According to this splitting of the contour γ, we write w_{j}(ω′, t, Λ) = w_{j}^{(1)}(ω′, t, Λ)+w_{j}^{(2)}(ω′, t, Λ) with

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

From Lemma 3.5 we know that

|τ_{j}(ω′, Λ)| ≤ C  (|ω′| = 1, Λ ≥ Λ_{0},  j = 1, …, μ)
|τ_{j}(ω′, Λ)| ≤ CΛ  (|ω′| = 1, Λ ≥ Λ_{0}, j = μ + 1, …, m) .

As A_{2μ} is elliptic we have, with the notation of Lemma 3.5, |τ_{j}(ω′, Λ)| ≥ C for j = 1, …, μ and |ω′| = 1, Λ ≥ Λ_{0}. With our additional assumption we also have

|τ_{j}(ω′, Λ)| ≥ CΛ (|ω′| = 1, Λ ≥ Λ_{0}, j = μ + 1, …, m),

as \frac{τ_{j}(ω′,Λ)}{Λ} → τ_{j}^{1} and Im τ_{j}^{1} > 0, cf. Lemma 3.5 (ii). Therefore

|A_{+}(ω′, τ, Λ)| = ∏_{k=1}^{m} |τ − τ_{k}(ω′, Λ)| ≥ \{ \begin{matrix}CΛ^{m−μ} & on γ^{(1)} , \\ CΛ^{m} & on γ^{(2)} \end{matrix}  (4.12)

(note that |τ| ≈ C on γ^{(1)} and |τ| ≈ CΛ on γ^{(2)}). Now we have to estimate |M_{j}(ω′, τ, Λ)| in (4.4). For this we use the fact that according to (4.7)

|a_{k}(ω′, Λ)| ≤ ∑_{l_{1}<...<l_{k}} |τ_{l_{1}}| · . . . · |τ_{l_{k}}| ≤ \{ \begin{matrix}CΛ^{k}, & k ≤ m − μ , \\ CΛ^{m−μ}, & k ≥ m − μ . \end{matrix} (4.13)

On γ^{(1)} we have

M_{j}(ω′, τ, Λ)| ≤ \{ \begin{matrix}CΛ^{m−μ}, & j ≤ μ , \\ CΛ^{m−j}, & j ≥ μ . \end{matrix}

As length(γ^{(1)}) ≤ C we obtain

|∫_{γ^{(1)}}(iτ)^{l} \frac{M_{j}(ω′, τ, Λ)}{A_{+}(ω′, τ, Λ)}e^{itτ} dτ| ≤ \{ \begin{matrix}C exp(−Ct) , & j ≤ μ , \\ CΛ^{μ−j} exp(−Ct), & j ≥ μ , \end{matrix} (4.14)

and therefore

∥(D_{t}^{l}w_{j}^{(1)})(ω′, ·, Λ)∥_{L_{2}(ℝ_{+})} ≤ \{ \begin{matrix}C , & j ≤ μ , \\ CΛ^{μ−j}, & j ≥ μ , \end{matrix} (|ω′| = 1, Λ ≥ Λ_{0}) .

(4.15)

For an estimation on γ^{(2)} we first remark that for every l ≥ 0 we have

|τ^{l}M_{j}(ω′, τ, Λ)| ≤ ∑_{k=0}^{m−j} |a_{k}| |τ^{m−j+l−k}| ≤ CΛ^{m−j+l} .

Therefore the inequalities

|D_{t}^{l}w_{j}^{(2)}(ω′, t, Λ)| ≤ CΛ^{l−j+1} exp(−CΛt)

and

∥D_{t}^{l}w_{j}^{(2)}(ω′, ·, Λ)∥_{L_{2}(ℝ_{+})} ≤ CΛ^{l−j+ \frac{1}{2}} (l ≥ 0)  (4.16)

hold. To find a sharper estimate in the case j ≤ μ we use the relation

τ^{l}M_{j}(ω′, τ, Λ) = τ^{l−j} ∑_{k=0}^{m−j} a_{k}(ω′, Λ)τ^{m−k}

= τ^{l−j}(A_{+}(ω′, τ, Λ) − ∑_{k=m−j+1}^{m} a_{k}(ω′, Λ)τ^{m−k})

which yields

D_{t}^{l}w_{j}^{(2)}(ω′, t, Λ) = −\frac{1}{2πi} ∫_{γ^{(2)}} \frac{∑_{k=m−j+1}^{m} a_{k}(ω′, Λ)τ^{m−k+l−j}}{A_{+}(ω′, τ, Λ)} e^{itτ} dt .

Here we used the fact that the contour γ^{(2)} does not enclose the origin, and therefore τ^{l−j}e^{itτ} is holomorphic inside γ^{(2)}.

We obtain for the case j ≤ μ and for every l ≥ 0 that

| ∑_{k=m−j+1}^{m} a_{k}τ^{m−k+l−j}| ≤ CΛ^{m−μ}Λ^{m−(m−j+1)+l−j} = CΛ^{m−μ+l−1}

and

∥D_{t}^{l}w_{j}^{(2)}(ω′, ·, Λ)∥_{L_{2}(ℝ_{+})} ≤ CΛ^{l−μ− \frac{1}{2}} (j ≤ μ, l ≥ 0)  (4.17)

in view of Remark 2.3 for, say, Λ ≥ 1. Now we compare the right-hand sides of (4.15)–(4.17) with the right-hand side of (4.11).

a) For j, l ≤ μ the norm of D_{t}^{l}w^{(1)} is O(1) and the norm of D_{t}^{l}w^{(2)} is estimated by Λ^{l−μ− \frac{1}{2}} ≤ Λ^{− \frac{1}{2}} .

b) For j ≤ μ and l > μ according to (4.17) the norm of D_{t}^{l}w^{(2)} is estimated by Λ^{l−μ− \frac{1}{2}} ≥ Λ^{\frac{1}{2}} and the norm of D_{t}^{l}w^{(1)} is estimated by a constant.

c) For j > μ and l ≤ μ according to (4.15) and (4.17) the norm of D_{t}^{l}w^{(1)} is estimated by Λ^{μ−j} and the norm of D_{t}^{l}w^{(2)} is estimated by Λ^{l−j+ \frac{1}{2}} ≤ Λ^{μ−j+ \frac{1}{2}} .

d) For j, l > μ the norm of D_{t}^{l}w^{(2)} is estimated by Λ^{l−j+ \frac{1}{2}} and the norm of D_{t}^{l}w^{(1)} is estimated by Λ^{μ−j} < Λ^{l−j+ \frac{1}{2}} .

Thus the inequality (4.11) holds, which finishes the proof of the theorem. □

5. The main results

Now we want to prove an a priori estimate for the Dirichlet boundary value problem corresponding to the elliptic pencil A(x, D, λ) defined in (1.1). First we consider model problems in ℝ^{n} and ℝ_{+}^{n}.

Let A be a polynomial of the form (3.1). As it was already mentioned at the beginning of Section 4, the Newton polygon N_{2m,2μ} of A(ξ, λ) has the form indicated in Figure 2 with r = 2m and s = 2μ. The a priori estimates which we will obtain below, however, do not use the Sobolev spaces corresponding to this Newton polygon but the “energy spaces” which are defined as the Sobolev spaces corresponding to the Newton polygon N_{m,μ}. For this Newton polygon we have

Ξ(ξ, λ) := Ξ_{N_{m,μ}}(ξ, λ) ≈ (1 + |ξ|)^{μ}(λ + |ξ|)^{m−μ} .  (5.1)

In the notation of the Introduction, we have H^{Ξ}(ℝ^{n}) = H^{(m,μ)}(ℝ^{n}). As in Section 2, we will denote by Ξ^{(−l)}(ξ, λ) the weight function corresponding to the shifted Newton polygon (with a shift of length l to the left). The space H^{(−m,−μ)}(ℝ^{n}) which appears in the Introduction is equal to the space H^{\frac{1}{Ξ}} (ℝ^{n}).

From the trace results of Theorem 2.9 we immediately obtain the continuity of the corresponding operators:

Lemma 5.1 a) The operator A(D, λ) acts continuously from H^{Ξ}(ℝ^{n}) to H^{\frac{1}{Ξ}} (ℝ^{n}).

b) The boundary operator D_{n}^{j−1} (j ≤ m) acts continuously from H^{Ξ}(ℝ^{n})

to H^{Ξ^{(−j+ \frac{1}{2} )}}(ℝ^{n−1}).

Here and in the following, the continuity of the operator means that the norm of this operator can be estimated by a constant independent of λ.

Proposition 5.2 (A priori estimate in ℝ^{n}.) Let A(ξ, λ) be N-elliptic with parameter in [0, ∞). Then for every λ_{0} > 0 the inequality

∥u∥_{Ξ,ℝ^{n}} ≤ C(∥A(D, λ)u∥_{\frac{1}{Ξ} ,ℝ^{n}} + λ^{m−μ}∥u∥_{L_{2}(ℝ^{n})})  (5.2)

holds for all λ ≥ λ_{0} with a constant C = C(λ_{0}) independent of u and λ.

Proof. By changing the constant in (3.2) we can rewrite the Nellipticity condition in the form

λ^{2m−2μ} + C_{1}^{−1} \frac{|A(ξ, λ)|^{2}}{(1 + |ξ|^{2})^{μ}(λ^{2} + |ξ|^{2})^{m−μ}}

≥ λ^{2m−2μ} + |ξ|^{4μ} (1 + |ξ|^{2})^{−μ}(λ^{2} + |ξ|^{2})^{m−μ} .

For |ξ| ≥ 1 the right-hand side can be estimated from below by

(1 + |ξ|^{2})^{μ}(λ^{2} + |ξ|^{2})^{m−μ}

For |ξ| ≤ 1 and λ ≥ λ_{0} the right-hand side can be estimated from below by

λ^{2m−2μ} = (1 + λ^{−2})^{−m+μ} (1 + λ^{2})^{m−μ}
≥ (1 + λ_{0}^{−2})^{−m+μ} 2^{−2μ} (1 + |ξ|^{2})^{μ} (λ^{2} + |ξ|^{2})^{m−μ} .

Combining these estimates we obtain for λ ≥ λ_{0}

(1 + |ξ|^{2})^{μ}(λ^{2} + |ξ|^{2})^{m−μ} ≤ C(λ_{0})(\frac{|A(ξ, λ)|^{2}}{(1 + |ξ|^{2})^{μ}(λ^{2} + |ξ|^{2})^{m−μ}} + λ^{2m−2μ}) .

Multiplying both sides by |F u(ξ)|^{2} and integrating with respect to ξ we obtain the inequality

∥u∥_{Ξ,ℝ^{n}}^{2} ≤ C(λ_{0})(∥A(D, λ)u∥_{\frac{1}{Ξ} ,ℝ^{n}}^{2} + λ^{2m−2μ} ∥u∥^{2})

equivalent to (5.2). □

Theorem 5.3 (A priori estimate in ℝ_{+}^{n}.) Let A(ξ, λ) be N-elliptic with parameter in [0, ∞) and degenerate regularly for λ → ∞. Then for every λ_{0} > 0 there exists a constant C = C(λ_{0}) such that for all λ ≥ λ_{0} and all u ∈ H^{Ξ}(ℝ_{+}^{n}) the estimate

∥u∥_{Ξ,ℝ_{+}^{n}} ≤ C(∥A(D, λ)u∥_{\frac{1}{Ξ} ,ℝ_{+}^{n}}
+ ∑_{j=1}^{m} ∥D_{n}^{j−1}u∥_{Ξ^{(−j+ \frac{1}{2} )},ℝ^{n−1}} + λ^{m−μ}∥u∥_{L_{2}(ℝ_{+}^{n})}) (5.3)

holds.

Proof. We will follow a standard plan in elliptic theory. In the first part of the proof we reduce (5.3) to the case f ≡ 0. Then using Theorem 4.1, we treat the case of the homogeneous equation.

1) Denote by E a linear operator of extension of functions defined in ℝ_{+}^{n} to functions in ℝ^{n}. If we use the well-known Hestenes construction then the operator E : L_{2}(ℝ_{+}^{n}) → L_{2}(ℝ^{n}) and its restriction E : H^{Ξ}(ℝ_{+}^{n}) → H^{Ξ}(ℝ^{n}) are bounded operators. We will denote by R the operator of restriction of functions on ℝ^{n} onto ℝ_{+}^{n}.

2) Let ψ(ξ) ∈ C^{∞}(ℝ^{n}) be a cut-off function, i.e. ψ(ξ) = 1 for |ξ| ≤ 1 and ψ(ξ) = 0 for |ξ| ≥ 2. We write

u = u_{1} + u_{2} + v = Rψ(D)Eu + R(1 − ψ(D))A^{−1}(D, λ)Ef + v (5.4)

where we have set Ef = A(D, λ)Eu.

First of all we show that u_{1} and u_{2} belong to H^{Ξ}(ℝ_{+}^{n}) and their norms in this space can be estimated by a constant times

∥f∥_{\frac{1}{Ξ} ,ℝ_{+}^{n}} + λ^{m−μ} ∥u∥_{L_{2}(ℝ_{+}^{n})} .

3) Since the operator ψ(D) is infinitely smoothing we get for λ ≥ λ_{0} that

∥u_{1}∥_{Ξ,ℝ_{+}^{n}} ≤ ∥ψ(D)Eu∥_{Ξ,ℝ^{n}} ≤ Cλ^{m−μ}∥Eu∥_{L_{2}(ℝ^{n})} ≤ C_{1}λ^{m−μ}∥u∥_{L_{2}(ℝ_{+}^{n})} .

4) Using the Fourier transform we obtain

∥u_{2}∥_{Ξ,ℝ_{+}^{n}} ≤ ∥(1 − ψ(D))A^{−1}(D, λ)Ef∥_{Ξ,ℝ^{n}}
= ∥Ξ(ξ, λ)(1 − ψ(ξ))A^{−1}(ξ, λ)(F Ef)(ξ)∥_{L_{2}(ℝ^{n})} .

Since 1 − ψ(ξ) = 0 for |ξ| ≤ 1, we obtain from the N-ellipticity condition that

Ξ(ξ, λ) |1 − ψ(ξ)| |A^{−1}(ξ, λ)| ≤ C Ξ^{−1}(ξ, λ)

and

∥u_{2}∥_{Ξ,ℝ_{+}^{n}} ≤ const ∥Ef∥_{\frac{1}{Ξ} ,ℝ^{n}} .

If the norm in H^{Ξ^{−1}}(ℝ^{n}) is defined by means of the pseudodifferential operator

((1 + |D′|^{2})^{1/2} + iD_{n})^{−μ}((λ^{2} + |D′|^{2})^{1/2} + iD_{n})^{−m+μ} ,

then according to Section 2

∥Ef∥_{\frac{1}{Ξ} ,ℝ^{n}} = ∥f∥_{\frac{1}{Ξ} ,ℝ_{+}^{n}} .

5) Now we begin the estimation of v defined in (5.4). We have v = u − u_{1} − u_{2} ∈ H^{Ξ}(ℝ_{+}^{n}) and

A(D, λ) v = 0 ,  (5.5)
D_{n}^{j−1}v(x)|_{x_{n}=0} = h_{j}(x′) ,  (5.6)

where we set h_{j}(x′) := D_{n}^{j−1}u(x′, 0) − D_{n}^{j−1}u_{1}(x′, 0) − D_{n}^{j−1}u_{2}(x′, 0). We shall prove the inequality

∥v∥_{Ξ,ℝ_{+}^{n}} ≤ const ( ∑_{j=1}^{m} ∥h_{j}∥_{Ξ^{(−j+1/2)},ℝ^{n−1}} + λ^{m−μ}∥u∥_{L_{2}(ℝ^{n})}) (5.7)

The a priori estimate (5.3) follows from this inequality because, due to Theorem 2.9,

∥D_{n}^{j−1}u_{i}∥_{Ξ^{(−j+1/2)},ℝ^{n−1}} ≤ const ∥u_{i}∥_{Ξ,ℝ_{+}^{n}} (i = 1, 2) .

The right-hand side of this inequality is already estimated by the righthand side of (5.3).

6) We define

Φ(ξ, λ) := ∑_{i,k} |ξ|^{i}λ^{k} ,  (5.8)

where the sum extends over all integer points (i, k) belonging to the side of N_{m,μ} which is not parallel to the coordinate lines. From this definition it follows that

Φ(ξ, λ) ≈ |ξ|^{μ}(λ + |ξ|)^{m−μ} .  (5.9)

and ∥v∥_{Ξ,ℝ_{+}^{n}} is equivalent to

∥v∥_{Φ,ℝ_{+}^{n}} + λ^{m−μ}∥v∥_{L_{2}(ℝ_{+}^{n})} .

The second term can be estimated by λ^{m−μ}(∥u∥_{L_{2}(ℝ_{+}^{n})} + ∥u_{1}∥_{L_{2}(ℝ_{+}^{n})} + ∥u_{2}∥_{L_{2}(ℝ_{+}^{n})}) ≤ λ^{m−μ}∥u∥_{L_{2}(ℝ_{+}^{n})} + ∥u_{1}∥_{Ξ,ℝ_{+}^{n}} + ∥u_{2}∥_{Ξ,ℝ_{+}^{n}}. Therefore, it is enough to estimate ∥v∥_{Φ,ℝ_{+}^{n}} by the right-hand side of (5.7). Repeating the argument in Section 2 (see (2.19)) we reduce our problem to the estimation of

∫_{0}^{∞} ∥(D_{n}^{l}v)(·, x_{n})∥_{Φ^{(−l)},ℝ^{n−1}}^{2} dx_{n} (l = 0, …, m)

or after the Fourier transform with respect to x′

∫_{0}^{∞} ∫_{ℝ^{n−1}} |Φ^{(−l)}(ξ, λ)(D_{n}^{l}F′v)(ξ′, x_{n})|^{2} dξ′dx_{n} (l = 0, …, m) . (5.10)

The function F′v(ξ′, x_{n}) =: w(ξ′, x_{n}) is (for almost every ξ′ ∈ ℝ^{n−1}) a solution of

A(ξ′, D_{n}, λ)w(x_{n}) = 0 ,  (5.11)
D_{n}^{j−1}w(x_{n})|_{x_{n}=0} = (F′h_{j})(ξ′) .  (5.12)

Due to Theorem 4.1, this solution is unique and given by

w(ξ′, x_{n}) = ∑_{j=1}^{m} w_{j}(ξ′, x_{n}, λ)(F′h_{j})(ξ′)  (5.13)

with w_{j}(ξ′, x_{n}, λ) being the solution of (4.1)–(4.2).

8) To obtain the estimate for w = F′v we reformulate Theorem 4.1. It follows from the definition of N_{m,μ} that

Φ^{(−r)}(ξ, λ) ≤ \{ \begin{matrix}|ξ|^{μ−r}(λ + |ξ|)^{m−μ}, & r ≤ μ, \\ (λ + |ξ|)^{m−r}, & r > μ. \end{matrix}

From this it follows that

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

Comparing the right-hand sides of these inequalities with the right-hand side of (4.3) we see that

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

From (5.13) and the last inequality it follows that

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

Integrating this inequality with respect to ξ′ we obtain the desired estimate. □

Now we consider the Dirichlet boundary value problem for differential operators with parameter acting on a smooth compact manifold M with smooth boundary Γ. In this case we can choose a finite number of coordinate systems. In each of these systems the operator is of the form (1.1). The principal part of the operator is invariantly defined at each of these systems and at every fixed point x^{0} ∈ M it is of the form

A^{(0)}(x^{0}, D, λ) = A_{2m}^{(0)}(x^{0}, D) + … + λ^{2m−2μ}A_{2μ}^{(0)}(x^{0}, D)  (5.14)

(here A_{j}^{(0)} denotes the principal part of A_{j}). We suppose that for each x^{0} ∈ \overline{M} our operator is N-elliptic with parameter. From the reason of continuity and compactness the constant C in inequality (3.2) can be chosen independent of x^{0}.

We can suppose without loss of generality that the coefficients of A(x, D, λ) are of the form

a_{αj}(x) = a_{αj} + a′_{αj}(x), a_{αj} ∈ D(ℝ^{n}) .  (5.15)

Now we fix a point x^{0} ∈ Γ and a coordinate system in the neighborhood of x^{0} such that in this system locally the boundary Γ is given by the equation x_{n} = 0. In this case we can define an analog of the polynomial (3.8):

Q(x^{0}, τ) = τ^{−2μ}A^{(0)}(x^{0}, 0, τ, 1)  (5.16)

Suppose that at a point x^{0} ∈ Γ and in a fixed coordinate system this polynomial has m−μ roots in the upper half-plane of the complex plane. It easily follows from this fact that every polynomial (5.16) corresponding to an arbitrary x^{0} ∈ Γ has the same property. In this case we say that the operator A(x, D, λ) degenerates regularly at the boundary Γ.

Lemma 5.4 For a(x) = a + a′(x) with a′ ∈ D(ℝ^{n}) and f ∈ H^{\frac{1}{Ξ}} (R^{n}) we have af ∈ H^{\frac{1}{Ξ}} (R^{n}), and the following statements hold:

a) There exists a constant C(a) depending on a but not on f or λ such that

∥af∥_{\frac{1}{Ξ} ,ℝ^{n}} ≤ C(a)∥u∥_{\frac{1}{Ξ} ,ℝ^{n}} .  (5.17)

b) There exists a constant C′(a) depending only on a such that the inequality

∥af∥_{\frac{1}{Ξ} ,ℝ^{n}} ≤ sup_{x∈ℝ^{n}} |a(x)|∥f∥_{\frac{1}{Ξ} ,ℝ^{n}} + C′(a)∥f∥_{Ψ,ℝ^{n}}  (5.18)

holds, where we have set

∥f∥_{Ψ,ℝ^{n}} := (∫ (1 + |ξ|)^{−2μ−2}(λ + |ξ|)^{−2m+2μ}|f̂(ξ)|^{2} dξ)^{\frac{1}{2}} . (5.19)

Proof. Part a) is a special case of the following more general result which is taken from [15], Section I.2.4. Let σ be a weight function which satisfies

σ(ξ)σ^{−1}(η) ≤ C(1 + |ξ − η|^{m}) .

Then we have for a′ ∈ D(ℝ^{n}) the inequality

∥a′f∥_{H^{σ}(ℝ^{n})} ≤ c(a′)∥f∥_{H^{σ}(ℝ^{n})}  (5.20)

with c(a′) := C ∫ (1 + |ξ|^{m})|(F a′)(ξ)|dξ.

Part b) can be shown by standard arguments similar to those used in [11], Section 1.7.1, and [8], Lemma 1.4.5. □

Using the above mentioned covering of M by local coordinate systems and a partion of unity subordinated to this covering we can define the spaces H^{Ξ}, H^{\frac{1}{Ξ}} and H^{Ξ^{(−j+3/2)}}. From Lemma 5.4 and the trace results for model problems in ℝ^{n} and ℝ_{+}^{n} we immediatly obtain

Lemma 5.5 Let D_{Γ}(u) := (u|_{Γ}, \frac{∂}{∂ν} u|_{Γ}, …, (\frac{∂}{∂ν})^{m−1}u|_{Γ}) be the Dirichlet boundary operator. Then

(A(x, D, λ), D_{Γ}) : H^{Ξ}(M) ⟶ H^{\frac{1}{Ξ}} (M) × ∏_{j=1}^{m} H^{Ξ^{(−j+ \frac{1}{2} )}}(Γ)

is continuous with norm bounded by a constant independent of λ.

Theorem 5.6 Let A(x, D, λ) be an operator pencil of the form (1.1), acting on the manifold M with boundary Γ. Let A be N-elliptic with parameter in [0, ∞) and assume that A degenerates regularly at the boundary Γ. 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∥_{\frac{1}{Ξ} ,M} + ∑_{j=1}^{m} ∥(\frac{∂}{∂ν} )^{j−1}u∥_{Ξ^{(−j+ \frac{1}{2} )},Γ}(5.21)
+ λ^{m−μ}∥u∥_{L_{2}(M)}) .

Proof. For the proof we use the standard technique of localization (“freezing the coefficients”). We only indicate the main steps. By means of a partition of unity it is sufficient to prove (5.21) for u ∈ H^{Ξ}(M) with small support supp u ⊂ U. In the case U ∩ Γ = ∅, we fix x_{0} ∈ U and use local coordinates. We obtain from the a priori estimate for the model problem in ℝ^{n} that

∥u∥_{Ξ,ℝ^{n}} ≤ C_{1}(∥A^{(0)}(x_{0}, D)u∥_{\frac{1}{Ξ} ,ℝ^{n}} + λ^{2m−2μ}∥u∥_{L_{2}(ℝ^{n})})
≤ C_{1}(∥A(x, D)u∥_{\frac{1}{Ξ} ,ℝ^{n}} + λ^{2m−2μ}∥u∥_{L_{2}(ℝ^{n})})
+ C_{1}∥(A(x, D) − A^{(0)}(x_{0}, D))u∥_{\frac{1}{Ξ} ,ℝ^{n}}  (5.22)

with a constant C_{1} independent of u and λ.

We fix ε > 0. From Lemma 5.4 b) we obtain if the support of u is sufficiently small that

∥(A(x, D) − A^{(0)}(x_{0}, D))u∥_{\frac{1}{Ξ} ,ℝ^{n}} ≤ ε∥u∥_{Ξ,ℝ^{n}} + C∥u∥_{Ξ^{(−1)},ℝ^{n}} . (5.23)

Here we have used that

∑_{α,k} λ^{k}∥D^{α}u∥_{\frac{1}{Ξ} ,ℝ^{n}} ≤ C∥u∥_{Ξ,ℝ^{n}}  (5.24)

and

∑_{α,k} λ^{k}∥D^{α}u∥_{Ψ,ℝ^{n}} ≤ C∥u∥_{Ξ^{(−1)},ℝ^{n}}  (5.25)

Now we use the interpolation inequality

∥u∥_{Ξ^{(−1)},ℝ^{n}} ≤ ε∥u∥_{Ξ,ℝ^{n}} + Cλ^{m−μ}∥u∥_{L_{2}(ℝ^{n})}  (5.26)

which is a consequence of the interpolation inequality for the Sobolev spaces H^{s}(ℝ^{n}) because of

∥u∥_{Ξ^{(−1)},ℝ^{n}} ≈ ∥u∥_{H^{m−1}(ℝ^{n})} + λ^{m−μ}∥u∥_{H^{μ−1}(ℝ^{n})} .  (5.27)

If we choose ε with C_{1}ε < 1 we obtain

∥u∥_{Ξ,ℝ^{n}} ≤ C(∥A(x, D, λ)u∥_{\frac{1}{Ξ} ,ℝ^{n}} + λ^{m−μ}∥u∥_{L_{2}(ℝ^{n})}) .  (5.28)

In the case U ∩ Γ ≠ ∅ we choose x_{0} ∈ U ∩ Γ, use local coordinates, and obtain in the same way as above

∥u∥_{Ξ,ℝ_{+}^{n}} ≤ C(∥A(x, D, λ)u∥_{\frac{1}{Ξ} ,ℝ_{+}^{n}} + ∑_{j=1}^{m} ∥D_{n}^{j−1}u∥_{Ξ^{(−j+ \frac{1}{2} )},ℝ^{n−1}}
+ λ^{m−μ}∥u∥_{L_{2}(ℝ^{n})}) ,  (5.29)

where we used the a priori estimate for (A^{(0)}(x_{0}, D), (D_{n}^{j−1})_{j=1}^{m}). □

Now to finish the existence theory we present the construction of the right (rough) parametrix of the Dirichlet problem.

Suppose the assumptions of Theorem 5.6 to hold. We will see below that the solution constructed above for constant coefficients is a right parametrix with respect to the Sobolev spaces defined by the Newton polygon. We will define this parametrix, as usual, with the help of local coordinates. First of all we present the construction in the case of model domains ℝ^{n} and ℝ_{+}^{n}.

Proposition 5.7 Suppose A(x, ξ, λ) satisfies the N-ellipticity condition and the coefficients are of the form (5.15). Then there exists a bounded operator

B : H^{\frac{1}{Ξ}} (ℝ^{n}) → H^{Ξ}(ℝ^{n})  (5.30)

such that

A(x, D, λ)B = I + T  (5.31)

where I denotes the identity operator in H^{\frac{1}{Ξ}} (ℝ^{n}) and

T : H^{\frac{1}{Ξ}} (ℝ^{n}) → H^{Θ}(ℝ^{n})  (5.32)

is continuous with norm bounded by a constant independent of λ. Here we posed Θ(ξ, λ) = (1 + |ξ|)/Ξ(ξ, λ).

Proof. We define B as a classical ps.d.o with symbol

B(x, ξ, λ) := ψ(ξ)\frac{1}{A_{0}(x, ξ, λ)},  (5.33)

where ψ ∈ C^{∞}(ℝ^{n}) is a cut-off function with ψ ≡ 0 for |ξ| ≤ 1 and ψ ≡ 1 for |ξ| ≥ 2.

The continuity of operator (5.30) is equivalent to the statement that the L_{2} − L_{2} norm of the operator

(1 + |D|^{2})^{\frac{μ}{2}} (λ^{2} + |D|^{2})^{\frac{m−μ}{2}} B(x, D, λ)(1 + |D|^{2})^{\frac{μ}{2}} (λ^{2} + |D|^{2})^{\frac{m−μ}{2}}

can be estimated by a constant independent of λ. Using standard results on the L_{2}-boundedness of ps.d.o. (cf. [10], Section 2.4) we have to show the inequalities

ψ(ξ)|D_{x}^{α}A_{0}^{−1}(x, ξ, λ)| ≤ C_{α}(1 + |ξ|)^{−2μ}(λ + |ξ|)^{−2m+2μ} .

For |α| = 0 this inequality directly follows from N-ellipticity with parameter, to prove it for arbitrary α we must use the chain rule.

To prove (5.31)–(5.32) we write the operator T in the form

T = T̃ + (A(x, D, λ) − A_{0}(x, D, λ))B

with T̃u = A_{0}(x, D, λ)Bu − u. Noting that

A(x, D, λ) − A_{0}(x, D, λ) : H^{Ξ}(ℝ^{n}) → H^{Θ}(ℝ^{n})

is continuous, it is sufficient to prove (5.32) with T replaced by T̃. As above, this is equivalent to the uniform L_{2} − L_{2} boundedness of

(1 + |D|^{2})^{− \frac{μ−1}{2}} (λ^{2} + |D|^{2})^{− \frac{m−μ}{2}} T̃ (1 + |D|^{2})^{\frac{μ}{2}} (λ^{2} + |D|^{2})^{\frac{m−μ}{2}} .

For this it is enough to show that the symbol T̃(x, ξ, λ) of T̃ satisfies

(1 + |ξ|)|D_{x}^{β}T̃(x, ξ, λ)| ≤ C_{β} .  (5.34)

The last inequality follows easily from the fact that for |ξ| ≥ 2 we have

T̃(x, ξ, λ) = ∑_{0<|α|≤2m} \frac{1}{α!}∂_{ξ}^{α}A_{0}(x, ξ, λ) D_{x}^{α} \frac{1}{A_{0}(x, ξ, λ)}  (5.35)

and from the estimates

|D_{x}^{β} \frac{1}{A_{0}(x, ξ, λ)}| ≤ C (Ξ_{P} (ξ, λ))^{−1} (|ξ| ≥ 2)  (5.36)

and

|D_{x}^{γ}∂_{ξ}^{α}A_{0}(x, ξ, λ)| ≤ C Ξ_{P}^{(−|α|)}(ξ, λ) (0 ≤ |α| ≤ 2m) .  (5.37)
□

Proposition 5.8 Suppose the conditions of Theorem 5.6 are satisfied and the coefficients of A(x, ξ, λ) are of the form (5.15). Then there exists a bounded operator

B : H^{\frac{1}{Ξ}} (ℝ_{+}^{n}) × ∏_{j=1}^{m} H^{Ξ^{(−j+1/2)}}(ℝ^{n−1}) → H^{Ξ}(ℝ_{+}^{n})  (5.38)

such that

(A, γ_{0}, γ_{0}D_{n}, …, γ_{0}D_{n}^{m−1})^{t}B = I + T ,  (5.39)

where I denotes the identity operator in

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

γ_{0} is the operator of taking the trace of the function at x_{n} = 0, and

T : H^{\frac{1}{Ξ}} (ℝ_{+}^{n}) × ∏_{j=1}^{m} H^{Ξ^{(−j+1/2)}}(ℝ^{n−1}) → H^{Θ}(ℝ_{+}^{n}) × ∏_{j=1}^{m} H^{Ξ^{(−j+3/2)}}(ℝ^{n−1})

(5.40)

is continuous with norm bounded by a constant independent of λ.

Proof. We set

B(f, g_{1}, …, g_{m}) := B_{0}f + ∑_{j=1}^{m} B_{j}(g_{j} − γ_{0}D_{n}^{j−1}B_{0}f) .  (5.41)

Here (compare Proposition 5.7)

B_{0}f := Rψ(D)A_{0}^{−1}(x, D, λ)Ef  (5.42)

for f ∈ H^{\frac{1}{Ξ}} (ℝ_{+}^{n}). As in (5.4) R and E are restriction and extension operators, respectively, ψ(ξ) is the cut-off function from the proof of Proposition 5.7 and B_{j} for j = 1, …, m is a ps.d.o. in ℝ^{n−1} (with x_{n} as parameter) defined by

(B_{j}g_{j})(x′, x_{n}) := ψ′(D′)w_{j}(x′, x_{n}, D′, λ)g_{j} .  (5.43)

The symbol w_{j} in (5.43) is given by (compare (4.4))

w_{j}(x′, x_{n}, ξ′, λ) := \frac{1}{2πi} ∫_{γ(ξ′,λ)} \frac{M_{j}(x′, 0, ξ′, τ, λ)}{A_{+}(x′, 0, ξ′, τ, λ)}e^{ix_{n}τ} dτ ,  (5.44)

where A_{+}(x′, 0, ξ′, τ, λ) and M_{j}(x′, 0, ξ′, τ, λ) are given by (4.5)–(4.6) with A(ξ′, λ) replaced by A^{(0)}(x′, 0, ξ′, λ). The function ψ′(ξ′) ∈ C^{∞}(ℝ^{n−1}) is defined by ψ′(ξ′) := ψ(ξ′, 0).

First of all we check that the operator (5.41) is bounded. The boundness of B_{0} follows easily from the proof of Proposition 5.7. Now we proof the continuity of operators

B_{j} : H^{Ξ^{(−j+1/2)}}(ℝ^{n−1}) → H^{Ξ}(ℝ_{+}^{n}) .  (5.45)

If the condition (5.15) is satisfied, then the operator B_{j} can be represented in the form B_{j} = B_{j}^{0} + B′_{j}, where B_{j}^{0} is a ps.d.o with symbol independent of x and the symbol of B′_{j} is independent of x, when the modulus of x is large enough.

In fact, using the norm

( ∑_{l=0}^{m} ∫_{0}^{∞} ∥D_{n}^{l}u(·, x_{n})∥_{Ξ^{(−l)},ℝ^{n−1}}^{2} dx_{n})^{1/2}  (5.46)

in H^{Ξ}(ℝ_{+}^{n}) and Theorem 4.1 we can easily prove the continuity of operator B_{j}^{0}. To prove the continuity of B′_{j} we show the estimates

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

(5.47)

The case β′ = 0 was treated in the proof of Theorem 4.1, the general case follows by the same method after differentiating in (5.44) under the integral sign.

Now we prove the continuity of the operator (5.40). The main step of the proof is reduced to showing that AB_{j} = 0, j = 1, …, m, up to a lower order operator, i.e. the operator

A(x, D, λ)B_{j} : H^{Ξ^{(−j+1/2)}}(ℝ^{n−1}) → H^{Θ}(ℝ_{+}^{n})  (5.48)

is continuous.

Before proving this statement we finish the proof of (5.39). Denote by T_{0}, T_{1}, …, T_{m} the components of operator T. Operator T_{0} maps the space on the left-hand side of (5.40) into H^{Θ}(ℝ_{+}^{n}) and T_{k}, k > 0, maps the space on the left-hand side of (5.40) into H^{Ξ^{(−k+3/2)}}(ℝ^{n−1}). We have

T_{0}{f, g_{1}, …, g_{m}} = (AB_{0}f − f) + ∑_{j=1}^{m} AB_{j}(g_{j} − γ_{0}D_{n}^{j−1}B_{0}f) .

The first term on the right-hand side belongs to H^{Θ}(ℝ_{+}^{n}) according to Proposition 5.7, the sum belongs to this space according to (5.48).

Turning to estimates of other components of T we note that according to (4.2)

γ_{0}D_{n}^{k−1}B_{j}(x′, x_{n}, D_{n}, λ)h = δ_{jk}ψ′(D′)h .

From this follows that for k ≥ 1 we have

T_{k}{f, g_{1}, …, g_{m}}
= γ_{0}D_{n}^{k−1}B_{0}f + ∑_{j=1}^{m} γ_{0}D_{n}^{k−1}B_{j}(g_{j} − γ_{0}D_{n}^{k−1}B_{0}f) − g_{k}
= (1 − ψ′(D′))(γ_{0}D_{n}^{k−1}B_{0}f − g_{k}) .  (5.49)

The function 1 − ψ′(ξ′) belongs to D(ℝ^{n−1}) and, consequently, function (5.49) belongs to H^{∞}(ℝ^{n−1}) for arbitrary {f, g_{1}, …, g_{m}} from the space on the left-hand side of (5.40).

To prove (5.48) we can suppose, without loss of generality, that the operator A is replaced by its principal part

A^{(0)}(x, D, λ) = A^{(0)}(x′, 0, D, λ) + (A^{(0)}(x, D, λ) − A^{(0)}(x′, 0, D, λ)) .

The composition of A^{(0)}(x′, 0, D, λ) and B_{j} is a ps.d.o. in ℝ^{n−1} (depending on x_{n}) with the symbol

C_{j}(x′, x_{n}, ξ′, λ)

= ψ′(ξ′) ∑_{|α′|=1}^{2m} \frac{1}{(α′)!}∂_{ξ′}^{α′}A^{(0)}(x′, 0, ξ′, D_{n}, λ)D_{x′}^{α′}w_{j}(x′, x_{n}, ξ′, λ) .

The summand corresponding to α′ = 0 is identically zero due to the definition of w_{j}. We can rewrite the symbol above in the form

∑_{l=0}^{2m−1} c_{l}(x′, ξ′, λ)D_{n}^{l}w_{j}(x, ξ′, λ),

where the coefficients c_{l}(x′, ξ′, λ) and their derivatives with respect to x′ admit the estimates

|D_{x′}^{β′}c_{l}(x′, ξ′, λ)| ≤ C Ξ_{P}^{(−l−1)}(ξ′, λ) .

Differentiating under the integral sign and repeating the argument of Theorem 4.1 we obtain

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

Remembering that Ξ_{P} (ξ, λ) ≡ Ξ^{2}(ξ, λ), we easily deduce

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

From this follows the continuity of operator (5.48) with A replaced by A^{(0)}(x′, 0, D, λ).

Applying the Lagrange formula to the coefficients of the operator A^{(0)}(x, D, λ) − A^{(0)}(x′, 0, D, λ), we can rewrite this operator in the form

∑_{l=0}^{2m} c′_{l}(x, ξ′, λ)x_{n}D_{n}^{l} ,

where

|D_{x′}^{β′}c′_{l}(x, ξ′, λ)| ≤ C Ξ(ξ′, λ) Ξ^{(−l)}(ξ′, λ) .

Using the relation

x_{n} exp(ix_{n}τ) = −i\frac{∂}{∂τ} exp(ix_{n}τ)

and integrating under the sign of the contour integral in (4.4) we obtain that

x_{n}D_{n}^{l}w_{j}(x′, x_{n}, ξ′, λ) = il D_{n}^{l−1}w_{j}(x′, x_{n}, ξ′, λ).

Now we easily come to the inequalities

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

The above estimates permit us to finish the proof of the Proposition. □

Now we return to the case of a manifold as considered in Theorem 5.6. To construct a right parametrix for (A, D_{Γ}), we will use local coordinates.

Let {O_{j}}_{j=1,...,N} be a covering of M with coordinate neighborhoods where O_{j} is homeomorphic to an open ball in ℝ^{n} for j = 1, …, N′ and homeomorphic to a semi-ball {x ∈ ℝ^{n} : |x| < r_{j}, x_{n} > 0} for j = N′ + 1, …, N. In local coordinates corresponding to the neighborhood O_{k} we obtain an operator A_{k} in ℝ^{n} for k = 1, …, N′ and in ℝ_{+}^{n} for k = N′ + 1, …, N. We may assume that the coefficients of A in these local coordinates have the form (5.15).

For k = 1, …, N′ we define a local right parametrix B_{k} to the operator A_{k} according to Proposition 5.7; for k = N′ + 1, …, N this can be done due to Proposition 5.8. We then set

B := ∑_{j=1}^{N} ψ_{k}B_{k}(ϕ_{k} ·)

where {ϕ_{k}}_{k} is a partition of unity subordinated to the covering of M and ψ_{k} has support in O_{k} and is equal to 1 in a neighborhood of supp ϕ_{k}. From the previous two propositions we obtain the following theorem.

Theorem 5.9 The operator B defined above is a right rough parametrix to (A, D_{Γ}) in the sense that

(A, D_{Γ})B = I + T ,  (5.50)

where I denotes the identity operator in H^{\frac{1}{Ξ}} (M) × ∏_{j=1}^{m} H^{Ξ^{(−j+1/2)}}(Γ) and

T : H^{\frac{1}{Ξ}} (M) × ∏_{j=1}^{m} H^{Ξ^{(−j+1/2)}}(Γ) → H^{Θ}(M) × ∏_{j=1}^{m} H^{Ξ^{(−j+3/2)}}(Γ) (5.51)

is continuous with norm bounded by a constant independent of λ.

As the norms considered in (5.51) are for every fixed λ equivalent to the corresponding norms in the standard Sobolev spaces, we obtain the compactness of T and the Fredholm property of (A, D_{Γ}). Note also that for fixed λ the operator (A, D_{Γ}) is elliptic in the usual sense. From the last remark we also see that the index of this operator is equal to zero.

Some elliptic operator pencils 39

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