Benzoni-Gavage S., Serre D. Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications

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A practical tool 135

automatically. Since the set of such points is algebraic, its equation must

divide Lop or merely a power of it, according to Hilbert’s Nullstellensatz.

Since the corresponding factor of Lop does not involve the coeﬃcients of

the boundary operator B at all, it is easily identiﬁed and can be removed

immediately. An alternative method consists in the replacement of the ﬁrst

column of M by another one. The choice of the jth column of M yields a

polynomial Lop

. These polynomials diﬀer only by these spurious factors.

Taking their g.c.d., one obtains a simpler polynomial Lop

that vanishes

everywhere ∆ does.

Points where A displays a Jordan block also form an algebraic variety,

whose equation enters automatically as a factor in Lop. This factor may

be identiﬁed since it does not involve B, and is then removed. However, it

is not removed by taking the g.c.d. Lop

, since it is present in every Lop

Example Consider the system ∂

u + A(∇

)u = 0, with d = n =2and

A(ξ):=



−ξ



The spectrum of A(ξ) consists in ±|ξ|. Hence the boundary condition at x

must be scalar (one incoming characteristics): Bu = b

+ b

.Givenan

eigenvalue µ of A(τ,η), that is a root of τ

+ η

= µ

, a typical eigenvector is

R(τ,η):=(iη − τ, µ)

. The only exception is the point given by τ = iη (a bound-

ary point) and µ = 0; near such a point, a convenient choice of an eigenvector

would be R



(τ,η):=(−µ, τ + iη)

The Lopatinski˘ı determinant is ∆(τ,η)=b

(iη − τ)+b

µ. Eliminating µ

between ∆ = 0 and τ

+ η

= µ

, we obtain the equation

(τ

+ η

)=b

(iη − τ)

Therefore,

Lop(z,η)=b

(η + z)

+ b

(η

− z

)=(η + z)(b

(η + z)+b

(η − z)).

The fact that Lop vanishes at the point z = −η, regardless of the value of



reﬂects the fact that R does not span an eigenspace at this point. A computation

with the choice R



(τ,η) would have given a similar result with the factor

z − η instead of z + η. This factor is thus irrelevant, and the vanishing of the

Lopatinski˘ı determinant must imply that of the simpler polynomial

Lop

(z,η)=b

(η + z)+b

(η − z).

This formula shows that the IBVP satisﬁes the Lopatinski˘ı condition for every



b such that b

= ±b

. What the formula hides is that the IBVP still satisﬁes

the Lopatinski˘ı condition when b

= b

, while it does not if b

= −b

.The

replacement of the half-space x

> 0 by the domain x

< 0, or the reversal of

the time arrow, exchanges the roles. Hence, it is important to keep in mind

136 Initial boundary value problem in a half-space with constant coeﬃcients

that the vanishing set of Lop

encodes not only the vanishing of the Lopatinski˘ı

determinant, but also a number of other properties.

4.6.3 A geometrical view of (UKL) condition

Denote by W (k, n) the Grassmannian manifold, consisting of the linear subspaces

of dimension k in C

. For instance, W (1,n) is just the projective space P

n−1

(C).

A correct deﬁnition is the following one. Recall that Λ

) is the set of elements

of degree k in the exterior algebra Λ(C

). Consider the subset Λ

)ofnon-

zero simple k-vectors, that is of the form X

,whereF is some linear subspace of

dimension k, in the notations of Section 4.6.1. This set is a cone, and W (k,n)isits

quotient by the relation X ∼ Y iﬀ X and Y are parallel. In other words, W (k, n)

is the projective space associated to Λ

). The Grassmannian manifolds are

compact and endowed with an analytic structure. For a theory of Grassmannian

manifolds, we refer to Harris’ book [81].

Let M be a linear subspace in C

, of dimension n − p. One easily sees that

the subset M

◦

in W (p, n) consisting of p-dimensional subspaces that meet M

non-trivially, is closed and therefore compact.

Assume now that the IBVP (4.1.1)–(4.1.3) satisﬁes the uniform Kreiss–

Lopatinski˘ı condition. Let W be the subset of W (p, n), consisting of the spaces

−

(τ,η)forη ∈ R

d−1

and Re τ>0. The (non-uniform) Lopatinski˘ı condition

tells us that W does not meet (kerB)

◦

. Uniformity obviously tells us more. If

F belongs to the closure of W, then there exists a sequence (τ

,η

), such that

−

(τ

,η

) converges towards F . Then the inequality |V |≤C|BV | passes to

the limit and therefore holds on F . This proves that F ∈ (kerB)

◦

Conversely, assume that (kerB)

◦

does not meet the closure of W.When

F ∈

W,letC

be the best constant in the inequality |V |≤C|BV | for V ∈ F .

Obviously, F → C

is continuous. Since W is compact, F → C

is bounded. In

other words, the IBVP satisﬁes (UKL) condition. We can summarize as follows

Lemma 4.12 If the IBVP (4.1.1)–(4.1.3) is hyperbolic and has the correct

number of boundary conditions, then the uniform Kreiss–Lopatinski˘ı condition is

equivalent to

(kerB)

◦

W = ∅

in the Grassmann manifold W (p, n).

Remark Because of homogeneity, we may also deﬁne the set W by

W = {E

−

(τ,η); |τ|

+ η

=1}.

The closure of W diﬀers from W itself only by limits of sequences E

−

(τ

,η

)

when Re τ

→ 0

and η

converges.

A practical tool 137

4.6.4 The Lopatinski˘ı determinant of the adjoint IBVP

Recall (Theorem 4.2) that the Kreiss–Lopatinski˘ı condition is satisﬁed by the

adjoint IBVP if and only if it is satisﬁed by the original IBVP, and similarly

for the uniform KL condition. This suggests that a relation holds between the

respective Lopatinski˘ı determinants. We state it now. For the sake of simplicity,

we restrict ourselves to the non-characteristic case.

Theorem 4.4 Assume that (L, B) is normal and the boundary is non-

characteristic. Let (τ

,η

) (with Re τ

≥ 0, η

∈ R

d−1

) be a point in the neigh-

bourhood of which a Lopatinski˘ı determinant ∆ is well-deﬁned, meaning that

−

(τ,η) is locally continuous.

Then E

∗

−

is well-deﬁned in a neighbourhood of (−¯τ

, −η

), and one may take

the function

(θ, σ) →

∆(−

θ, −σ)

as a Lopatinski˘ı determinant for the dual IBVP.

Remark Taking the complex conjugate in the formula above is useful only in

that it preserves holomorphy in θ as Re θ<0.

Proof The ﬁrst statement is a consequence of the fact that

∗

−

(−¯τ,−η)=(A

−

(τ,η))

⊥

We now turn to the construction of the adjoint Lopatinski˘ı determinant ∆

∗

(θ, σ).

Recall the deﬁnition (4.6.40), where {X

(τ,η),...,X

(τ,η)} is a regular basis of

−

(τ,η). Let us choose constant vectors X

p+1

,...,X

such that

(τ

,η

),...,X

(τ

,η

),X

p+1

,...,X

}

is a basis of C

. Then, in a neighbourhood of (τ

,η

), the matrix

X(τ,η):=(X

(τ,η),...,X

(τ,η),X

p+1

,...,X

)

is non-singular. Denote by Y

(τ,η),...,Y

(τ,η) the column vectors of Y :=

X(τ,η))

−∗

.ThenY

p+1

,...,Y

forms a regular basis of E

∗

−

(−¯τ,−η).

Let us write these matrices blockwise:

X =(X

−

),Y=(Y

−

wheretheminusblocksaren × p and the plus blocks are n × (n − p). Our

Lopatinski˘ı determinants are deﬁned by the formula

∆(τ,η)=det(BX

−

(τ,η)), ∆

∗

(−¯τ,−η)=det(CY

(τ,η))

(note that both sides of the last equality are anti-holomorphic in τ).

Finally, we recall the equality

= C

N + M

138 Initial boundary value problem in a half-space with constant coeﬃcients

Deﬁning the n × n matrices

P :=





,Q:=





we have Q

P = A

. It follows that

(QY )

∗

(PX)=Y

∗

PX = Y

∗

X = I

thus QY =(PX)

−∗

. However, we have by deﬁnition

PX =



−

··



,QY=



··

· CY



Using Schur’s formula (see Proposition 8.1.2 and Corollary 8.1.1 in [187]), one

sees that

∆

∗

(−¯τ,−η)=

∆(τ,η)

det(PX)

Since P and X are locally non-singular, the function (τ,η) → det(PX) and its

inverse are smooth . Therefore, the formula above, together with a renormaliza-

tion of, say, the vector Y

, allow us to take simply

∆

∗

(−¯τ,−η):=∆(τ,η).



CONSTRUCTION OF A SYMMETRIZER UNDER (UKL)

This chapter is devoted to the proof of the existence of a Kreiss’ symmetrizer in

the non-characteristic case. A technical, though important ingredient in the proof

is to establish the so-called ‘block structure’ property for the matrix A at points

(τ,η)whereReτ = 0 (boundary points), of glancing type. Although we focus on

problems with constant coeﬃcients in a half-space, our construction is ﬂexible

enough to handle the case where the data (normal direction to the boundary,

entries of the symbol, boundary matrix) are parametrized. This fact is crucial

in the applications to variable-coeﬃcient problems, especially non-linear ones,

and/or in general domains.

5.1 The block structure at boundary points

5.1.1 Proof of Lemma 4.5

Let us recall the terms of Lemma 4.5:

Assume that the operator L is constantly hyperbolic and the boundary {x

0} is non-characteristic. Then the map (τ,η) → E

−

(τ,η) (already deﬁned for

Re τ>0) admits a unique limit in the Grassmannian G(n, p) at every boundary

point (iρ, η) (meaning that ρ ∈ R,η∈ R

d−1

), with the exception of the origin.

Proof We ﬁrst prove that the stable spectrum of A(τ,η) admits a continuous

extension up to the boundary. Let (iρ

,η

) be a non-zero boundary point, and

let ω(τ,η) be an eigenvalue of A(τ,η)forReτ>0, depending continuously on

(τ,η). As (τ,η) tends to (iρ

,η

), the omega-limit set of ω(τ,η) is connected, by

continuity and boundedness of ω, and by connectedness of the domain. However,

this omega-limit set is contained in the spectrum of A(iρ

,η

), a discrete set.

It is therefore a singleton. This shows that ω(τ,η) has a limit as (τ,η) tends to

(iρ

,η

Denote by ˆω

,...,ˆω

those distinct eigenvalues of A(iρ

,η

) that are limits,

as τ → iρ

and Re τ>0, of stable eigenvalues of A(τ,η). Obviously, Re ˆω

≤ 0,

although eigenvalues with non-positive real part need not belong to {ˆω

,...,ˆω

}

in general.

When (τ,η) is close to (iρ

,η

)andReτ>0, E

−

(τ,η) may be split as a

direct sum of invariant subspaces F

(τ,η),...,F

(τ,η), so that the eigenvalues

of A(τ, η)onF

have the single limit ˆω

as (τ, η) tends to (iρ

,η

). Each F

inherits the analyticity of E

−

140 Construction of a symmetrizer under (UKL)

It will be suﬃcient to prove that each of these spaces F

has a limit as (τ,η)

tends to (iρ

,η

). We select an index j and denote F (τ,η)=F

(τ,η), ˆω =ˆω

for the sake of simplicity. Let

F be a cluster point of F (τ,η)as(τ, η) tends to

(iρ

,η

). It will be enough to prove the uniqueness of

F . By continuity,

F is

invariant under A(iρ

,η

), and is associated to the sole eigenvalue ˆω. Therefore,

we have

F ⊂

G,where

G denotes the generalized eigenspace of A(iρ

,η

associated to the eigenvalue ˆω.

Let us begin with the easy case, when Re ˆω<0. Classically,

G locally extends

analytically as an invariant subspace G(τ,η)ofA(τ,η). But since the correspond-

ing eigenvalues will keep a negative real part, we ﬁnd that G(τ,η) ⊂ E

−

(τ,η), so

that F (τ,η) must be equal to G(τ, η)for(τ, η) close to (iρ

,η

), and therefore

F =

G. Notice that this argument can be used to prove that every eigenvalue ω

of A(iρ

,η

) with negative real part must belong to the list ˆω

,...,ˆω

,withits

full multiplicity.

There remains the case ˆω = iµ

, with µ

∈ R. By Proposition 1.7,

we decompose the characteristic polynomial P

τ,η

of A(τ,η)asP

τ,η

(X)=

(τ,iη,X)Q(τ,iη,X)

,whereP

and Q are homogeneous polynomials with real

entries and

is a simple root of Q(·,η

,µ

(ρ

,η

,µ

) =0.

Let N ≥ 1 be the multiplicity of µ

as a root of Q(ρ

,η

, ·). The eigenvalues of

A(τ,η) that are close to iµ

are roots of Q(−iτ, η, −i·) and their multiplicities are

multiples of q.SinceF (τ, η) is the sum of some of the corresponding generalized

eigenspaces, q divides its dimension. We shall denote l := (dim F (τ,η))/q.Then

dim

F = lq.

Let O denote the open set of pairs (τ,η) ∈ C × R

d−1

for which the factors

(τ,iη,·) in (1.5.56) have simple roots, distinct for distinct indices k. Likewise,

we denote by O

when allowing complex values for both τ and η. For instance,

(i, 0) ∈Oholds. The complement of O

, being the zero set of the discriminant ∆

of Π

(τ,η,·), is an algebraic variety of complex codimension one. Therefore,

is dense and arcwise connected. Likewise, O is dense in C × R

d−1

,for

otherwise the polynomial ∆, vanishing on a non-void open set, would vanish

identically, contradicting the fact that (i, 0) ∈O. We shall admit for a moment

the following

Lemma 5.1 For every pair (τ,η) in O

, the matrix A(τ,η) is diagonalizable.

In particular, there is a neighbourhood V of (iρ

,η

), such that if Re τ>0

and (τ,η) ∈V∩O,thenF (τ, η)isthesumofl eigenspaces, all of them being

of dimension q. Therefore, the minimal polynomial of A over F has the form

(X − ω

(τ,η)). Since O is dense, we obtain by continuity that a polynomial

of degree l annihilates the restriction of A on F , for every (τ,η) ∈V with

The block structure at boundary points 141

Re τ>0. Letting (τ,η) tend towards (iρ

,η

), we see that the polynomial

(X − iµ

)

annihilates the restriction of A(iρ

,η

)on

F .

Let us now deﬁne the Jordan tower of A(iρ

,η

), associated to the eigenvalue

iµ

:= ker(A(iρ

,η

) − iµ

)

:= dim G

− dim G

k−1

Classical linear algebra tells us that g

≥···≥g

. Also, G

is simply the kernel

of ρ

+ A(η

,µ

), which has dimension q by assumption, just because ρ

has

multiplicity q and A(η

,µ

) is diagonalizable. Thus g

= q.Since

F ⊂ G

,we

have ql =dim

F ≤ dim G

= g

+ ···+ g

≤ lg

= lq. Therefore dim G

=dim

F ,

and we conclude that

F = G

, which is the uniqueness property. 

Proof of Lemma 5.1 From Theorem 1.5, the matrix

A(iρ, η)=−i(A

)

−1

(ρI

+ A(η))

is diagonalizable for real ρ and η, provided ρ>>|η|.

In a small neighbourhood of (i, 0), let ω

(τ,η) be the distinct eigenvalues of

A(τ,η), each of them being of constant multiplicity m

, and therefore holomor-

phic. The generalized eigenspaces are holomorphic too and one can choose, using

Kato’s procedure, holomorphic bases {X

,...,X

}. Given two indices j and

k ≤ m

, the holomorphic map

V (τ,η):=(A(τ,η) −ω

(τ,η))X

(τ,η)

satisﬁes

(Re τ =0, Im η =0)=⇒ (V (τ,η)=0),

according to the beginning of the proof. Hence, V vanishes identically. This shows

that A(τ,η) is diagonalizable in a neighbourhood of (i, 0).

Given a point (τ,η)inO

, there exists a path Γ in O

, connecting (i, 0) to

(τ,η). From the deﬁnition of O

, the generalized eigenspaces of A can be followed

holomorphically along Γ. Using the same argument of holomorphic continuation

as above, we see that A(τ, η) is diagonalizable. 

5.1.2 The block structure

The proof of Lemma 4.5 tells us much more than actually stated. First,

−

(iρ

,η

) is the direct sum of subspaces that we denote by E

(possibly

many) and E

(only one), each one being invariant for the matrix A(iρ

,η

−iB(ρ

,η

), with

B(ρ

,η

):=(A

)

−1

(ρ

+ A(η

)) ∈ M

(R).

We notice that the spectrum of −iB remains unchanged under the symmetry with

respect to the imaginary axis. The component E

is exactly the stable subspace

of A(iρ

,η

). It may be trivial, or equal to E

−

, or something else in between.

Obviously, Lemma 4.1 does not apply up to τ = iρ

. The component E

is an

142 Construction of a symmetrizer under (UKL)

invariant subspace on which A(iρ

,η

) has a unique eigenvalue iµ

,theµ

sbeing

pairwise distinct real numbers. Thus the direct sum of the E

s is the ‘central’

part of E

−

. We warn the reader that E

does not coincide, in general, with the

generalized eigenspace ker(A−iµ

)

(see Proposition 5.1 below), so that the

central part of E

−

(iρ

,η

) is only a subspace of the central invariant subspace

of A. What the proof above tells us is that, for each index j, the restriction of

A to E

is similar to the very regular Jordan block

J(iµ

; q, l):=







iµ

...

... 0

iµ







where l stands for the number of diagonal blocks and q stands for the size of

each one (the multiplicity of −ρ

as an eigenvalue of A(η, µ

).)

What is even more interesting is that the proof can be adapted to the

study of the generalized eigenspace ker(A−iµ

)

. As a matter of fact, this

subspace extends analytically to nearby values of (τ,η), as an invariant subspace

H(τ,η)ofA(τ, η). The latter is the sum of generalized eigenspaces associated

to the eigenvalues of A that are close to iµ

. These are precisely the roots of

Q(−iτ, η, −i·) that are close to iµ

, and their multiplicities as eigenvalues are

q times their multiplicities as roots. Thus, the dimension of H equals qN, with

the notation of the previous section. Again, A being diagonalizable for (τ,η)

in O, we see that the minimal polynomial of the restriction of A to H is a

polynomial of degree at most N for (τ,η) close to (iρ

,η

). By continuity, this

still holds at point (iρ

,η

). Hence, E

is included in ker(A(iρ

,η

) − iµ

)

.The

same convexity argument about the Jordan tower shows that the spaces G

have

dimensions kq for every k up to N . Therefore, the Jordan block of A(iρ

,η

corresponding to its eigenvalue iµ

, is exactly the very regular J(iµ

; q, N).

Let us now investigate the link between the numbers l and N. The roots

of Q(−iτ, η, ·) behave, as (ρ, η) varies near (ρ

,η

), like N th roots of unity of

the discriminant D(−iτ, η) of the polynomial (Puiseux’s theory), modulo higher-

order terms. Therefore, they form an approximately regular N-agon, centred on

the real axis. Among the N vertices, exactly l have a negative imaginary part. In

order to evaluate l,wemayﬁxη to the value η

and let τ = iρ

+ γ vary along

iρ

+ R

. Using Newton’s polygon, plus the fact that ∂Q/∂ρ(ρ

,η

,µ

) =0,we

obtain that the N roots of Q(−iτ, η

, ·)nearµ

obey

(µ − µ

)

∼ iγ



∂Q

∂ρ

∂

∂µ



(ρ

,η

,µ

)=:ciγ.

One immediately concludes that |2l − N|≤1. In other words, l = N/2inthe

even case, while l =(N ± 1)/2intheoddcase.

The block structure at boundary points 143

The above results, which we summarize in the following proposition, express

the fact that a constantly hyperbolic operator, associated to a non-characteristic

boundary, satisﬁes the so-called block structure condition, introduced by Kreiss

[103] as an assumption, and proved by M´etivier [134] in our context.

Proposition 5.1 We assume that L is constantly hyperbolic and that A

invertible. Let (iρ, η) be a boundary point ((ρ, η) is real and non-zero). Then

i) Given a purely imaginary eigenvalue iµ of A(iρ, η), the corresponding

Jordan factor is a ‘regular’ Jordan block J(iµ; q, N), where q is precisely

the multiplicity of −ρ as an eigenvalue of A(η, µ).

ii) The space E

−

(iρ, η) is the direct sum of

the stable subspace of A(iρ, η),

the subspaces E

(ρ, η):=ker(A(iρ, η) − iµ

)

for some index l

∈ [(N

−

1)/2, (N

+1)/2], where N

is as in point i)).

We complete this information with the following.

Lemma 5.2 Let the operator ∂



∂

be constantly hyperbolic. Let

(η

,µ

) ∈ R

be a non-zero point, and let −ρ

be an eigenvalue of A(η

,µ

). Obvi-

ously, −ρ

is real, and we denote its multiplicity by q. We denote by R an n × q

matrix, whose columns span the eigenspace ker(ρ

+ A(η

,µ

)), and similarly

by L a q × n matrix, whose rows span the left-eigenspace of ρ

+ A(η

,µ

L(ρ

+ A(η

,µ

)) = 0, (ρ

+ A(η

,µ

))R =0, rk R = rk L = q.

Then it holds that

R = −

dρ

dµ

(µ

)LR (5.1.1)

in M

(R) (LR is non-singular, since A(η

,µ

) is diagonalizable), where ρ(µ) is

the root of P (·,η

,µ) such that ρ(µ

)=ρ

Finally, dρ/dµ(ρ

) is non-zero if and only if the multiplicity of iµ

,asan

eigenvalue of A(iρ

,η

) (from Proposition 1.7, it is a multiple of q), equals q,

that is if N =1. In that case, dρ/dµ(ρ

) is negative if l =1, positive if l =0.

Proof As above, we denote by P = P

a factorization in which ρ

is a

simple root of Q(·,η

,µ

). Classically, there is a unique locally deﬁned analytic

function µ → ρ that solves Q(·,η

,µ)=0,withρ(µ

)=ρ

. Also, the right- and

left-eigenspaces depend analytically upon µ. This means that the bases of these

eigenspaces can be chosen analytically. In other words, R and L can be extended

analytically with the properties

L(µ)(ρ(µ)I

+ A(η

,µ)) = 0, (ρ(µ)I

+ A(η

,µ))R(µ)=0,

rk R(µ)=rkL(µ)=q.

144 Construction of a symmetrizer under (UKL)

Diﬀerentiating with respect to µ, we obtain

(ρ

+ A(η

,µ

))

dµ



dρ

dµ

+ A



R =0.

Multiplying on the left by L, we obtain relation (5.1.1).

Obviously, the multiplicity of iµ

as an eigenvalue of A(iρ

,η

) equals q times

the multiplicity of µ

as a root of Q(ρ

,η

, ·). It equals q if and only if

∂Q

∂µ

(ρ

,η

,µ

) =0,

which amounts to saying that dρ/dµ(µ

) =0.

We ﬁnally consider the case N =1.For(τ,η) close to (iρ

,µ

), A(τ,η) admits

a unique eigenvalue ω close to iµ

. It is an analytic function of (τ,η), and

∂ω

∂τ



dρ

dµ



−1

the latter being a non-zero real number. Since Re ω =0whenReτ>0, we see

that ω is a stable eigenvalue for Re τ>0, if and only if dρ/dµ(µ

) < 0. 

Important remark All the results in Sections 4.3.3 and 5.1 do not really

use the fact that (τ, η) →A(τ, η) is a polynomial function. The properties are

valid also when its coeﬃcients are rational fractions, provided that the following

assumptions hold true:

i) When τ = iρ with ρ ∈ R, the matrix B(ρ, η):=iA(iρ, η) has real entries,

ii) The rational fraction P (τ, η,ω):=det(ωI

−A(η,τ)) is homogeneous,

iii) When ω = iξ with ξ ∈ R, the roots of P (·,η,iξ) are purely imaginary,

and their multiplicities do not vary with (η, ξ) = 0 (note that, because of

Property i), the polynomial (ρ, η, ξ) → P (iρ,η,iξ) has real coeﬃcients),

iv) Given such an eigenvalue iλ

, with multiplicity m

,thekernelofξI

B(λ

,η) has dimension m

v) The boundary points (iρ, η) under consideration are not poles of A.

In practice, Properties i)andii) will be ensured by the construction of A, while

Properties iii)andiv) will express a form of constant hyperbolicity.

5.2 Construction of a Kreiss symmetrizer under (UKL)

We prove in this section the following important result.

Theorem 5.1 Let a normal hyperbolic IBVP be deﬁned by the domain

Ω={x ∈ R

; x

> 0},

with a linear ﬁrst-order operator L and a boundary matrix B, both with constant

coeﬃcients. Assume that L is constantly hyperbolic and that the boundary is