High-order elliptic operators: Carleman
estimates at boundaries and interfaces
J´erˆ
ome Le Rousseau
F´ed´eration Denis-Poisson
Universit´e d’Orl´eans
In collaboration with
Mourad Bellassoued (Facult´e des Sciences de Bizerte, Tunisia),
based in parts on works in collaboration with
Nicolas Lerner (Universit´e Pierre-et-Marie-Curie, Paris, France),
and Luc Robbiano (Universit´e de Versailles-Saint Quentin, France).
Inverse Problems and Imaging Conference
Institut Henri Poincar´e, Paris, France
April 2014
1/ 35
J. Le Rousseau
high-order elliptic operators
Form and examples of Carleman estimate
Notation
We consider a bounded domain Ω of Rd .
P differential operator of order m (need not be elliptic here)
ϕ = ϕ(x) a smooth weight function to be determined.
τ > 0: large parameter
Form of the estimates away from boundaries
For all K compact in Ω, there exist C > 0, τ0 > 0 such that
P 2(m−|β|)−1 τ ϕ β 2
τ
ke Dx ukL2 ≤ Ckeτ ϕ P uk2L2 ,
D = ∂/i
(1)
|β|<m
for u ∈ Cc∞ (Ω), supp(u) ⊂ K and τ ≥ τ0
There are necessary conditions on both P and ϕ.
There are sufficient conditions on both P and ϕ.
2/ 35
J. Le Rousseau
high-order elliptic operators
Form and examples of Carleman estimate
Notation
We consider a bounded domain Ω of Rd .
P differential operator of order m (need not be elliptic here)
ϕ = ϕ(x) a smooth weight function to be determined.
τ > 0: large parameter
Form of the estimates away from boundaries
Example for the Laplace operator, P = −∆
τ 3 keτ ϕ uk2L2 + τ keτ ϕ ∇x uk2L2 ≤ Ckeτ ϕ P uk2L2 ,
for u ∈ Cc∞ (Ω), supp(u) ⊂ K and τ ≥ τ0
There are necessary conditions on both P and ϕ.
There are sufficient conditions on both P and ϕ.
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J. Le Rousseau
high-order elliptic operators
Applications
Some domains of applications
Unique continuation
Carleman, H¨
ormander, Zuily, Alinhac, Robbiano, Lerner, Jerison,
Kenig, Tataru, Escauriaza, and many others.
Inverse problems : Identification of coefficients including stability
results
Bukhgeim, Klibanov, Imanuvilov, Yamamoto, Puel, Uhlmann,
Kenig, Isakov, Eller, Dos-Santos Ferreira, Salo, Rosier, Baudouin,
Osses, Doubova, Fern´
andez-Cara, Bellassoued, Chouli, Mercado,
Benabdallah, Cristofol, Gaitan, Bourgeois, and many others
Control theory
Lebeau, Robbiano, Fursikov, Imanuvilov, Fern´
andez-Cara, Zuazua,
Guerrero, de Teresa, Gonzales-Burgos, Puel, Rosier, Benabdallah,
Ammar-Khodja, Miller, Cepa, Ervedoza, Vancostenoble, Martinez,
Cannarsa, and many others
3/ 35
J. Le Rousseau
high-order elliptic operators
Convexity of the weight function and more precise form of the estimates
A Carleman estimate of the form
P
τ 2(m−|β|)−1 keτ ϕ Dxβ uk2L2 . keτ ϕ P uk2L2 ,
|β|<m
can be insufficient to achieve some results, especially for inverse
problems and unique continuation.
4/ 35
J. Le Rousseau
high-order elliptic operators
Convexity of the weight function and more precise form of the estimates
A Carleman estimate of the form
τ 3 keτ ϕ uk2L2 + τ keτ ϕ ∇x uk2L2 . keτ ϕ ∆uk2L2 ,
can be insufficient to achieve some results, especially for inverse
problems and unique continuation.
4/ 35
J. Le Rousseau
high-order elliptic operators
Convexity of the weight function and more precise form of the estimates
A Carleman estimate of the form
τ 3 keτ ϕ uk2L2 + τ keτ ϕ ∇x uk2L2 . keτ ϕ ∆uk2L2 ,
can be insufficient to achieve some results, especially for inverse
problems and unique continuation.
Some authors introduce a second large parameter
τ 3 α4 kϕ3/2 eτ ϕ uk2L2 + τ α2 kϕ1/2 eτ ϕ ∇x uk2L2 . keτ ϕ ∆uk2L2 ,
with ϕ(x) = exp(αψ(x)) and α ≥ α0 .
4/ 35
J. Le Rousseau
high-order elliptic operators
Convexity of the weight function and more precise form of the estimates
A Carleman estimate of the form
P
τ 2(m−|β|)−1 keτ ϕ Dxβ uk2L2 . keτ ϕ P uk2L2 ,
|β|<m
can be insufficient to achieve some results, especially for inverse
problems and unique continuation.
Some authors introduce a second large parameter
P
(τ α)2(m−|β|)−1 kϕm−|β|−1/2 eτ ϕ Dxβ uk2L2 . keτ ϕ P uk2L2 ,
ϕ = eαψ
|β|<m
For parabolic/elliptic operators: Fursikov-Imanuvillov
Parabolic operators at the boundary: Imanuvilov-Puel-Yamamoto
Hyperbolic operators at the boundary: Bellassoued-Yamamoto
For second-order operators: Isakov, Eller, Kim
4/ 35
J. Le Rousseau
high-order elliptic operators
Example of applications
Bellassoued-Yamamoto, 10
Result: A carleman estimate for ∂t2 − ∆g with two large parameters
(global estimate, ie, up to the boudary)
Thanks to the second large parameter it yields Carleman estimates
for
the plate equation
the thermoelasticity plate equations
thermoelasticity system with residual stress
5/ 35
J. Le Rousseau
high-order elliptic operators
Setting at a boundary
We consider
P be a smooth elliptic of order m = 2µ.
P
P =
aα (x)Dα ,
|α|≤m
with complex valued coefficients.
m/2 linear smooth boundary operators of order less than m
P k
Bk =
bα (x)Dα , k = 1, . . . , µ = m/2,
|α|≤βk
with complex-valued coefficients, defined in some neighborhood
of ∂Ω.
Consider the elliptic boundary value problem
(
P u(x) = f (x),
x ∈ Ω,
k
k
B u(x) = g (x), x ∈ ∂Ω, k = 1, . . . , µ.
6/ 35
J. Le Rousseau
high-order elliptic operators
Setting at a boundary
Consider the elliptic boundary value problem
(
P u(x) = f (x),
x ∈ Ω,
B k u(x) = g k (x), x ∈ ∂Ω, k = 1, . . . , µ.
We wish to obtain an estimate of the form
µ
P
keτ ϕ uk2 + |eτ ϕ T (u)|2 . keτ ϕ P (x, D)uk2 +
|eτ ϕ B k (x, D)u|∂Ω |2 ,
k=1
for u supported near a point at the boundary
T (u) is the trace of (u, Dν u, . . . , Dνm−1 u)
If we set
Pϕ = eτ ϕ P (x, D)e−τ ϕ ;
Bϕk = eτ ϕ B k (x, D)e−τ ϕ ;
v = eτ ϕ u.
then the Carleman estimate reads:
µ
P
kvk2 + |T (v)|2 . kPϕ vk2 +
|Bϕk v|∂Ω |2 ,
k=1
Estimates of this form were obtained by Tataru. We give more
precise estimates here and include the complex coefficient case.
J. Le Rousseau
high-order elliptic operators
7/ 35
2nd order operator at a boundary
2nd-order operators at the boundary were precisely treated by
G. Lebeau and L. Robbiano (95, 97).
Set
P
P = −∆ = j Dj2 ;
∂Ω = {xn = 0} and Ω = {xn > 0}.
We write x = (x0 , xn ) and ξ = (ξ 0 , ξn ).
Take ϕ = ϕ(xn ) such that ϕ0 > 0.
Then
M2
Pϕ = Dxn + iτ ϕ0 (x)
2
+
z P}|
{
Dj2
1≤j≤n−1
0
= Dxn + i(τ ϕ (x) + M ) Dxn + i(τ ϕ0 (x) − M ) ,
where M = Op(|ξ 0 |).
Consider the principal symbol
pϕ (x, ξ) = p(x, ξ + iτ ϕ0 ) = ξn + i(τ ϕ0 (x) + |ξ 0 |) ξn + i(τ ϕ0 (x) − |ξ|)
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J. Le Rousseau
high-order elliptic operators
2nd order operator at a boundary
Principal symbol
pϕ (x, ξ) = p(x, ξ + iτ ϕ0 ) = ξn + i(τ ϕ0 (x) + |ξ 0 |) ξn + i(τ ϕ0 (x) − |ξ|)
= ξn − ρ1 ξn − ρ2
In the low-frequency regime, |ξ 0 | small,
Im(z)
ρ1
ρ2
Re(z)
We have (a microlocal perfect elliptic estimate)
kvk2,τ + |T (v)|1,1/2,τ . kPϕ vkL2
(+ · · · )
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J. Le Rousseau
high-order elliptic operators
2nd order operator at a boundary
Principal symbol
pϕ (x, ξ) = p(x, ξ + iτ ϕ0 ) = ξn + i(τ ϕ0 (x) + |ξ 0 |) ξn + i(τ ϕ0 (x) − |ξ|)
= ξn − ρ1 ξn − ρ2
In the high-frequency regime, |ξ 0 | large,
Im(z)
Im(z)
ρ2
ρ2
Re(z)
ρ1
Re(z)
ρ1
We have
τ −1/2 kvk2,τ + |T (v)|1,1/2,τ . kPϕ vkL2 + boundary norm
(+ · · · )
The boundary norm can be of Dirichlet, Neumann, Robin type...
9/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
Principal symbol
`ϕ (x, ξ) = p(x, ξ + iτ ϕ0 ) =
k
Q
ξn − ρj (x, τ, ξ 0 )
j=1
If all the roots have a negative imaginary part,
Im(z)
ρ1
ρk
Re(z)
ρ2
ρ3
we have (a microlocal perfect elliptic estimate)
kvkk,τ + |T (v)|k−1,1/2,τ . k`ϕ vkL2
(+ · · · )
10/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
We have (a microlocal perfect elliptic estimate)
kvkk,τ + |T (v)|k−1,1/2,τ . k`ϕ vkL2
(+ · · · )
Ideas of the proof: Write `ϕ = `1 + i`2 , `1 and `2 both self adjoint.
k`1 vk2L2 + k`2 vk2L2 ≥ Ckvk2k,τ − C 0 |T (v)|2k−1, 1 ,τ
2
(the roots of l1 and l2 are real and distinct)
With a generalized green formula we have
2(`1 v, `2 v)2L2 ≥ B(v) − Ckvk2k,− 1 ,τ
2
The position of the roots gives
B(v) & |T (v)|2k−1, 1 ,τ
2
11/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
Set %0 = (x, ξ 0 , τ )
We have
pϕ (%0 , ξn ) =
m
Y
0
− 0
0
0
ξn − ρj (%0 ) = p+
ϕ (% , ξn )pϕ (% , ξn )pϕ (% , ξn ),
j=1
with
0
p±
ϕ (% , ξn ) =
Q
(ξn − ρj ),
± Im ρj >0
p0ϕ (%0 , ξn ) =
Q
(ξn − ρj ).
Im ρj =0
p−
ϕ yields a prefect elliptic estimate.
We set
0
0
0
κϕ (%0 , ξn ) = p+
ϕ (% , ξn )pϕ (% , ξn )
12/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
Boundary operators: B k ,
Conjugated operators:
Bϕk
k = 1, . . . , µ
= eτ ϕ B k e−τ ϕ
Principal symbol: bkϕ (%0 , ξn )
Strong Lopatinskii condition:
The set {bkϕ (%0 , ξn )}k=1,...,µ is complete modulo κϕ (%0 , ξn ) as
polynomials in ξn .
For all f (ξn ) polynomial, there exist c1 , . . . , cµ ∈ C and q(ξn )
polynomial such that
f (ξn ) =
µ
P
ck bkϕ (%0 , ξn ) + q(ξn )κϕ (%0 , ξn )
k=1
13/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
With the strong Lopatinskii condition we obtain
|T (v)|m−1,1/2,τ .
µ
P
|Bϕk v|xn =0 |m−1/2−βk ,τ + kPϕ vkL2
(+ · · · )
k=1
Idea of the proof:
From the strong Lopatinskii condition we have
µ
P
k=1
|Bϕk v|xn =0 |m−1/2−βk ,τ +
−
mP
−1
|Dnk Op(κϕ )v|xn =0 |m− −1/2−k,τ
k=0
& |T (v)|m−1,1/2,τ
Then use the perfect elliptic estimate for Pϕ− writing pϕ = p−
ϕ κϕ :
k Op(κϕ )vkm− ,τ + |T (Op(κϕ )v)|m− −1,1/2,τ . kPϕ vkL2
(+ · · · )
14/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
We write Pϕ = A + iB with A and B both selfadjoint.
Principal symbols:
pϕ (%0 , ξn ) = a(%0 , ξn ) +i b(%0 , ξn ),
| {z } | {z }
=Re pϕ
%0 = (x, ξ 0 , τ ).
=Im pϕ
Sub-ellipticity property:
pϕ (%, ξn ) = 0
⇒
{a, b}(%, ξn ) > 0.
Classically we then obtain
Cτ −1 kvk2m,τ ≤ C 0 τ −1 kAvk2L2 + kBvk2L2 + |T (v)|2m−1,1/2,τ
+ Re (Av, iBv) − Ba,b (v)
15/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
Assuming the Strong Lopatinskii condition and the sub-ellipticity
condition
We recall
|T (v)|2m−1,1/2,τ .
µ
P
k=1
|Bϕk v|xn =0 |2m−1/2−βk ,τ + kPϕ vk2L2
(+ · · · )
and
Cτ −1 kvk2m,τ ≤ C 0 τ −1 kAvk2L2 + kBvk2L2 + |T (v)|2m−1,1/2,τ
+ Re (Av, iBv) − Ba,b (v)
We have |B(v)| . |T (v)|2m−1,1/2,τ .
Combining the two blue estimates we obtain
τ −1 kvk2m,τ + |T (v)|2m−1,1/2,τ . kPϕ vk2L2 +
µ
P
k=1
|Bϕk v|xn =0 |2m−1/2−βk ,τ
for τ large.
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J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
We have thus obtained
Theorem (Bellassoued, LR)
Under
sub-ellipticity condition,
strong Lopatinskii condition,
Let x0 ∈ ∂Ω. There exist W a neighborhood of x0 , C > 0, and τ0 > 0
such that at the boundary
τ −1 keτ ϕ uk2m,τ + |eτ ϕ T (u)|2m−1,1/2,τ
µ
P
|eτ ϕ B k (x, D)u|∂Ω |2m−1/2−βk ,τ ,
≤ C keτ ϕ P (x, D)uk2L2 +
k=1
for τ ≥ τ0 and u = w|Ω with w ∈ Cc∞ (W ).
17/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
Let ϕ = eαψ . With proper pseudo-differential calculus [LR12] we can
obtain
Theorem (Bellassoued, LR)
Under
strong pseudo-convexity condition,
strong Lopatinskii condition,
Let x0 ∈ ∂Ω. There exist W a neighborhood of x0 , C > 0, τ0 > 0, α0
such that at the boundary
1
k˜
τ − 2 eτ ϕ uk2m,˜τ + |eτ ϕ T (u)|2m−1,1/2,˜τ
µ
P
|eτ ϕ B k (x, D)u|∂Ω |2m−1/2−βk ,˜τ ,
≤ C keτ ϕ P (x, D)uk2L2 +
k=1
for τ ≥ τ0 , α ≥ α0 and u = w|Ω with w ∈ Cc∞ (W ).
Here τ˜ = τ αϕ.
18/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary
Let ϕ = eαψ . With proper pseudo-differential calculus [LR12] we can
obtain
Theorem (Bellassoued, LR)
Under
the simple characteristic condition,
strong Lopatinskii condition,
Let x0 ∈ ∂Ω. There exist W a neighborhood of x0 , C > 0, τ0 > 0, α0
such that at the boundary
1
αk˜
τ − 2 eτ ϕ uk2m,˜τ + |eτ ϕ T (u)|2m−1,1/2,˜τ
µ
P
|eτ ϕ B k (x, D)u|∂Ω |2m−1/2−βk ,˜τ ,
≤ C keτ ϕ P (x, D)uk2L2 +
k=1
for τ ≥ τ0 , α ≥ α0 and u = w|Ω with w ∈ Cc∞ (W ).
Here τ˜ = τ αϕ.
19/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary – unique continuation
Application: unique continuation at the boundary under strong
pseudo-convexity condition
∂Ω
{f (x) = f (x0 )}
{ϕ(x) = ϕ(x0 ) − δ}
x0
B
{ϕ(x) = ϕ(x0 )}
u=0
20/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary – unique continuation
Theorem (Bellassoued-LR)
Let x0 ∈ ∂Ω, f ∈ C ∞ (Ω), and V a neighborhood of x0 such that
1
the strong pseudo convexity property is fulfilled in V
the strong Lopatinskii condition holds at x0
Assume that u ∈ H m (Ω) satisfies
2
|P u(x)| ≤ C
X
|Dα u(x)|,
a.e. in V ;
|α|≤m−1
for k = 1, . . . , µ
|B k u(x)| ≤ C
X
|Dα u(x)|, a.e. in a neighborhood of V ∩ ∂Ω,
|α|≤βk −1
and u vanishes in {x ∈ V ; f (x) ≥ f (x0 )}.
Then u vanishes in a neighborhood of x0 .
21/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator at a boundary – unique continuation
In the same setting, let P1 and P2 be of order m1 and m2 respectively.
If P1 , f , B1k , k=1, . . . , m1 /2 satisfy
the strong pseudo convexity property
the strong Lopatinskii condition holds at x0
If P2 , f , B2k , k=1, . . . , m2 /2 satisfy
the single characteristic property
the strong Lopatinskii condition holds at x0
Then a similar unique continuation result holds for P = P1 P2 .
22/ 35
J. Le Rousseau
high-order elliptic operators
Elliptic operators across an interface
We wish to obtain similar results for the case of transmission problem
across an interface.
Setting Ω = Ω1 ∪ S ∪ Ω2
S = interface between Ω1 and Ω2 .
P1 and P2 elliptic operators of order m1 = 2µ1 and m2 = 2µ2
(possibly different).
m1 + m2 linear transmission operators
P j
Skj =
sk,α (x)Dα , k = 1, 2, j = 1, . . . , m = µ1 + µ2 . (1)
|α|≤βkj
We consider the following elliptic transmission problem
(
Pk uk = fk
in Ωk , , k = 1, 2
S1j u1 + S2j u2 = g j , in S, j = 1, . . . , m.
in addition with boundary conditions.
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J. Le Rousseau
high-order elliptic operators
Elliptic operators across an interface
The type of estimate that we wish to obtain is of the form
P
k=1,2
τ −1 keτ ϕk uk k2mk ,τ + |eτ ϕk T (uk )|2mk −1,1/2,τ
.
P
keτ ϕk Pk (x, D)uk k2L2
k=1,2
+
m
P
j=1
|eτ ϕ|S (S1j (x, D)u1 + S2j (x, D)u2 )|S |2m−1/2−β j ,τ ,
24/ 35
J. Le Rousseau
high-order elliptic operators
2nd order operator across an interface
2nd-order operators at an interface were treated by [LR–Robbiano, 10;
LR–Lerner 13].
Set
P1 = −∇c1 ∇; P2 = −∇c2 ∇;
S = {xn = 0}, Ω1 = {xn < 0}, and Ω2 = {xn > 0}.
We write x = (x0 , xn ) and ξ = (ξ 0 , ξn ).
Take ϕ = ϕ(xn ) such that ϕ0 > 0.
Then
pj,ϕ (x, ξ) = pj (x, ξ + iτ ϕ0 ) = ξn − ρj1 ξn − ρj2
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J. Le Rousseau
high-order elliptic operators
2nd order operator across an interface
Im(z)
Im(z)
ρ12
ρ11
ρ21
no info
ρ22
Re(z)
two traces are determined
Im(z)
Im(z)
ρ12
ρ11
ρ21
ρ22
no info
One relation
J. Le Rousseau
high-order elliptic operators
Re(z)
26/ 35
2nd order operator across an interface
Im(z)
Im(z)
ρ12
ρ11
two traces are dertermined
one relation
Im(z)
Re(z)
ρ22
ρ21
Im(z)
ρ12
ρ22
Re(z)
ρ11
ρ21
one relation
J. Le Rousseau
one relation
high-order elliptic operators
27/ 35
2nd order operator across an interface
Conditions to always be able to solve for the traces at the interface
are put forward in [LR–Robbiano].
These conditions are proven sharp in [LR–Lerner] A quasi-mode is
constructed otherwise.
These conditions are generalized here for the high-order transmission
problem.
28/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator across an interface
we set
Pj,ϕ = eτ ϕj Pj (x, D)e−τ ϕj ;
k
Sj,ϕ
= eτ ϕj Sjk (x, D)e−τ ϕj ;
vj = eτ ϕj uj .
We have
pj,ϕ (%0 , ξn ) =
m
Qj
k=1
−
0
0
0
0
ξn − ρjk (%0 ) = p+
j,ϕ (% , ξn )pj,ϕ (% , ξn )pj,ϕ (% , ξn ),
with
0
p±
j,ϕ (% , ξn ) =
Q
(ξn − ρjk ),
± Im ρjk >0
p0ϕ (%0 , ξn ) =
Q
(ξn − ρjk ).
Im ρjk =0
We set
0
0
0
κj,ϕ (%0 , ξn ) = p+
j,ϕ (% , ξn )pj,ϕ (% , ξn )
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J. Le Rousseau
high-order elliptic operators
High-order operator across an interface
The transmission condition for Pj , Sjj , ϕj , j = 1, . . . , m reads:
for all pairs of polynomials, qj (ξn ), there exist Uj , polynomials, and
ck ∈ C, k = 1, . . . , m = m1 + m2 , such that:
q1 (ξn ) =
m
P
ck sk1,ϕ (%0 , ξn ) + U1 (ξn )κ1,ϕ (%0 , ξn ),
k=1
and
q2 (ξn ) =
m
P
ck sk2,ϕ (%0 , ξn ) + U2 (ξn )κ2,ϕ (%0 , ξn ).
k=1
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J. Le Rousseau
high-order elliptic operators
High-order operator across an interface
The important proposition then becomes
Proposition
Assume that the transmission condition is satisfied then
C |T (v1 )|m1 −1,1/2,τ + |T (v2 )|m2 −1,1/2,τ
m
P
k
k
|S1,ϕ
v1|xn =0 + S2,ϕ
v2|xn =0 |m−1/2−β k ,τ
≤
k=1
+ kP1,ϕ v1 kL2 + kP2,ϕ v2 kL2 + (...),
for τ ≥ τ0 , v1 , v2 ∈ C ∞ .
Then the Carleman estimate follows.
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J. Le Rousseau
high-order elliptic operators
High-order operator across an interface
Theorem (LR–Bellassoued)
Let x0 ∈ S and let ϕ ∈ C 0 (Ω) be such that ϕk = ϕ|Ωk ∈ C ∞ (Ωk ) for
k = 1, 2 and such that the pairs {Pk , ϕk } have the sub-ellipticity
property
in a neighborhood of x0 in Ωk . Moreover, assume that
Pk , ϕ, Skj , k = 1, 2, j = 1, . . . , µ satisfies the transmission condition
at x0 . Then there exist a neighborhood W of x0 in Rn and two
constants C and τ∗ > 0 such that
P
k=1,2
τ −1 keτ ϕk uk k2mk ,τ + |eτ ϕk T (uk )|2mk −1,1/2,τ
≤C
P
keτ ϕk Pk (x, D)uk k2L2
k=1,2
+
m
P
j=1
|eτ ϕ|S (S1j (x, D)u1 + S2j (x, D)u2 )|S |2m−1/2−β j ,τ ,
for all uk = wk|Ωk with wk ∈ Cc∞ (W ) and τ ≥ τ∗ .
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J. Le Rousseau
high-order elliptic operators
High-order operator across an interface
Version with two large parameters.
Theorem (LR–Bellassoued)
Let x0 ∈ S and let ψ ∈ C 0 (Ω) be such that ψk = ψ|Ωk ∈ C ∞ (Ωk ) for
k = 1, 2 and such that ψk have the strong pseudo-convexity property
with respect to Pk in a neighborhood of x0 in Ωk . Moreover, assume
that Pk , ψ, Skj , k = 1, 2, j = 1, . . . , µ satisfies the transmission
condition at x0 . Then there exist a neighborhood W of x0 in Rn and
three constants C, τ∗ > 0, and α∗ > 0 such that for ϕk = exp(αψk )
and τ˜k = τ αϕk :
P
k=1,2
−1/2 τ ϕ
2
τ˜
e k uk m
k
≤C
τk
k ,˜
+ |eτ ϕ|S T (uk )|2mk −1,1/2,˜τ
P
keτ ϕk Pk (x, D)uk k2L2
k=1,2
+
m
P
j=1
|e
τ ϕ|S
(S1j (x, D)u1|S + S2j (x, D)u2|S )|2m−1/2−β k ,˜τ ,
for all uk = wk|Ωk with wk ∈ Cc∞ (W ), τ ≥ τ∗ and α ≥ α∗ .
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J. Le Rousseau
high-order elliptic operators
High-order operator across an interface
Version with two large parameters.
Theorem (LR–Bellassoued)
Let x0 ∈ S and let ψ ∈ C 0 (Ω) be such that ψk = ψ|Ωk ∈ C ∞ (Ωk ) for
k = 1, 2 and such that ψk have the simple characteristic property with
respect
to Pk in a neighborhood of x0 in Ωk . Moreover, assume that
Pk , ψ, Skj , k = 1, 2, j = 1, . . . , µ satisfies the transmission condition
at x0 . Then there exist a neighborhood W of x0 in Rn and three
constants C, τ∗ > 0, and α∗ > 0 such that for ϕk = exp(αψk ) and
τ˜k = τ αϕk :
P
k=1,2
−1/2 τ ϕ
2
ατ˜k
e k uk m
≤C
τk
k ,˜
P
+ |eτ ϕ|S T (uk )|2mk −1,1/2,˜τ
keτ ϕk Pk (x, D)uk k2L2
k=1,2
+
m
P
j=1
|e
τ ϕ|S
(S1j (x, D)u1|S + S2j (x, D)u2|S )|2m−1/2−β k ,˜τ ,
for all uk = wk|Ωk with wk ∈ Cc∞ (W ), τ ≥ τ∗ and α ≥ α∗ .
34/ 35
J. Le Rousseau
high-order elliptic operators
High-order operator across an interface
With the above estimates we can deduce unique continuation results
near an interface.
S
{f (x) = f (x0 )}
{ϕ(x) = ϕ(x0 ) − δ}
{ϕ(x) = ϕ(x0 )}
x0
B
u=0
35/ 35
J. Le Rousseau
high-order elliptic operators
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