Parametrices for Symmetric Systems

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10 Οκτ 2013 (πριν από 4 χρόνια και 28 μέρες)

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Historical Background Normal Form Hermite Transform Intertwining Solution Operator
Parametrices for Symmetric Systems
Clifford Nolan

and Gunther Uhlmann

∗ Department of Mathematics & Statistics
University of Limerick,Ireland
† Department of Mathematics,University of Washington,U.S.A.
µ-local analysis and harmonic analysis in inverse problems,CIRM,
Marseilles,March 28,2007
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform Intertwining Solution Operator
Outline
1
Historical Background
2
Normal Form
3
Hermite Transform
4
Intertwining
5
Solution Operator
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Parametrices for Symmetric Systems
Historical Background
Normal Form Hermite Transform Intertwining Solution Operator
Historical Background
Wave propagation in anisotropic media is very challenging and
of great practical importance.
The propagation of singularities and the construction of
microlocal parametrices for scalar PDO's of real
principal-type
,
like the acoustic wave equation is well understood.
This is consequence of Egorov's theorem,which says one may
conjugate the operator via an invertible Fourier integral operator
to the operator D
x
n
.
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Parametrices for Symmetric Systems
Historical Background
Normal Form Hermite Transform Intertwining Solution Operator
Historical Background
However,most systems of PDE have characteristics with
variable multiplicity and do not t the principal-type mode l
In such instances,striking phenomena occur when the
characteristic (wave) speeds coalesce,e.g.,
conical refraction
Melrose and Uhlmann ('79) constructed a parametrix for the
Cauchy problem for Maxwell's equations in a biaxial crystal with
double involutive characteristic sheets:
P = ∂
2
t



2
x
1
+∂
2
x
2

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Parametrices for Symmetric Systems
Historical Background
Normal Form Hermite Transform Intertwining Solution Operator
Historical Background
The propagation of singularities for a class of symmetric
systems with double characteristics satisfying a generic
condition has been extensively studied in Ivrii ('77,'81)
Melrose ('85) gave a parametrix for scalar pde with
non-involutive double characterisitics
The propagation of polarization has been studied by Denker
('88)
A step toward the construction of parametrices for generic
symmetric systems was realized Braam and Duistermaat ('93)
by giving a pair of normal forms
Goal of this talk is to describe a parametrix for one of these
normal forms - the hyperbolic case
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Parametrices for Symmetric Systems
Historical Background
Normal Form Hermite Transform Intertwining Solution Operator
Symmetric Systems of Interest fromMath
Physics
Maxwell's equations (crystal optics)

ǫ(x)∂
2
t
+∇×∇×

E(x,t) = 0.
Linear elasticity

2
u
i
∂t
2

1
ρ
∂(c
ijkl
∂u
k
∂x
l
)
∂x
j
= 0
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Parametrices for Symmetric Systems
Historical Background
Normal Form Hermite Transform Intertwining Solution Operator
Overview
Intertwine B-D normal formto operator of principal-type
This is achieved by using a FIO asociated to a singular canonical
transformation
The transformed variables live on the circle times reduced
space-time
Solve this simplied Cauchy problem globally on the circle
The resulting parametrix (solution operator) is a mixed
Fourier-Integral-Series Operator
The zero Fourier frequencies corresponding to the angular
variable are associated to multiple characteristic data
Within the double characteristic variety,there is no propagation
of singularity,but there is propagation of polarisation
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Parametrices for Symmetric Systems
Historical Background
Normal Form
Hermite Transform Intertwining Solution Operator
Normal Formof Braam-Duistermaat
Under generic conditions on the principal symbol of a symmetric
systemQ,one may conjugate to one of two normal forms:
MQK =
˜
P =
˜
P
±
1
+
˜
R
±
1
+
˜
R
±
0
near ξ
2
6= 0,where
˜
P
±
1
:=


D
t
+D
x
1
x
1
D
x
2
0
x
1
D
x
2
±(D
t
−D
x
1
) 0
0 0 E
±
1


where E
±
1
is elliptic and M,K are elliptic FIO's,and the symbol
of
˜
R
±
1
is at at double characteristic set:
Σ
2
:= {τ = x
1
= ξ
1
= 0}
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Parametrices for Symmetric Systems
Historical Background
Normal Form
Hermite Transform Intertwining Solution Operator
Cofactor System
We will consider the associated (2 ×2) Cauchy problem
˜
P
±
u = 0
u|
t=0
= u
0
,u
t
|
t=0
= u
1
Multiplying across the negative cofactor matrix:
P
±
w:=

P
±
2
+R
±
2
+P
±
1
+P
±
0

w = 0
w|
t=0
= (Ku)|
t=0
;w
t
|
t=0
= (Ku)
t
|
t=0
where
P
±
2
= ±∂
2
t
∓∂
2
x
1
−x
2
1

2
x
2
,P
±
1
= A
±

x
2
+R
±
1
A
±
=

0 ±1
−1 0

and R
±
2
,R
±
1
are scalar and non-scalar operators resp.,with at
and vanishing symbols resp.at Σ
2
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Parametrices for Symmetric Systems
Historical Background
Normal Form
Hermite Transform Intertwining Solution Operator
Cofactor System
Need to assume that the co-factor systemis well-posed -
checked explicitly for Crystal Optics and Linear Elasticity with
cubic symmetry
Observe that the co-factor systemfor B-D normal formis a
symmetric hyperbolic system
Fromnow on,only deal with the + normal formof B-D
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Parametrices for Symmetric Systems
Historical Background Normal Form
Hermite Transform
Intertwining Solution Operator
Singular canonical transformation
Notation:x

= (x
2
,...,x
n
),x
′′
= (x
3
,...,x
n
),etc
Consider the homogeneous symplectic transformation T
x
1
=
p
2k/η
2
cos θ
ξ
1
= −

2kη
2
sinθ
x
2
= y
2
+(k/2η
2
) sin2θ
ξ

= η

x
′′
i
= y
′′
i
where (x;ξ),(θ,y

;k,η

) are domain and range coords resp.
The range is a conic subset of T

(S
1
θ
×R
n−1
y

),where k ≥ 0 is
the dual angular (θ) variable.Observe that this transformation
fails to be smooth at the hypersurface
T (Σ
2
) = {(θ,y

,k = 0,η

):η
2
6= 0 }.
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Parametrices for Symmetric Systems
Historical Background Normal Form
Hermite Transform
Intertwining Solution Operator
Remark:Singular canonical transformation
Remark:
The point of this transformation is that it pulls back the
principal symbol p
2
= τ
2
−ξ
2
1
−x
2
1
ξ
2
2
(which has characteristics
with variable multiplicity) to the simple characteristic operator
with principal symbol τ
2
−2kη
2
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Parametrices for Symmetric Systems
Historical Background Normal Form
Hermite Transform
Intertwining Solution Operator
Hermite Operator corresponding to T
We dene an operator T by a pair of representatives of its dist
kernel in a pair of charts covering

R
t
×S
1
×R
n−1
y


×(R
t
×R
n
x
):
K
T
1
((t

,θ,y

),(t,x)) =
Z
dξ e
i
{
−(x
2
1
ξ
2
/2)cot(θ)+(y

−x

)ξ

}
λ
1
(θ,y

,t,x,ξ) δ(t

−t)
K
T
2
((t

,θ,y

),(t,x)) =
Z
dξ e
i
{

2
1
/2ξ
2
)tan(θ)+y



−xξ
}
λ
2
(θ,y

,t,x,ξ) δ(t

−t)
The wavefront set of the distribution kernel of T is designed to
be the graph of T,and a left-parametrix for T exists
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Parametrices for Symmetric Systems
Historical Background Normal Form
Hermite Transform
Intertwining Solution Operator
Hermite Operator corresponding to T
Note that
T:E

(R
t
′ ×R
n
x
) →E

(R
t
×S
1
θ
×R
n−1
y

;L)
where the right hand side represents the space of distributional
sections of a complex line bundle L over R
t
×S
1
θ
×R
n−1
y

and
has transition functions exp(i π/4).
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform
Intertwining
Solution Operator
Intertwining
Earlier remark (in red) implies
TP
2
=


2
t
−2L
θ

y
2
+
˜
R
2

T
and in fact we can combine T with another intertwining operator
to remove at term
˜
R
2
by equivalence of operators of
principal-type.
A lengthy calculation shows that we can simplify the rst ord er
terms by interwining with a ΨDO G:
GP −

L
2
+E
1
(t,x
1
D
x
1
D
−1
x
2
/2 +x
2
,x
′′
,D
x
′ )

G = 0
avoiding singular (square root) symbols.At the same time,we
can remove rst order at terms
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform
Intertwining
Solution Operator
Representation of x
1
-odd Cauchy data
The kernel of T consists of x
1
-odd distributions
Fortunately,we can rewrite these as odd data as
f (x) = (∂
x
1
−ix
1

x
2
) g(x)
f (x
1
,x

) = −f (−x
1
,x

)
g(x) = g(−x
1
,x

)
where a left-parametrix is available for ∂
x
1
−ix
1

x
2
Therefore we split Cauchy data into even and odd components
and solve separately
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform
Intertwining
Solution Operator
Summary so far...
At this stage the operator P has been conjugated to the form
B = ∂
2
t
−2L
θ

y
2
+A


x
2
+E
0
where A

= A for even data and A

= A−2iI for odd data.
We will assume that A

has been diagonalized at this stage,with
diagonal values i ν
1
,i ν
2
,where ν
1

2
have values −1,1 for even
data and 1,3 for odd data
We'll consider the Cauchy problem with data (w
0
,0),and the
complimentary data can be handled in a similar manner
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform Intertwining
Solution Operator
Solution Operator
We seek a solution operator w
0
7→w in the form of a
Fourier-Integral-Series pair:
w =
X
±,k≥0
Z
"
e
i φ
(1)
±
(t,k,η

)
0
0 e
i φ
(2)
±
(t,k,η

)
#
(2π)
−n
e
i
(
(4k+1)(θ−
˜
θ)+(y

−˜y

)η

)
e
±
(t,y

,k,η

) w
0
(
˜
θ,
˜
y

) d
˜
θd
˜
y



where φ
(j )
±
are phase functions to be determined and e
±
is a
2 ×2 matrix of symbols also to be determined.
Note:T-Transformed initial even data is odd under shift of
1/4-cycle,hence the (4k +1) factor in phase
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform Intertwining
Solution Operator
Eikonal and Transport Equations
Plugging in this ansatz,the resulting system,we obtain Eikonal
equations which have the explicit solutions
φ
(j )
±
= ± t
q
(4k +1 +ν
j

2
,j = 1,2
which are symbol-valued symbols of type (1/2,1/2) in the
frequencies (k,η

)
We write
e
±
=
"
e
(1)
±
e
(2)
±
#
where e
(j )
±
,j = 1,2 are 1 ×2 row vectors of symbols.
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform Intertwining
Solution Operator
Transport Equations
For odd data,the transport equations and initial conditions are
M
±
e
±
= 0:
M
±
=

p
(4k +1 +ν
1

2
0
0
p
(4k +1 +ν
2

2


t

1
2

4k +1 +ν
1
0
0 4k +1 +ν
2


y
2

i
2
(∂
2
t
−r
±
0
)
(e
+
+e

)|
t=0
∼ I,

p
(4k +1 +ν
1

2
0
0
p
(4k +1 +ν
2

2

(e
+
−e

)|
t=0

i ∂
t
(e
+
+e

)|
t=0
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform Intertwining
Solution Operator
Transport Equations
For even data,a`similar system'obtains
For odd data,we then solve a hierarchy of such (possibly
inhomogeneous) transport equations with
e
±


X
m=0
e
±,m
and
e
±
∈ S
0,0
;e
±,m
∈ S
−m/2,−m/2
We set e
±,0
=
1
2
I and e
−,m
= −e
+,m
for m ≥ 1 at t = 0,and
recursively solve initial conditions for the transport equations
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform Intertwining
Solution Operator
Transport Equations
For even data with k ≥ 1,we can use the same procedure
For even data with k = 0,the transport equations are solved by
a different method with
e
(j )
±
∈ S
0
;e
(j )
±,m
∈ S
−m/2
and the initial conditions are modied appropriately
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform Intertwining
Solution Operator
Propagation of Singularities and Polarisation
The k = 0 Fourier coefcient termdoes not propagate
singularities within the double characteristic variety.
The underlying reason for this is due to the fact that the
time-derivative coefcient in the transport operator M
±
contains
a factor


2
(for large k),whereas in Melrose-Uhlmann ('79),
M
±
contains a factor analogous to k which necessitated a
conical refraction correction termthat led to propagation of
singularities within the double characteristic variety.
While there is no propagation of singularities in Σ
2
,there is a
propagation of polarisation - interesting?
The k > 0 terms propagate singularities and polarisation along
characteristics of the principal type-operator
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform Intertwining
Solution Operator
Concluding Remarks
We need to make use of the structure of Maxwell's equations
and linear elasticity to make sure that the co-factor system is
well-posed
We also had to check that all the canonical transformations used
in B-D,for removing at terms,etc,can be arranged to preser ve
t = 0 for the Cauchy problem
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Parametrices for Symmetric Systems
Historical Background Normal Form Hermite Transform Intertwining
Solution Operator
Summary
The Parametrix E is
E = MJ
−1
G
−1
T

YSTGπ
even
JK +
MJ
−1
G
−1
(D
x
1
−ix
1
D
x
2
)T

YSTRGπ
odd
JK
is a solution operator for the Cauchy problem.Here,M,J,L are
microlocally elliptic Fourier integral operators associated to canonical
transformations.The operator Y is a product of pseudodifferential
operators in the θ and y
2
variables respectively.G is a
pseudodifferential operator.π
odd

even
project data to the odd and
even components with respect to the involution x
1
7→−x
1
.T is a
Fourier integral operator associated to a singular canonical
transformation.The operator S is the solution operator for the
Cauchy problem associated to the simple characteristic operator B.
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Parametrices for Symmetric Systems