Lagrangian Solutions
to
Systems of
Real Principal Type
Dissertation
zur Erlangung des Doktorgrades
des Fachbereichs Mathematik-Informatik
der Universit¨
at Paderborn
vorgelegt von Daniel R¨
ohrig
Paderborn, Dezember 2001
Gutachter: Prof. Dr. S¨onke Hansen
Prof. Dr. Gunther Uhlmann
Prof. Dr. Hermann Sohr
Tag der Promotion: 14. Februar 2002
Contents
1. Introduction 1
2. Preliminaries 4
2.1. Systems of Pseudodifferential Operators 4
2.2. Lagrangian Distributions 8
2.3. Systems of Real-Principal Type 11
3. Lagrangian Solutions 17
3.1. Statement of the Results 17
3.2. Special Inhomogeneous Equations. 19
3.3. Proof of the Theorems 24
4. Application to the Elastodynamics Equation 26
References 34
1
1. Introduction
We consider the wave equation
v=c−2(x)∂2
tv−∆v=0,
on Rt×R3
x. This is the prototype of the class of hyperbolic operators,
which describe wave-like propagation phenomena.
To find solutions to the wave equation, one can try the classical
ansatz of geometrical optics: Consider a function
v(t, x)=a(x, ω)eiω(φ(x)−t),a(x, ω)=
∞
k=0
(iω)−kak(x).(1.1)
with amplitude a. Here the principal part a0of the amplitude should
be unequal to 0.
By inserting (1.1) into the wave equation, an elementary calcula-
tion yields the following two conditions: φneeds to solve the eikonal
equation,
c|∇φ|=1,
and a0needs to solve the transport equation,
0=2(∇φ·∇)a0+∆φa
0,
here ∇is the formal differential operator (∂x1,∂
x2,∂
x3); ∇φis the gra-
dient of λ,∇φ·∇ is the derivation in ∇φ-direction and so forth.
If these two equations are satisfied, one has that v=O(ω2−2)=
O(ω0)forω→∞.
The linear transport equation can be solved by reduction to ordinary
differential equations along rays, which are the orthogonal trajectories
of the wavefronts φ= constant. By iterating this method with special
inhomogeneous equations, one gets an asymptotic solution vsuch that
for all k∈N
v=O(ω2−k)forω→∞.(1.2)
In general, the ansatz of geometrical optics does not provide global
solutions. In so-called caustics this method breaks down, cf. for exam-
ple Duistermaat [2, section 5.2].
The ansatz (1.1) translates into special Lagrangian distributions u∈
I0(X,Λ): Here X:= Rt×R3
xand uis of the form
u(t, x)=(2π)−3/2eiω(ϕ(x)−t)a(x, ω)dω.
Λ is the Lagrangian submanifold
{(t, x, τ, ξ)∈TX\0|t=ϕ(x),τ=−ωand ξ=−τ∇φ(x)},
2
where TXis the cotangent bundle of X. The solvability condition
(1.2) translates into
u∈C∞(X,Ω1
2).(1.3)
A solution to the caustics problem was first given by Maslov [8] in
1965. His ideas were included in the development of general Lagrangian
distributions, which sense no difficulties with caustics. This is part of
the modern theory of linear partial differential equations; the most
extensive presentation of this is by H¨ormander [6]. In the set-up of
this theory, it is natural to look for solutions in the larger class of
Lagrangian distributions.
The wave equation belongs to the class of (scalar) real-principal-type
operators. The solution theory of these operators has been thoroughly
treated by H¨ormander and Duistermaat in 1972, see for example their
original work [3, section 6] or H¨ormander [6, section 26.1]. Their results
did not cover systems of equations.
In 1982 Dencker [1] generalized the real-principal-type property to
systems of pseudodifferential operators and studied their propagation of
singularities. Several important equations from physics classify as such
systems of real-principal type, for example the Maxwell-equations of
electrodynamics and the Lam´e-equations of isotropic elastodynamics.
In this thesis, we investigate Lagrangian solutions to general sys-
tems of real-principal type. In particular, we derive the generalized
transport equation for these systems, which comprises a quantitative
description of the propagation of amplitudes along rays, the bicharac-
teristic curves. We shall show the necessity and the sufficiency of the
transport equation, for Lagrangian solutions. Many of the techniques
we use come from Dencker’s paper.
This is the structure of the following sections: The preliminaries are
given in section 2: In subsections 2.1 and 2.2, we state some facts about
systems of pseudodifferential operators and Lagrangian distributions.
Subsection 2.3 deals with systems of real-principal type, according to
Dencker. We shall show that the system of isotropic elastodynamics is
of real-principal type.
The reader who is familiar with the topics of section 2 might directly
head to section 3: The statement of the main results is to be found
in subsection 3.1. Subsection 3.2 deals with special inhomogeneous
equations. The results are required for the proof of the main theorems
in subsection 3.3.
3
Finally, in section 4, we apply the results to the elastodynamics sys-
tem. We determine the transport equation and we show that it corre-
sponds to a result of Karal and Keller [7], if applied to the geometrical
optics ansatz.
I wish to thank my supervisor, Professor S¨onke Hansen, for his ex-
cellent support and guidance during the work on this thesis.
4
2. Preliminaries
The purpose of this section is to state definitions, notations and re-
sults, to make this treatise more self-contained. It is practical if the
reader has some knowledge about the modern theory of linear partial
differential operators, namely about distributions, (scalar) pseudodif-
ferential operators, Fourier integral operators and generally the meth-
ods of microlocal analysis. The most extensive presentation of this
theory is by H¨ormander [6].
Manifolds and vector bundles are always meant to be C∞.
2.1. Systems of Pseudodifferential Operators. We assume that
the pseudodifferential operators we use are properly supported and
polyhomogeneous; the latter means that their full symbol is an asymp-
totic sum of homogeneous terms.
H¨ormander [6][Definition 18.1.32] defines systems of pseudodifferen-
tial operators, that act between sections of vector bundles. We restate
his definition using frames of the bundles:
Definition 2.1. Let Eand Fbe complex vector bundles over a man-
ifold X. A pseudodifferential operator of order m, from sections of E
to sections of F, is a continuous linear map
P:C∞
0(X,E)→C∞(X, F),
that satisfies the following local condition: For every open Y⊆X,
with local frames
e1,... ,e
NE:Y→E|Yand f1,... ,f
NF:Y→F|Y,
there is an NF×NE-matrix of pseudodifferential operators Pij ∈Ψm(Y),
such that for all u∈C∞
0(Y,E), u(x)=juj(x)ej(x)
(Pu(x))i=
j
(Pijuj)(x),x∈Y. (2.1)
We shall then write P∈Ψm(X;E,F).
We call the matrix (Pij) the trivialization of P,accordingtothe
chosen frames.
Example 2.1. The special case of trivial vector bundles.
A pseudodifferential operator P∈Ψm(X;CN,CM) corresponds to
its trivialization, an M×N-matrix of operators Pij ∈ψm(X). The
image of u=(u1,... ,u
N)∈C∞
0(X,CN)isgivenby
(Pu)i=
j
Pijuj.
5
Example 2.2. Operators P∈Ψm(X;Ω1
2,Ω1
2), which act between the
half-density bundle. For a definition of the one-dimensional half-density
bundle, cf. H¨ormander [6, vol. III, page 92].
Let Y⊆Xwith local coordinates x=(x1,... ,x
n):Y→Rn.The
corresponding frame of the half-density bundle is usually denoted by
|dx|1
2:Y→Ω1
2|Y.
Now P∈Ψm(X;Ω1
2,Ω1
2) if and only if, for every such choice of
local coordinates, there exists a P∈Ψm(Y) such that for every u∈
C∞
0(Y,Ω1
2), u(x)=u(x)|dx|1
2,
Pu(x)=Pu(x)|dx|1
2.
Next, we observe how the definition of the principal symbol carries
over to systems of pseudodifferential operators:
Theorem 2.1. Let P∈Ψm(X;E,F); the principal symbol σ0(P)of
Pis invariantly defined as an element of
Sm(TX,Hom(ˆ
E, ˆ
F)).
Here ˆ
E, ˆ
Fare the vector bundles over TXwith fiber at γ∈TX
equal to the fiber of E,F at π(γ);π:TX→Xis the projection of
the cotangent bundle.
Proof. Let γ∈TXand x:= π(γ)∈X.Letv∈Ex, the fiber of E
over x; we need to define p(γ)v∈Fx.
We choose frames
e1,... ,e
NE:Y→E|Y,f
1,... ,f
NF:Y→F|Y.
in a neighbourhood Y⊆Xof xand write vin the form v=iviei(x).
Let (Pij) be the trivialization of P, according to these bases and let
pij ∈S(T
YX) be the principal symbol of any Pij ∈ψm(Y). We define
(p(γ)v)i:=
j
pij(γ)vj.(2.2)
The following calculation shows that this definition is invariant under
changes of the frames: We choose an u∈C∞
0(Y,E)withv=u(x)and
6
φ∈C∞(X) such that γ=φ(x), then
lim
t→∞ t−me−itφ(x)P(eitφu)(x)
= lim
t→∞ t−me−itφ(x)
i
(
j
Pij(eitφuj)) fi
=
i
(
j
lim
t→∞ t−me−itφ(x)Pij(eitφuj)) fi
=
i
(
j
pij(γ)vj)fi
=p(γ)v.
(2.3)
Here we used a formula for the principal symbol of the scalar oper-
ators Pij, which follows from the so-called Fundamental Asymptotic
Expansion Lemma, that is for example treated by Taylor [10, page 184
ff.].
Now the invariance of the principal symbol follows from the invari-
ance of the first term in (2.3).
We shall write Sm(TX, Hom(E,F)) instead of Sm(TX, Hom( ˆ
E, ˆ
F)),
for ease of notation.
Let (Pij) be the trivialization of P, according to an arbitrary choice
of local frames. Then equation (2.2) means that the trivialization of
σ0(P), according to the same choice of local bases, is equal to the
matrix (pij).
Operators A∈ψm(X) have an asymptotic expansion of its full sym-
bol σ(A)intheform
σ(A)∼am+am−1+am−2+... ,
with unique, i-homogeneous symbols ai=σi(A). In general, the prin-
cipal symbol amis the only one of these which is invariantly defined as
an element of Sm(TX).
Operators A∈ψm(X;Ω1
2,Ω1
2) have a unique principal symbol a∈
Sm(TX), too. In addition one gets an invariant subprincipal symbol
σs(A)∈Sm−1(TX) for them, which is, in local coordinates (x, ξ)on
TX,givenby
am−1−1
2i
n
j=1
∂xj∂ξja,
cf. H¨ormander [6, Theorem 18.1.33] or Duistermaat [2, Proposition
4.3.1].
7
In section 3 we shall calculate with the trivializations of systems of
pseudodifferential operators. Therefore, we need the following, easy
consequence of the calculus of scalar pseudodifferential operators.
Lemma 2.2. Let X⊆Rnopen. Let A=(Aij)∈ψm(X;CN,CN),
B=(Bij)∈ψn(X;CN,CN)be pseudodifferential operators with prin-
cipal symbols a=(aij),b=(bij)and matrices of subprincipal symbols
as=(σs(Aij)),bs=(σs(Bij)) respectively. Then
AB ∈ψm+n(X;CN,CN),
its principal symbol is equal to ab and its matrix of subprincipal symbols
σs(AB)=abs+asb+1
2i{a, b}.(2.4)
Here {·,·} is the Poisson bracket
{a, b}=
j
∂ξja∂
xjb−∂xja∂
ξjb.
Proof. The operator AB is given by the matrix with the entries
(AB)ij =
k
AikBkj;
every entry is an element of ψm+n(X).
The expansion-formula for the full symbol of a product of scalar
pseudodifferential operators, cf. H¨ormander [6, Theorem 18.1.8] or
Folland [4, Theorem 8.37], easily carries over to these special systems:
σ(AB)∼
|α|≥0
1
α!∂α
ξσ(A)(x, ξ)Dα
xσ(B)(x, ξ),
here the differentiations are component-wise. As a direct consequence
we get that ab is the principal symbol of AB.
8
An elementary calculation then yields equation (2.4):
abs+asb+1
2i{a, b}
=abn−1+am−1b−1
2i
n
j=1
(a(∂xj∂ξjb)+(∂xj∂ξja)b)+ 1
2i{a, b}
=abn−1+am−1b+1
i
n
j=1
∂ξja∂
xjb
−1
i
n
j=1
∂ξja∂
xjb−1
2i
n
j=1
(a(∂xj∂ξjb)+(∂xj∂ξja)b)
+1
2i
n
j=1
(∂ξja∂
xjb−∂xja∂
ξjb)
=σm+n−1(ab)
−1
2i
n
j=1
(a(∂xj∂ξjb)+(∂xj∂ξja)b+∂xja∂
ξjb+∂ξja∂
xjb)
=σm+n−1(ab)−1
2i
n
j=1
∂xj∂ξj(ab)
=σs(AB)
The formula for the principal symbol of AB above remains valid in the
general case of operators between sections of vector bundles.
2.2. Lagrangian Distributions. Lagrangian distributions are invari-
antlydefinedbyH¨ormander [6, Definition 25.1.1]:
Definition 2.2. Let Xbe a manifold, Λ ⊆TX\0 a closed, conic
Lagrangian submanifold and Ea complex vector bundle over X.The
space Im(X, Λ; E) of Lagrangian distribution sections of E,oforderm,
is the set of all u∈D(X, E), such that
L1...L
Ku∈∞
Hloc
(−m−n/4)(X,E),
for all K∈Nand all properly supported Lj∈ψ1(X;E,E), with
principal symbols σ0(Lj) vanishing on Λ.
These distributions are characterized microlocally, as oscillatory in-
tegrals, in [6, Theorem 25.1.5]:
9
Theorem 2.3. Let φ(x, θ)be a non-degenerate phase function in an
open, conic neighbourhood of (x0,θ
0)∈Rn×(RN\0), such that
(x0,θ
0)∈C={(x, θ)|φ
θ(x, θ)=0}.
We set ξ0:= φ
x(x0,θ
0).Letφparametrize the Lagrangian manifold Λ
in a neighbourhood Uof (x0,ξ
0):
Λ∩U={(x, ξ)|ξ=φ
x(x, θ)for any (x, θ)∈C}.
If a∈Sm+(n−2N)/4(Rn×RN)has support in the interior of a sufficiently
small, conic neighbourhood Γof (x0,θ
0), then the oscillatory integral
u(x)=(2π)−(n+2N)/4eiφ(x,θ)a(x, θ)dθ (2.5)
defines a distribution u∈Im(Rn,Λ).
Conversely, every Lagrangian distribution u∈Im(Rn,Λ) with WF(u)
in a small conic neighborhood of (x0,θ
0)can, modulo C∞, be written
in the form (2.5).
Here, the case N= 1 corresponds to the ansatz of geometrical optics.
The amplitude a, in the microlocal representation (2.5), leads to an
invariant definition of principal symbols; cf. H¨ormander [6, Theorem
25.1.9] or Duistermaat [2, Definition 4.1.1]:
Theorem 2.4. There exists an isomorphism
Im(X,Λ; Ω
1
2
X⊗E)/Im−1(X,Λ; Ω
1
2
X⊗E)
→Sm+n
4(Λ,M
Λ⊗Ω
1
2
Λ⊗ˆ
E)/Sm+n
4−1(Λ,M
Λ⊗Ω
1
2
Λ⊗ˆ
E),
where ˆ
Eis the lifting of the bundle Eto Λ. The image under this map
is called the principal symbol.
MΛis the Maslov bundle on Λ; cf. H¨ormander [6, Definition 21.6.5].
The principal symbol of u∈Im(X,Λ; Ω1
2⊗E)isgivenby
Λρ→ <u,e
−iψ(·,ρ)χ>,
Here χ∈C∞
0(X;Ω1
2⊗E)andψ∈C∞(X, Λ) with ψ(π(ρ),ρ)≡0and
ψ
x(π(ρ),ρ)=ρ,πis the projection of the cotangent bundle. We shall
abbreviate
Sm+n
4(Λ,M
Λ⊗Ω
1
2
Λ⊗ˆ
E)
to Sm+n
4(Λ,M ⊗Ω1
2⊗E).
The next two theorems have been derived from the calculus of Fourier
integral operators, to the special case of pseudodifferential operators
and Lagrangian distributions. Theorem 25.2.3 in H¨ormander [6] and
10
Theorem 4.2.2 in Duistermaat [2] comprise the behavior of Lagrangian
distributions, under operation with pseudodifferential operators:
Theorem 2.5. Let P∈ψm(X;Ω1
2⊗E,Ω1
2⊗F)be a pseudodifferential
operator with principal symbol p∈Sm(TX, Hom(E,F)) and let u∈
Iµ(X,Λ; Ω1
2⊗E)be a Lagrangian distribution with principal symbol
σ0(u)=w∈Sµ+n
4(Λ,M ⊗Ω1
2⊗E).Then
Pu∈Im+µ(X,Λ; Ω1
2⊗F)
and its principal symbol is
σ0(Pu)=pw.
To be strict, we had to write σ0(Pu)=idM⊗id Ω1
2⊗p|Λ(w)above,
but the given short form is common.
Theorem 25.2.4 in H¨ormander [6] comprises the next Theorem, which
is essential for the derivation of the transport equation in section 3:
Theorem 2.6. Let P∈ψm(X;Ω1
2,Ω1
2)be a pseudodifferential op-
erator with principal symbol p∈Sm(TX)and subprincipal symbol
ps∈Sm−1(TX).Letu∈Iµ(X,Λ; Ω1
2)be a Lagrangian distribution
with principal symbol w∈Sµ+n
4(Λ,M⊗Ω1
2).If
Λ⊆CharP ={γ∈TX\0|p(γ)=0},
then
Pu∈Im+µ−1(X,Λ; Ω1
2)
and its principal symbol, of this lower order, is
1
iLHpw+psw. (2.6)
Here, LHpis the Lie derivative of half densities, with respect to the
vector field Hpon Λ: Let a∈C∞(TX, Ω1
2) be a section of the half
density bundle. Let (x, ξ)=((x,x
),(ξ,ξ)) : U→R2nbe local
coordinates in TXsuch that (x,ξ):U∩Λ→Rnare arbitrary local
coordinates in Λ. Then a|U=a|dxdξ|1
2, with a suitable trivialization
a∈C∞(U), and
LHp(a|dxdξ|1
2)=(Hpa+1
2div (Hp)a)|dxdξ|1
2.
For the invariance of this definition under changes of coordinates cf.
H¨ormander [6, vol. IV, page 22].
The notation of the principal symbol in (2.6) is abbreviated; to be
precise, one would write id MΛ⊗LHp|Λ(w) instead of LHpw.
11
2.3. Systems of Real-Principal Type. Let Xbe an n-dimensional
manifold. A scalar pseudodifferential operator Qon Xis of real-
principal type, if its principal symbol qis real and, in canonical local
coordinates (x, ξ)onTX, its Hamilton field
Hq=
n
j=1
∂ξjq∂
xj−∂xjq∂
ξj
is never proportional to the radial vector
n
j=1
ξj∂ξj,
on the characteristic set {q(x, ξ)=0,ξ=0}. The last condition on q
implies that dq =0,ifq=0.
The most important special case of scalar real-principal-type opera-
tors are those, whose principal symbol qsatisfies q
ξ=0on{q=0}.
Dencker [1] expanded the real-principal-type property to systems
P∈ψm(X;CN,CN) and noted that his results immediately carry over
to the somewhat more general case of operators P∈ψm(X;E,F), with
complex vector bundles E,Fover X. We formulate things directly in
the latter case.
Definition 2.3. Let Eand Fbe complex vector bundles over X.An
operator P∈ψm(X;E,F) with homogeneous principal symbol
p∈Sm(TX,Hom(E,F)),
is of real-principal type at γ∈TX\0 if it satisfies the following two
conditions:
1. For l∈Narbitrarily, there exist symbols ˜p∈Sl(TX,Hom(F,E))
and q∈Sl+m(TX), qof real-principal type, such that, in a neigh-
bourhood Uof γ,
˜pp =qI,
where Iis pointwise the identity on the fibers.
2. The conic, closed characteristic set
Char P={γ∈TX\0|det p(γ)=0}
is, locally in U,equalto
{γ∈TX\0|q(γ)=0}.
We say that Pis of real-principal type in Ω ⊆TX\0, if it is so at
every γ∈Ω.
12
The condition on Char Pimplies that it is locally a hypersurface
with non-radial Hamilton field.
Since pis homogeneous, ˜pand qcan be chosen homogeneous, too.
Therefore the set where Pis of real-principal type is conical and open
in TX\0.
If Pis elliptic, i.e. det p= 0, it is trivially of real-principal type;
take ˜p=qp−1. Thus, in general, we only have to check the existence
of ˜p,qmicrolocally on Char P.
The condition ˜pp =qI is equivalent to p˜p=qI: Thisistrivialif
q= 0, since then ˜p=qp−1holds, and follows from continuity in the
case q= 0, because dq =0.
Dencker [1, Proposition 3.2] gives a more geometric characterization
of real-principal-type operators, which is independent from the choice
of symbols ˜pand q:
Theorem 2.7. A pseudodifferential operator P∈Ψm(X;E,F)is of
real-principal type at γ∈Char Pif and only if the following two con-
ditions are satisfied in a neighborhood of γ:
1. Char P is a hypersurface with non-radial Hamilton field; the di-
mension of Ker p is constant on Char P.
2. Let πbe the quotient bundle-mapping
π:F→F/Im p=Coker p
and ρ∈N(Char P), the normal bundle of Char P.Then
π∂
ρp|Ker p:Ker p→Coker p
is a bijection, on Char P.
Proof. The proof is rather technical. We first show the necessity of the
two conditions:
Let ˜p∈Sl(TX, Hom(F, E)) and q∈Sl+m(TX), qof real-principal
type, be symbols with
Char P={q=0}
and
˜pp =qI (2.7)
in an open neighbourhood Uof γ. We can assume that Uis sufficiently
small, in order to get a chart
(x, ξ)=(x1,... ,x
n,ξ
1,... ,ξ
n):TU→R2n.
with q(x, ξ)=ξn. We calculate in these local coordinates, on
Char P={q(x, ξ)=ξn=0}.
13
From (2.7) we get that Im p⊆Ker ˜pand
Rank (p)+Rank(˜p)≤N. (2.8)
We differentiate (2.7) in the ξn-direction:
(∂ξn˜p)p+˜p(∂ξnp)=I. (2.9)
It follows that
Rank (p)+Rank(˜p)≥N.
Together with (2.8), we have that
Rank (p)+Rank(˜p)=Nand Im p=Ker˜p.
The rank of a symbol is lower semi-continuous, so
Rank p=N−Rank ˜p
is continuous and integer valued. Thus the dimension of Ker pis locally
constant.
By equation (2.9)
˜p∂
ξnp|Ker p=Iand consequently ˜p∂
ξnp(Ker p)=Kerp;
the equality Ker ˜p=Imptherefore yields ∂ξnp(Ker p)∩Im p={0}and
Dim (Ker p)=Dim(π∂
ξnp(Ker p)) ≤Dim (Coker p).(2.10)
Together with the fact that
Dim (Coker p)=Dim(F/Im p)=N−Dim (Im p)=Dim(Kerp),
we get equality in (2.10).
To prove the sufficiency of conditions 1 and 2, we can choose a symbol
q∈Sl+m(TX) of real-principal type, such that, in an open neighbour-
hood Uof γ,wehave
Char P={q=0}.
Again, we can assume that Uis sufficiently small, in order to get a
chart
(x, ξ):U→R2n,
where ξ=(ξ,ξ
n)withq(x, ξ)=ξn. We calculate in these local coor-
dinates and abbreviate (x, (ξ,0)) to (x, ξ).
Taylor-expansion yields
p(x, ξ)=p(x, ξ)+ξn∂ξnp(x, ξ)+O(ξ2
n).(2.11)
Condition 2 means that
π∂
ξnp|Ker p:Kerp→Coker P
14
is invertible, on Char P={(x, ξ)}. Therefore we can define
˜p0(x, ξ):=[(π∂
ξnp|Ker P)−1π](x, ξ):F→Ker p.
Then
˜p0(x, ξ)p(x, ξ)≡0,
because πp≡0, and
˜p0(x, ξ)∂ξnp(x, ξ)v=vfor v∈Ker p(x, ξ).
The last equation implies that
[I−˜p0∂ξnp](x, ξ)v=0ifv∈Ker p
v−˜p0(x, ξ)∂ξnp(x, ξ)vif v∈ Ker p
and we can choose a ˜p1, defined in Char P, such that
˜p1p=I−˜p0∂ξnp.
Equation (2.11) yields
[˜p0(x, ξ)+ξn˜p1(x, ξ)] p(x, ξ)
=˜p0(x, ξ)p(x, ξ)+ξnE(x, ξ)
=ξnE(x, ξ)
=q(x, ξ)E(x, ξ),
with
E(x, ξ)=˜p0∂ξnp+˜p0O(ξn)+˜p1p+ξn˜p1∂ξnp+˜p1O(ξ2
n)
which is equal to ˜p0∂ξnp+˜p1p=Iin Char P,soEis elliptic in a
neighbourhood. Without restriction, Eis elliptic in U. Define
˜p(x, ξ):=E−1(x, ξ)(˜p0(x, ξ)+q(x, ξ)˜p1(x, ξ)),
then ˜pp =qI.
The following, easy consequence shows the connection between kernel
and image of ˜pand p.
Corollary 2.8. Let P∈Ψm(X;E,F)be of real-principal type, with
homogeneous principal symbol
p∈Sm(TX,Hom(E,F)).
Let
˜p∈Sl(TX,Hom(F, E))
and
q∈Sl+m(TX),
15
of real-principal type, be homogeneous symbols with
˜pp =qI and Char P={q=0},
in a conical, open set Γ⊆TX\0.Then
Ker ˜p=Im pand Ker p=Im ˜pin Char P∩Γ.(2.12)
Proof. Let γ∈Char P∩Γ. The equality
Ker ˜p(γ)=Imp(γ)
was shown in the proof of the necessity in Theorem (2.7). The symme-
try
˜pp =qI =p˜p
then yields the second equality.
We turn towards some examples:
Example 2.3. Let P∈Ψm(X;CN,CN) be an operator, with principal
symbol
p=qIK0
0IN−K,
where 0 ≤K≤Nand qis an arbitrary real-principal-type symbol. P
is of real-principal type in TX\0; take
˜p=IK0
0qIN−K.
Every system of real-principal type can microlocally be transformed
to this form, by multiplication with elliptic systems, cf. Dencker [1,
page 359].
Dencker [1] also shows that Maxwell’s equations correspond to a
system of real-principal type. Another important example is the Lam´e-
equation of isotropic elastodynamics; this is used by Rachele [9] and,
in the more general case with residual stress, by Hansen and Uhlmann
[5]:
Example 2.4. Let Ω ⊆R3be a bounded domain with smooth bound-
ary ∂Ω, let 0 <ρ∈C∞(Ω); we consider an elastic medium with density
ρin Ω. The linear differential operator
L:C∞(Rt×Ω,C3)→C∞(Rt×Ω,C3)
of isotropic elastodynamics is given by
Lv =−ρ∂2
tv+(λ+µ)∇(∇·v)+µ∇2v+(∇·v)(∇λ)
+(∇µ)×(∇×v)+2(∇µ·∇)v,
16
with Lam´e parameters λ, µ ∈C∞(Ω), λ>0; these parameters repre-
sent the elasticity of the medium.
The displacement of the medium is a time-dependent vector field
v(t, ·)onΩ. Small displacements satisfy, in a source-free medium, the
homogeneous equation
Lv ≡0.
We want to determine the full symbol of L.Leta∈C3;anelemen-
tary calculation shows that
e−i(τt+ξx)L(ei(τt+ξx)a)
=ρτ2a−(λ+µ)(a·ξ)ξ−µ|ξ|2a
+i(ξ·a)∇λ+i(∇µ·a)ξ+i(∇µ·ξ)a.
Therefore the full symbol of Lis
σ(L)=l+l1
with principal symbol
l=(ρτ2−µ|ξ|2)I−(λ+µ)ξ⊗ξ
and
l1=i(∇µ·ξ)I+i∇λ⊗ξ+iξ⊗∇µ.
The elastodynamics operator is of real-principal type: We define two
scalar real-principal type symbols qp,q
sby
qs(t, x, τ, ξ):=ρ(x)τ2−µ(x)|ξ|2
qp(t, x, τ, ξ):=ρ(x)τ2−(λ(x)+2µ(x))|ξ|2.
Further let
π:= ξ⊗ξ
|ξ|2,πa=ξ·a
|ξ|2ξ
be the orthogonal projection in ξ-direction, then
l=qs(I−π)+qpπ.
If we take
˜
l:= qp(I−π)+qsπ,
we get
˜
ll =qsqpI.
So we can choose q:= qsqp.
17
3. Lagrangian Solutions
In this section, let Xbe an n-dimensional manifold and E,Fbe
complex, N-dimensional vector bundles over X.Let
P∈Ψm(X;Ω1
2⊗E,Ω1
2⊗F)
be a pseudodifferential operator of real-principal type, with principal
symbol
p∈Sm(TX,Hom(E,F)).
3.1. Statement of the Results. Let Λ ⊆TX\0 be a closed, conic,
Lagrangian submanifold. We are looking for Lagrangian distributions
u∈Iµ(X,Λ; Ω1
2⊗E),
with non-zero principal symbol
w∈Sµ+n
4(Λ,M⊗Ω1
2⊗E),
that solve the homogeneous equation
Pu∈C∞(X,Ω1
2⊗F)orPu≡0modI−∞(X,Λ; Ω1
2⊗F).
We always factor out a half-density bundle here, because this is appro-
priate for the symbol calculus of Lagrangian distributions, cf. Theorem
2.4.
In particular, we demand that usolves the equation to the highest
order, Pu ≡0modIm+µ−1, which means that the principal symbol
σ0(Pu)=pw vanishes, cf. Theorem 2.5. That implies the condition
Λ⊆Char P={γ∈TX\0|det p(γ)=0}and w∈Ker p,
which generalizes the eikonal equation in the ansatz of geometrical
optics.
Next, we want to declare the generalized transport equation for P.
We express it microlocally, by using the real-principal-type property:
Let γ∈Λ arbitrarily. Then there exist homogeneous symbols
˜p=˜pγ∈Sl(TX,Hom(F,E))
and
q=qγ∈Sl+m(TX)
such that
˜pp =qI,
in a conical neighbourhood Γ of γ. By choosing local frames of the
involved bundles, over a coordinate neighbourhood of γ, we can trivi-
alize the operator Pand the symbols ˜p,p,qand w.Letps:= σs(P)be
18
the subprincipal-symbol matrix of P, according to such a trivialization.
We define a linear partial differential operator
T=Tp,˜p,q :Sµ+n
4(X,Λ; Ω1
2⊗E)→Sl+m+µ−1+ n
4(X,Λ; Ω1
2⊗E)
microlocally by
Tw =1
iHqw+1
2idiv(Hq)w+1
2i{˜p, p}w+˜ppsw. (3.1)
Definition 3.1. Let rbe the trivialization of a symbol in
Sl+m+µ−1+ n
4(Λ,M⊗Ω1
2⊗E).
We say that wsatisfies the inhomogeneous, microlocal transport equa-
tion in γ∈Λ, with respect to the right side
r∈Sm+µ−1+ n
4(X,Λ; Ω1
2⊗E),
if, for every choice of ˜p=˜pγand q=qγand every choice of trivi-
alizations, over a coordinate neighbourhood of γ, there exists a conic
neighbourhood Γ1⊆Γofγ, such that wsolves the linear pde
Tw =˜pr in Γ1.
We say that wsatisfies the homogeneous, microlocal transport equa-
tion in γ, if it satisfies the inhomogeneous, microlocal transport equa-
tion in γ, with respect to the right side r=0.
Let M⊆Λ. We say that wsatisfies the homogeneous or inhomoge-
neous, microlocal transport equation in M,ifwsatisfies the respective
equation in every γ∈M.
Remark 3.1.If e∈S·(X,Λ) is elliptic in Γ, then
Tp,e˜p,eq =eT
p,˜p,q.
That means, the transport equation is invariant under changes of the
choice of functions ˜pand q.
Now we are able to state the main results. The first theorem shows
that the principal symbol of umust necessarily satisfy the transport
equation:
Theorem 3.1. Let Λ⊆Char Pbe a closed, conic, Lagrangian sub-
manifold. Let u∈Iµ(X,Λ; Ω1
2⊗E)be a Lagrangian distribution, with
principal symbol w∈Ker p, that solves
Pu≡0modI−∞(X, Λ; Ω1
2⊗F).
Then, wsatisfies the homogeneous, microlocal transport equation in Λ.
Conversely, the second theorem shows the sufficiency of the transport
equation, if Λ is the bicharacteristic flow-out of a suitable submanifold:
19
Theorem 3.2. Let Λ⊆Char Pbe a closed, conic, Lagrangian sub-
manifold. Let Λ0⊆Λbe a conic submanifold of codimension 1,such
that any bicharacteristic curve in Λintersects Λ0transversal and ex-
actly once. Let
w∈Sµ+n
4(Λ,M⊗Ω1
2⊗E),
be a homogeneous symbol that maps into Ker pand satisfies the ho-
mogeneous, microlocal transport equation in Λ. Then there exists a
Lagrangian distribution u∈Iµ(X, Λ; Ω1
2⊗E), with principal symbol
w, that solves
Pu≡0modI−∞(X, Λ; Ω1
2⊗F).
In particular, one can always find a non-trivial Lagrangian solution
u∈Iµ(X,Λ; Ω1
2⊗E)to this equation.
The additional condition on Λ is used to assure the global solvabil-
ity of the transport equation. For that purpose, one can start with
arbitrary, homogeneous values on Λ0.
The proof of Theorems 3.1 and 3.2 is given in section 3.3. We prepare
these proofs with several auxiliary results in section 3.2.
3.2. Special Inhomogeneous Equations. Let γ∈Λ arbitrarily.
Let ˜p=˜pγ∈Sl(TX,Hom(F,E)) and q=qγ∈Sl+m(TX)beho-
mogeneous symbols, such that ˜pp =qI in a conical neighbourhood Γ
of γ. We choose an operator
˜
P=˜
Pγ∈Ψl(X;Ω1
2⊗F,Ω1
2⊗E)
with principal symbol ˜pand define
Q:= ˜
PP ∈Ψl+m(X;Ω1
2⊗E,Ω1
2⊗E).
Let Dq∈ψl+m(X;Ω1
2,Ω1
2) be a scalar pseudodifferential operator with
principal symbol qand vanishing subprincipal symbol; let
DqI∈Ψl+m(X;Ω1
2⊗CN,Ω1
2⊗CN)
be the diagonal operator which is defined by operating with Dqon the
Nhalf-density components of any
v∈C∞
0(X,Ω1
2⊗CN)=(C∞
0(X,Ω1
2))N.
We transform Qinto the diagonal operator DqI:
Lemma 3.3. There exists a pseudodifferential operator
B∈ψ0(X;Ω1
2⊗CN,Ω1
2⊗E)
20
such that
QB ≡BD
qImod ψl+m−2(X;Ω1
2⊗CN,Ω1
2⊗E),(3.2)
and Bis elliptic, in Γ1∩Char P,whereΓ1⊆Γis a conical neighbour-
hood of γ.
Proof. We translate (3.2) into an equation for the homogeneous prin-
cipal symbol
b∈S0(TX,Hom(CN,E))
of B. We calculate in Γ. The principal symbols of QB and of BDqI
are both equal to qb; therefore equation (3.2) is equivalent to
σ0(QB −BD
qI)=0; (3.3)
the left side of this equation is meant to be the principal symbol of
QB −BD
qI, as operator of order l+m−1.
We evaluate this equation locally: Let U⊆TXbe an open coordi-
nate neighbourhood of γ, with local frames of Ω1
2and Eover U.Then
we calculate with the corresponding trivializations of the operators und
symbols. Lemma 2.2 yields that (3.3) is equivalent to
0=σs(QB)−σs(BDqI)
=qσ
s(B)+σs(Q)b+1
2i{qI,b}−(bσs(DqI)+qσ
s(B)+ 1
2i{b, qI})
=σs(Q)b+1
i{qI,b}
=σs(Q)b+1
iHqb.
Here we used the fact that σs(DqI)=0.
Therefore (3.2) is equivalent to the following linear, first-order pde,
in a conical neighbourhood of γ:
Hqb=1
iσs(Q)b. (3.4)
We get a 0-homogeneous, elliptic C∞solution bto this equation
in the intersection of Char qwith a conical neighbourhood of γ,by
locally solving linear, first-order ordinary differential equations along
the bicharacteristic curves. For that purpose, one can start with arbi-
trary, elliptic 0-homogeneous values on a suitable, conical hypersurface
transversal to Hq.
Transformation (3.2) enables us to deduce the principal symbol of Q
applied to a Lagrangian distribution. Later, this provides the transport
equation.
21
Lemma 3.4. Let Λ⊆Char Pbe a closed, conic, Lagrangian subman-
ifold. Let u∈Iµ(X, Λ; Ω1
2⊗E)be a Lagrangian distribution with
principal symbol w∈Ker p.Then
Qu ∈Il+m+µ−1(X,Λ; Ω1
2⊗E).
and, after choosing local frames of the bundles over a coordinate neigh-
bourhood of γ, the trivialization of its principal symbol is equal to
Tw =1
iHqw+1
2idiv (Hq)w+˜ppsw+1
2i{˜p, p}w,
in the intersection of Char Pwith a conical neighbourhood of γ.
Proof. Let Bbe the transformation operator of Lemma 3.3, B−1its
microlocal parametrix and b,b−1their principal symbols accordingly.
Without restriction we can assume that (3.2) is valid in Char P∩Γ.
Then the principal symbol of Qu is
σ0(Qu)=σ0(QB (B−1u))
=σ0(B(DqI)(B−1u))
=bσ0((DqI)(B−1u))
=1
ibLHq(b−1w),
For the last equation, notice that we’ve got the scalar operator Dq
acting on the Ncomponents of
B−1u∈Iµ(X,Λ; Ω1
2⊗CN)=(Iµ(X, Λ; Ω1
2))N.
In this situation, we can apply Theorem 2.6 to get a formula for the
required principal symbol of one order lower; take into account that
Dqhas vanishing subprincipal symbol.
We choose local frames of the involved bundles, over a coordinate
neighbourhood U⊆TXof γ, and calculate with the corresponding
trivializations of the symbols in a conical, open neighbourhood Γ2⊆Γ
of γ:
We eliminate band b−1from the last term. Observe that
0=Hq(bb
−1)=Hqbb
−1+bH
qb−1.
Together with equation (3.4) we get
1
ibH
qb−1w=iHqbb
−1w=σs(Q)w=˜ppsw+1
2i{˜p, p}w;
22
the last equation is Lemma 2.2 and w∈Ker p. Therefore
σ0(Qu)=1
ib(Hq(b−1w)+1
2div (Hq)b−1w)
=1
ib(Hqb−1w+b−1Hqw)+ 1
2idiv (Hq)w)
=1
iHqw+1
2idiv (Hq)w+˜ppsw+1
2i{˜p, p}w.
The transport equation has non-zero solutions in the Lagrangian
manifold Λ, if Λ is the bicharacteristic flow-out of a suitable submani-
fold:
Lemma 3.5. Let f∈Im+µ−1(X,Λ; Ω1
2⊗F).LetΛ⊆Char Pbe a
closed, conic, Lagrangian submanifold. Let Λ0⊆Λbe a conic sub-
manifold of codimension 1, such that any bicharacteristic curve in Λ
intersects Λ0transversal and exactly once. Then there exists a non-zero
symbol
w∈Sµ+n
4(Λ,M⊗Ω1
2⊗E)
that maps into Ker pand solves the inhomogeneous, microlocal trans-
port equation in Λ, with respect to the right side σ0(f).
Proof. We get a µ-homogeneous C∞solution wof the transport equa-
tion by solving linear, first-order ordinary differential equations along
the bicharacteristic curves. For that purpose, we can start with arbi-
trary µ-homogeneous values in Ker p, on the conic submanifold Λ0⊆Λ.
To verify that this method yields a solution wwhich maps into Ker p,
let γ=γ(t) be a bicharacteristic curve of Hqin Char Pand assume
that w∈Ker pat γ0=γ(t0)∈Λ0.Weshallshowthatwmaps into
Ker p,onallofγ:
0=qσ
0(f)=p˜pσ0(f)=pTw
=1
ipHqw+1
2idiv (Hq)(pw)+qpsw+1
2ip{˜p, p}w
=1
ipHqw+1
2idiv (Hq)(pw)+ 1
2ip{˜p, p}w.
Dencker [1, page 366] shows that
p{˜p, p}=2Hqp+{˜p, p}p,
which yields
0=1
iHq(pw)+ 1
2idiv (Hq)(pw)+ 1
2i{˜p, p}(pw).
23
This means that pw solves a first-order ordinary differential equation
along γwith initial value pw(γ0) = 0; by uniqueness we get that pw ≡0
on γ.
The next result is an easy consequence of Lemma 3.4. It implies
Theorem 3.1 in the special case f≡0.
Theorem 3.6. Let Λ⊆Char Pbe a closed, conic, Lagrangian sub-
manifold. Let f∈Im+µ−1(X, Λ; Ω1
2⊗F)and u∈Iµ(X,Λ; Ω1
2⊗E)be
Lagrangian distributions, the latter with principal symbol w∈Ker p.If
usolves
Pu≡fmod Im+µ−2(X,Λ; Ω1
2⊗F) (3.5)
then wsatisfies the inhomogeneous, microlocal transport equation in Λ,
withrespecttotherightsideσ0(f).
Proof. By applying ˜
Pto both sides of equation (3.5) we get
Qu ≡˜
Pf mod Il+m+µ−2(X,Λ; Ω1
2⊗E).
Therefore
σ0(Qu)=σ0(˜
Pf)=˜pσ0(f)
and Lemma 3.4 yields the claim.
We will iterate the following converse result, in the proof of Theorem
3.2.
Theorem 3.7. Let Λ⊆Char Pbe a closed, conic, Lagrangian sub-
manifold. Let f∈Im+µ−1(X, Λ; Ω1
2⊗F)be a Lagrangian distribution
and
w∈Sµ+n
4(Λ,M⊗Ω1
2⊗E)
a homogeneous symbol that maps into Ker pand satisfies the inhomo-
geneous, microlocal transport equation in Λ, with respect to the right
side σ0(f). Then there exists a Lagrangian distribution
u∈Iµ(X,Λ; Ω1
2⊗E)
with principal symbol wthat solves
Pu≡fmod Im+µ−2(X,Λ; Ω1
2⊗F).(3.6)
Proof. Let u∈Iµ(X, Λ; Ω1
2⊗E) be a Lagrangian distribution with
principal symbol w. We shall show that an u ∈Iµ−1(X,Λ; Ω1
2⊗E)
exists, such that
P(u+u)≡fmod Im+µ−2(X,Λ; Ω1
2⊗F).
24
From the assumptions on wand Lemma 3.4, we get that microlocally
σ0(Qu)=˜pσ0(f), which is equivalent to
Qu−˜
Pf ∈Il+m+µ−2(X,Λ; Ω1
2⊗E)
⇔˜
P(Pu−f)∈Il+m+µ−2(X,Λ; Ω1
2⊗E)
⇔σ0(Pu−f)∈Ker ˜p;
here σ0is the principal-symbol mapping for Lagrangian distributions
in Im+µ−1(X,Λ; Ω1
2⊗F). Corollary 2.8 yields
Ker ˜p=Imp
microlocally in Λ, which implies that
σ0(Pu−f)∈Im p
on all of Λ. So we find an u ∈Iµ−1(X,Λ; Ω1
2⊗E)with
σ0(Pu−f)=p(−σ0(u)) = σ0(−Pu),
which implies the desired equation.
The inhomogeneous equation (3.6) is always solvable if Λ is the
bicharacteristic flow-out of a suitable submanifold:
Corollary 3.8. Let Λ⊆Char Pbe a closed, conic, Lagrangian sub-
manifold. Let Λ0⊆Λbe a conic submanifold of codimension 1,such
that any bicharacteristic curve in Λintersects Λ0transversal and ex-
actly once. Let f∈Im+ν−1(X,Λ; Ω1
2⊗F)be a Lagrangian distribution.
Then there exists a Lagrangian distribution
u∈Iν(X,Λ; Ω1
2⊗E)
that solves
Pu≡fmod Im+ν−2(X,Λ; Ω1
2⊗F).
Proof. This is Theorem 3.7 combined with Lemma 3.5.
3.3. Proof of the Theorems.
Proof of Theorem 3.1. Follows from Theorem 3.6, with f≡0.
We iterate the results of Theorem 3.7 and Corollary 3.8 in the next
proof:
Proof of Theorem 3.2. Taking f≡0, we get from Theorem 3.7 a La-
grangian distribution u1∈Iµ(X, M ⊗Ω1
2⊗E) with principal symbol
w,thatsolves
Pu1∈Im+µ−2(X,M ⊗Ω1
2⊗F).
25
We set f1:= −Pu1. An application of Corollary 3.8, with ν=µ−1,
yields an u2∈Iµ−1(X,M ⊗Ω1
2⊗E)with
P(u1+u2)=Pu2−f1∈Im+µ−3(X, M ⊗Ω1
2⊗F).
By iteration we get a sequence of such uiwith
Pk
i=1
ui∈Im+µ−(k+1)(X,M ⊗Ω1
2⊗F).
From these ui, we can compose an u∈Iµ(X,M ⊗Ω1
2⊗E), whose
full symbol is the asymptotic sum
σ(u)∼
∞
i=1
σ(ui).
Then the principal symbol σ0(u)isequaltoσ0(u1)=wand
Pu∈I−∞(X,M ⊗Ω1
2⊗F).
26
4. Application to the Elastodynamics Equation
We use the notations of example 2.4.
To apply the results of section 3, we interpret the differential operator
L:C∞(R×Ω,C3)→C∞(R×Ω,C3)
of isotropic elastodynamics,
Lv =−ρ∂2
tv+(λ+µ)∇(∇·v)+µ∇2v+(∇·v)(∇λ)
+(∇µ)×(∇×v)+2(∇µ·∇)v,
as the trivialization of an operator P∈ψ2(R×Ω; Ω1
2⊗C3,Ω1
2⊗C3).
Inexample2.4,wehaveseenthatthefullsymbolofLis
σ(L)=l+l1
with principal symbol
l=(ρτ2−µ|ξ|2)I−(λ+µ)ξ⊗ξ
and
l1=i(∇µ·ξ)I+i∇λ⊗ξ+iξ⊗∇µ.
Recall that Lis of real-principal type: If we define qs:= ρτ2−µ|ξ|2
and qp:= ρτ2−(λ+2µ)|ξ|2we get l=qs(I−π)+qpπ.Then˜
ll =qI
holds for ˜
l:= qp(I−π)+qsπand q:= qsqp.
First, we evaluate the terms of the transport equation in this special
case, namely the matrix ls=σs(L), of subprincipal symbols, the term
˜
ll
sand the Poisson-bracket {˜
l, l}:
Lemma 4.1. The subprincipal-symbol matrix lsof Lis given by
2ils=ξ⊗∇(λ−µ)−∇(λ−µ)⊗ξ. (4.1)
Proof. From the equation
∂xj∂ξj(l)=−2(∂xjµ)ξjI−∂xj(λ+µ)(ej⊗ξ+ξ⊗ej),
where ej=(δij)1≤i≤3is the j-th unit vector of the canonical basis of
R3,weget
j
∂xj∂ξj(l)=−2(∇µ·ξ)I−(∇λ+∇µ)⊗ξ−ξ⊗(∇λ+∇µ).
27
Therefore
ls=l1−1
2i
j
∂xj∂ξj(l)
=i(∇µ·ξ)I+i∇λ⊗ξ+iξ⊗∇µ
−i(∇µ·ξ)I+1
2i(∇λ+∇µ)⊗ξ+1
2iξ⊗(∇λ+∇µ)
=1
2i(ξ⊗(∇λ−∇µ)−(∇λ−∇µ)⊗ξ).
Lemma 4.2. The subprincipal-symbol term of the transport equation
satisfies
˜
llsπ=0on Char qp
and
˜
lls(I−π)=0on Char qs.
Proof. On Char qp
˜
l=qp(I−π)+qsπ=qsπ
and, with the abbreviation b:= ∇λ−∇µ, we get from Lemma 4.1 that
2i˜
llsπ=qsπ(ξ⊗b−b⊗ξ)π.
The matrix ξ⊗b−b⊗ξis skew-symmetric and
π=ξ⊗ξ
|ξ|2
is symmetric; therefore π(ξ⊗b−b⊗ξ)πis skew-symmetric, too. The
rank of the latter matrix is even and ≤1, because the rank of πis
equal to 1; therefore it has to be 0. So ˜
llsπ= 0 on Char qp.
Analogous, we get that ˜
l=qp(I−π) holds on Char qs,and
2i˜
lls(I−π)=qp(I−π)(ξ⊗b−b⊗ξ)(I−π)
=−qp(I−π)(b⊗ξ)(I−π).
The last equation implies that the rank of the skew-symmetric matrix
(I−π)(ξ⊗b−b⊗ξ)(I−π)is≤1; then it has to be 0. In consequence,
˜
lls(I−π)=0onCharqs.
Lemma 4.3. The Poisson-bracket term of the transport equation sat-
isfies
{˜
l, l}π=−2qsHqpππ+{qs,q
p}πon Char qp(4.2)
28
and
{˜
l, l}(I−π)=2qpHqsπ(I−π)+{qp,q
s}(I−π)on Char qs.(4.3)
Proof. We calculate on Char qp:
From π2=πwe get
{rI, π}={rI,π2}={rI,π}π+π{rI,π},
which implies the following two equations:
π{rI, π}={rI,π}(I−π),
(I−π){rI, π}={rI,π}π.
Then we get
{˜
l, l}={qp(I−π)+qsπ,qs(I−π)+qpπ}
={qp(I−π),q
s(I−π)}+{qp(I−π),q
pπ}
+{qsπ,qs(I−π)}+{qsπ,qpπ}.
{qp(I−π),q
s(I−π)}π=qs(I−π){qpI,I −π}π
=−qs(I−π){qpI,π}π
=−qs{qpI,π}π
=−qsHqpππ.
{qp(I−π),q
pπ}π=0,because qp=0.
{qsπ,qs(I−π)}π=qsπ{qsI,I −π}π
=−qsπ{qsI,π}π
=−qs{qsI,π}(I−π)π
=0.
{qsπ,qpπ}π={qs,q
p}π+qs{π,qpI}π
={qs,q
p}π−qsHqpππ.
Summed up this gives equation (4.2).
Equation (4.3) follows by symmetry if we exchange πwith I−πand
qpwith qs.
Now we are able to derive the transport equation:
Theorem 4.4. The transport equation of isotropic elastodynamics, for
symbols w∈Ker lon a Lagrangian manifold Λ⊆Char q,is
πHqpw+1
2div (Hqp)w=0on Λ∩Char qp,(4.4)
29
and is
(I−π)Hqsw+1
2div (Hqs)w=0on Λ∩Char qs.(4.5)
Or, in a more compact form, it is
Hqw+1
2div (Hq)w±Hqπw=0on Λ∩Char q, for q=qs
p.(4.6)
Proof. We calculate on Char qp:Therel=qs(I−π)andqs=0.
The condition w∈Ker ltherefore yields (I−π)w=0andπw =w.
Together with Lemma 4.2, we get
˜
ll
sw=˜
ll
sπw =0.
Then
(1
qs
˜
l)l=qpI
yields the transport equation
0=Hqpw+1
2div (Hqp)w+1
21
qs
˜
l,lw
=Hqpw+1
2div (Hqp)w+1
2qs
{˜
l, l}w−1
2qs
{qs,q
p}w.
The last equation follows from a formula in Dencker [1, page 366]:
{f˜
l, l}=f{˜
l, l}+{f,q}Ion Char q.
Now we apply equation (4.2) an get:
0=Hqpw+1
2div (Hqp)w−Hqpπw.
By using the fact that πw =wwe get the transport equation:
0=πHqpw+1
2div (Hqp)w.
The transport equation on Char qsfollows analogous.
Karal and Keller [7] calculated solutions to the isotropic elastody-
namics equation on the basis of the classical ansatz of geometrical op-
tics: Consider solutions of the form
v(t, x)=a(x, ω)eiω(φ(x)−t),a(x, ω)∼
∞
k=0
(iω)−kak(x).(4.7)
with amplitude a.
30
These special solutions translate into the theory of Lagrangian distri-
butions as the trivializations, with respect to the half-density bundle,
of distributions u∈I0(Rt×R3
x,Λ,Ω1
2),
u(t, x)=(2π)−3/2eiω(ϕ(x)−t)a(x, ω)dω |dx dω|1
2.
Here, the Lagrangian manifold Λ, corresponding to the special phase
function
ψ(t, x, ω)=ω(ϕ(x)−t),
is equal to
{(t, x, τ, ξ)|ψ
ω(t, x, ω)=0and(τ,ξ)=ψ
t,x(t, x, ω)}
={(t, x, τ, ξ)|t=ϕ(x),τ=−ωand ξ=−τ∇φ(x)}
={(φ(x),x,τ,−τ∇φ(x))}.
We choose coordinates (x, τ) on Λ. The trivialization of the principal
symbol of uis equal to the principal part a0of the amplitude.
We’ll show that our method, if applied to these special Lagrangian
distributions, results in the same eikonal- and transport equation as
the elementary calculations of Karal and Keller. But in contrast to
their result, the generalized transport equation in Theorem 4.4 is in
addition correct and meaningful at caustics.
First, we evaluate the eikonal equation:
Theorem 4.5. The conditions Λ⊆Char Land a0∈Ker lare equiva-
lent to
|∇φ|2=ρ/(λ+2µ)and a0×∇φ=0 on Char qp
and
|∇φ|2=ρ/µ and a0·∇φ=0 on Char qs.
Proof. The characteristic set Char Lis
{(t, x, τ, ξ)|qp(t, x, τ, ξ)=0orqs(t, x, τ, ξ)=0}
=(t, x, τ, ξ)||ξ|2=τ2ρ
λ+2µor |ξ|2=τ2ρ
µ.
Therefore the condition Λ ⊆Char Lis equivalent to
|∇φ|2=ρ
λ+2µon Char qp
and
|∇φ|2=ρ
µon Char qs.
31
Next, we evaluate the condition a0∈Ker l.
First we calculate on Char qp: We saw in the proof of Theorem 4.4
that the condition a0∈Ker lis equivalent to πa
0=a0.Withξ=
−τ∇φon Λ we get that a0×∇φ=0.
On Char qs, the condition a0∈Ker lis equivalent to πa
0=0,which
implies a0·∇φ=0.
This corresponds to equations (9) to (12) in Karal and Keller [7].
Theorem 4.6. Assume that the eikonal equation holds. The transport
equation for a0on Char qp, expressed in α0such that a0=α0∇φ,is
equal to
0=2(∇φ·∇)α0+1
ρ∇·(ρ∇φ)α0
and on Char qsit is
0=2(∇φ·∇)a0+1
µ∇·(µ∇φ)a0+µ
ρ(a0·∇(ρµ−1))∇φ.
Proof. On Char qpthe transport equation is (4.4); we evaluate the com-
ponents:
In the coordinates (x, τ) on Λ, the Hamiltonian Hqpis
j
∂ξjqp∂xj−∂tqp∂τ
=−2(λ+2µ)
j
ξj∂xj−0
=−2(λ+2µ)(ξ·∇)
=2τ(λ+2µ)(∇φ·∇).
The divergence div Hqpis therefore equal to
∇·[2τ(λ+2µ)∇φ]=2τ∇·[(λ+2µ)∇φ].
By using that
|∇φ|2=ρ
λ+2µand ξ=−τ∇φ
on Λ ∩Char qpwe get that the projection
π=λ+2µ
ρ∇φ⊗∇φ.
We insert this into equation (4.4):
0=πHqpa0+1
2div (Hqp)a0
=2τ(λ+2µ)π[(∇φ·∇)a0]+τ∇·[(λ+2µ)∇φ]a0.
32
Division by τ(λ+2µ) yields
0=2π[(∇φ·∇)a0]+ 1
λ+2µ∇·[(λ+2µ)∇φ]a0
=2λ+2µ
ρ(∇φ·[(∇φ·∇)a0]) ∇φ+1
λ+2µ∇·[(λ+2µ)∇φ]a0
We next use equation (120) of Karal and Keller [7],
∇φ·[(∇φ·∇)a0]=
1
2α0∇φ·[∇(ρ(λ+2µ)−1]
+ρ
λ+2µ(∇φ·∇)α0,
and eliminate ∇φin the transport equation:
0=2(∇φ·∇)α0+α0(λ+2µ)
ρ∇φ·[∇(ρ(λ+2µ)−1]
+1
λ+2µ∇·[(λ+2µ)∇φ]α0
=2(∇φ·∇)α0+1
ρ∇·(ρ∇φ)α0.
The derivation of the transport equation on Char qsworks likewise:
In coordinates (x, τ) on Λ, the Hamiltonian Hqsis
Hqs=2τµ(∇φ·∇),
the divergence div Hqsis equal to 2τ∇·(µ∇φ) and the projection is
π=µ
ρ∇φ⊗∇φ.
On insertion into equation (4.5), we get
0=(I−π)Hqsa0+1
2div (Hqs)a0
=2τµ(I−π)(∇φ·∇)a0+τ∇·(µ∇φ)a0
We divide by τµ and get
0=2(∇φ·∇)a0−2π(∇φ·∇a0)+ 1
µ∇·(µ∇φ)a0.
Now, we use Karal and Keller [7, equation (69)],
∇φ·[(∇φ·∇)a0]=−1
2[a0·∇(ρµ−1)],
to get that
π(∇φ·∇a0)=µ
ρ(∇φ·[(∇φ·∇)a0]) ∇φ=−µ
2ρ[a0·∇(ρµ−1)] ∇φ.
33
That yields the transport equation
0=2(∇φ·∇)a0+µ
ρ(a0·∇(ρµ−1)) ∇φ+1
µ∇·(µ∇φ)a0.
This result corresponds to equations (116) and (72), in Karal and Keller
[7].
34
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