Carleman estimates for second order operators of real principal type

Carleman estimates for second order
operators of real principal type
under weak pseudo convexity
Hideki TAKUWA
(Doshisha University, Kyoto, Japan)
Seminar: Control theory in Lab. J.-L. Lions,
UPMC, Paris, 7 Nov 2014.
Joint work with
• David Dos Santos Ferreira (Université de Lorraine,
France)
• Jérôme Le Rousseau (Université d’Orléans, France)
Plan of the talk:
1. Carleman estimates and pseudo convexity
2. Weak pseudo convexity for the operator of second order
and real principal type
3. Limiting Carleman weight in Calderón problem
4. Main result
5. Idea and sketch of proof
(Note: The slides are rearanged after the talk.)
1 Carleman estimates and pseudo
convexity
In x ∈ Rn , let P = P (x, D) =
X
aα (x)Dα be a
|α|≤m
differential operator of order mX
whose principal symbol is
C ∞ function p = pm (x, ξ) =
aα (x)ξ α .
|α|=m
We use the notation Dxj = −i∂xj for 1 ≤ j ≤ n. For a
large parameter λ and a real valued function ϕ = ϕ(x),
we want to have the estimate
X
λ2(m−|β|)−1
|β|≤m−1
≤C
Z
e2λϕ |Dβ u|2 dx
Z
e2λϕ |P u|2 dx
(1)
for u ∈ C0∞ (Ω), where Ω is a neighborhood of a point x0 .
For two functions f = f (x, ξ) and g = g(x, ξ), Poisson
bracket between f and g is defined by
n ³
X
∂f ∂g
∂f ∂g ´
{f, g} =
−
.
∂ξ
∂x
∂x
∂ξ
j
j
j
j
j=1
The condition to get Carleman estimates above is called
pseudo convexity for the function ϕ that appears in the
estimate (1).
The hypersurface S = {x ∈ Rn | ψ(x) = ψ(x0 )} which is
defined by the function ψ is called strongly pseudo convex
(at x0 ) if
Re{p, {p, ψ}}(x, ξ) > 0
(2)
on the subset on the cotangent bundle T ∗ Rn
NR ={(x, ξ) ∈ T ∗ Rn \0 |
p(x, ξ) = {p, ψ}(x, ξ) = 0},
(3)
and
1
{p(x, ξ − iλdψ), p(x, ξ + iλdψ)} > 0
2iλ
for ξ ∈ Rn , λ > 0
(4)
on the set
NC = {(x, ξ) ∈ T ∗ Rn \0 | p(x, ξ + iλdψ)
= {p(x, ξ + iλdψ), ψ}(x, ξ) = 0}. (5)
X
aα (x)Dα is
The differential operator P = P (x, D) =
|α|≤m
m−1
called principally normal if |{p, p}| ≤ C|ξ|
|p| for x ∈ Ω,
ξ ∈ Rn Elliptic operators and operators with real principal
symbols are principally normal.
For the function ψ which define the hypersurface S, we set
X 1
ψε (x) =
∂ α ψ(x0 )(x − x0 )α − ε|x − x0 |2 ,
α!
(6)
|α|≤2
ϕ(x) = exp tψε (x)
(7)
for sufficiently small positive ε and sufficiently large t.
Under the strongly pseudo convexity of S defined by ψ we
have the uniqueness of the solutions by proving the
Carleman estimates with the weight ϕ.
Theorem 1.1 Let P be the operator
Xof principally normal
aα (x)ξ α . Let the
defiend by the symbol pm (x, ξ) =
|α|=m
hypersurface S is strictly convex at x0 , that is, we have (2)
and (4) on (3) and (5), respectively. Then there exist
positive constants λ0 > 0, C > 0 and a neighborhood Ω of
x0 such that we have the inequality (1) for λ ≥ λ0 .
The details of this result can be seen in Hörmander [3] and
Zuily [16].
Note: I found the nice lecture note written by N. Lerner
(UPMC) in his home page.
Roughly speaking, the information of the phase function is
considered through the operator
eλϕ P (x, D)e−λϕ = P (x, D + iλdϕ). The meaning of the
conditions about the strict pseudo convexity is that we
have the positivity of the symbol
(|ξ|2 + λ2 )m−1 ≤ C1 |p(x, ξ + iλdϕ)|2
C2
+
{p(x, ξ − iλdϕ), p(x, ξ + iλdϕ)} + C3 r(x, ξ, λ)
2iλ
where r = r(x, ξ, λ) is a small real symbol of in some sense
if P is of principally normal. It follows from the positivity of
the symbol above and the sharp Gårding inequality in the
theory of pseudo differential operators that Carleman
estimate (1) will be obtained.
2 Weak pseudo convexity for the
operator of second order and real
principal type
We restrict our attention for the operators of second order
and of real principal type. So m = 2 and the coefficients
aα are real valued smooth functions for |α| = 2. The
operator P is of principal type if
∇ξ p(x, ξ) 6= 0 on p(x, ξ) = 0, ξ 6= 0.
Since p(x, ξ) is of second order and satisfies
p(x, tξ) = t2 p(x, ξ) for t ∈ R, we have
(8)
n
X
∂p
ξk
(x, ξ) = 2p(x, ξ), so we have ∇ξ p(x, ξ) 6= 0 for
∂ξk
k=1
all (x, ξ) with ξ 6= 0. In our result the positivity of the
symbol
n ¯
¯2
X
∂p
¯
¯
2
|ξ| ≤ C
¯
¯
∂ξk
k=1
is useful. Let the hypersurface
S = {x ∈ R | ψ(x) = ψ(x0 )} be noncharacteristic for the
operator P = P (x, D), we have p(x, ∇ψ) 6= 0. It is not so
difficult to check that one of the conditions about strongly
pseudo convexity, that is (4), is automatically satisfied if
the hypersurface S is noncharacteristic for the operator of
second order with real principal symbol.
We repeat the classical result about Carleman estimates for
the operators of second order and real principal type.
Theorem 2.1 Let P = P (x, D) =
p = p(x, ξ) =
n
X
n
X
ajk (x)Dj Dk and
j,k=1
ajk (x)ξj ξk . Assume that the operator
j,k=1
P is of principal type, that is, ∇ξ p(x, ξ) 6= 0 when
p(x, ξ) = 0 and ξ 6= 0. Assume that the hypersurface
S = {x ∈ Rn | ψ(x) = ψ(x0 )} is noncharactersistic,
p(x, ∇ψ) 6= 0, and strongly pseudo convex
{p, {p, ψ}}(x, ξ) > 0
(9)
on the set
NR = {(x, ξ) ∈ T ∗ Rn \0 | p(x, ξ) = {p, ψ}(x, ξ) = 0}.
(10)
Set ϕ = ϕ(x) from the function ψ = ψ(x) as above. Then
there exist positive constants λ0 > 0, C > 0 and a
neighborhood Ω of x0 such that we have
Z
λ3
Z
n
X
e2λϕ |u|2 dx +
λ e2λϕ |Dj u|2 dx
j=1
≤C
for λ ≥ λ0 and u ∈ C0∞ (Ω).
Z
e2λϕ |P u|2 dx
(11)
Once we have the inequality (11), we can add the lower
order terms freely as
n
n
X
X
ajk (x)Dj Dk +
P = P (x, D) =
bj (x)Dj + c0 (x)
j,k=1
j=1
by replacing λ0 with more large one. Theorem 1.2 is stable
under first order perturbations.
When the condition (9) is degererate as
{p, {p, ϕ}}(x, ξ) ≥ 0
(12)
on NR , it is very difficult to get the Carleman estimate
(11). In fact, we need to impose more conditions on the
operators and hypersurface S or to hope weak Carleman
estimates.
Example (wave operator with conctant coefficients)
In R3 near the origin (0, 0, 0) we study
P (D) = Dx21 − Dx22 − Dx23 with symbol
p(ξ) = ξ12 − ξ22 − ξ32 . Since we have
p(ξ) = ξ12 − ξ22 − ξ32 ,
∂ψ
∂ψ
∂ψ
{p(ξ), ψ(x)} = 2ξ1
− 2ξ2
− 2ξ3
,
∂x1
∂x2
∂x3
³ ∂2ψ
2
2 ´
∂
ψ
∂
ψ
2
2
2
{p, {p, ψ}} = 4 ξ1 2 + ξ2 2 + ξ3 2 ,
∂x1
∂x2
∂x3
ψ1 (x) = x1 and ψ2 (x) = x21 + x22 + x3 are strictly pseudo
convex, but ψ3 (x) = x3 is not!
Historically, Lerner and Robbiano [8] studied the problem
about unique continuation under weak pseudo convexity.
This property is called compact uniqueness or Cauchy
compact. After the result [8] by Lerner and Robbiano,
Hörmander [3] in his book arranged the proof by Lerner
and Robbiano and got the new Carleman estimate under
weak pseudo convexity (12) on NR . We review his
Carleman estimate. Let P = P (x, D) = D12 − R(x, D0 ),
where x = (x1 , x0 ) ∈ R × Rn−1 and D0 = (D2 , · · · , Dn )
and ψ = ψ(x1 ) = x1 . This restriction looks the loss of
generality. But we can reduce the general case into this
1 2
form at least locally. For the function ϕ(x1 ) = x1 + x1
2
and sufficiently small ε > 0 we have for λ ≥ λ0 and
u ∈ C0∞ (Ω0 )
Z
Z
λ3 e2λϕ |u|2 dx + λ e2λϕ |D1 u|2 dx
+
Z
n
X
Z
e2λϕ |Dj u|2 dx ≤ C
e2λϕ |P (εx, D)u|2 dx.
j=2
(13)
We shall compare two inequalities (11) and (13). Smallness
of the parameter ε > 0 means that the support of the
function u should be very small and the operator P (x, D)
looks like as the operator with constant coefficients.
3 Limiting Carleman weight in Calderón
problem
In the theory of inverse problems, Calderón problem is well
known. This is the inverse problem that we determine the
information inside as potentials or coefficients in the
equations from the maps like the measurements on the
boundary. There the special solutions which are called the
complex geometrical optics solutions have important roles
in many approaches.
For the linear partial differential operator P (x, D) of order
2, we want to construct the solutions u = u(x; h) with
complex phase function Φ(x) = ϕ(x) + iψ(x) ∈ C ∞ (Ω, C)
of the form
u(x; h) = e
1
h (ϕ(x)+iψ(x))
(a0 (x) + hr(x, h)),
where h is a small positive parameter.
We can give a simple and important example for this
approach. Set a, b ∈ Rn with |a| = |b| and ha, bi = 0
n
X
∂ 2 ha+ib,xi
e
= 0.
2
∂x
j
j=1
First we shall construct the phase function
Φ(x) = ϕ(x) + iψ(x).
(14)
The complex eikonal equation is obtained as
0 =p(x, d(ϕ + iψ))
n
³ ∂ϕ
´³ ∂ϕ
´
X
∂ψ
∂ψ
g jk (x)
=
+i
+i
.
∂xj
∂xj
∂xk
∂xk
j,k=1
Two real valued functions ϕ and ψ should satisfy the
system of the nonlinear equations of first order
n
X
g
jk
j,k=1
n
X
j,k=1
¡
¢
∂xj ϕ(x)∂xk ϕ(x) − ∂xj ψ(x)∂xk ψ(x) = 0,
g jk (x)∂xj ϕ(x)∂xk ψ(x) = 0.
Once the function ϕ is fixed, the symbols a(x, ξ) and
b(x, ξ) are defined by
a(x, ξ) = −
b(x, ξ) =
n
X
j,k=1
n
X
¡
¢
g ∂xj ϕ(x)∂xk ϕ(x) − ξj ξk
jk
g jk (x)∂xj ϕ(x)ξk
j,k=1
For the function ϕ we want to construct ψ as the solution
to
a(x, dψ) = 0,
b(x, dψ) = 0.
(15)
We review this theory from the point of the pseudo
Riemann geometry. Set two symbols a(x, ξ) and b(x, ξ) of
real valued from p(x, ξ + idϕ(x)) = a(x, ξ) + ib(x, ξ) where
n
X
p(x, ζ) =
g jk (x)ζj ζk for x ∈ Rn and
j,k=1
ζ = (ζ1 , · · · , ζn ) ∈ Cn . We define the notion of limiting
Carleman weights.
Definition 3.1 A real valued function ϕ in Rn is said to
be a limiting Carleman weight if dϕ 6= 0 and it satisfy a
kind of Hörmander subelliptic condition
{pϕ , pϕ } = 0 when pϕ = 0,
(16)
where pϕ = a(x, ξ) + ib(x, ξ).
The symbols a(x, ξ) and b(x, ξ) are defined by the pseudo
distance and pseudo inner product on T ∗ Rn as
a(x, ξ) = |ξ|2g − |dϕ|2g ,
for dϕ =
n
X
(∂xj ϕ)dxj , ξ =
j=1
n
X
b(x, ξ) = 2hdϕ, ξi
ξj dxj . The condition of
j=1
the limiting Carleman weight means that the manifold
J = {(x, ξ) ∈ T ∗ Rn | a(x, ξ) = b(x, ξ) = 0} is involutive,
that is,
{a, b}(x, ξ) = 0
on J = {(x, ξ) ∈ T ∗ Rn | a(x, ξ) = b(x, ξ) = 0}.
We shall check this condition from pseudo convexity. For
the function ϕ with p(x, dϕ) = 1 on Ω, we have
a(x, ξ) = p(x, ξ) − 1,
b(x, ξ) = {p, ϕ}(x, ξ),
that is {a, b}(x, ξ) = {p, {p, ϕ}}(x, ξ).
Carleman estimate for the limiting Carleman weights had
obtained by Sylvester-Uhlmann [12] for the linear phase.
Kenig, Sjöstrand and Uhlmann [6] used the new limiting
Carleman weight like ϕ(x) = log |x| for the flat Laplacian.
Laplace Beltrami oparator, that is the second order elliptic
operator with real symmetric coefficients, has been studied
after their result. We can see them in [1]. They based on
elliptic Carleman estimate. In this talk we want to study
non-elliptic operators.
4 Main result
4.1 Nonlocal Carleman estimate under weak
pseudo convexity
We give the main result for the operator of second order
and real principal type.
Theorem 4.1 (Ferreira-Rousseau-Takuwa) Let
x = (x1 , x0 ), (x0 = (x2 , · · · , xn )). For the second order
operator
P (x, D)
2
=Dx1
− R(x, Dx0 )
=Dx21 −
n
X
j,k=2
Dxj (g jk (x)Dxk )
with coefficients g jk of real valued, g jk = g kj and
det(g jk ) 6= 0. Set the principal symbol
n
X
r(x, ξ 0 ) =
g jk (x)ξj ξk for the operator R(x, Dx0 ).
j,k=2
Assume that the operator P (x, D) is of principal type, that
is ∇ξ0 r(x, ξ) 6= 0 for ξ 0 6= 0. We also assume the weak
pseudo convexity
∂r
(x, ξ 0 ) ≥ 0
∂x1
when r(x, ξ 0 ) = 0.
(17)
Assume |∂xl g jk (x)| are small in Ω. (We do not give pricise
assumption here.)
Then there exists λ0 > 0 such that we have
Z
Z
λ3
e2λϕ(x) |u|2 dx + λ
e2λϕ(x) |Dx1 u|2 dx
Ω
Ω
Z
+
e2λϕ(x) |Dx0 u|2 dx
Ω
Z
2λϕ(x)
2
2
0
≤C
e
|{Dx1 − R(x, Dx )}u| dx,
(18)
Ω
where ϕ = ϕ(x1 ) = x1 for λ ≥ λ0 and u ∈ C0∞ (Ω).
Note: Non selfadjoint form can be treated in our proof:
R(x, Dx0 ) =
n
X
j,k=2
g jk (x)Dxj Dxk .
L. Hörmander [3] and N. Lerner-L. Robbiano [8] had
studied Carleman estimates that were obtained only for a
sufficiently small neighborhood Ω. Because of weakness of
the convexity of the weight functions. It is very delicate to
have Carleman type estimates. Theorem 4.1 shows the new
Carleman estimate without the restriction of the size of the
domain Ω. In fact, we can consider that Theorem 4.1 is the
improvement of the original Carleman estimates proved by
L. Hörmander for operators of second order real principal
type.
5 Idea and sketch of proof
Step 1: Convexification of the phase function ϕ(x1 ) = x1
¡
¢
1 2
ϕ
eδ (x1 ) = fδ ϕ(x1 ) = x1 + δx1 ,
2
1 2
1
fδ (s) = s + δs , 0 ≤ δ ≤ .
2
2
µ
By setting δ = , where µ > 0 is small enough, the
λ
commutators
[e
2
1
δx
1
2
, D1 ],
[e
2
1
δx
1
2
, D12 ],
[e
2
1
δx
1
2
, R(x, Dx0 )] = 0
are nice terms (by the boundedness in x1 ).
Note:
• ϕ
eδ (x1 ) is still weak pseudo convex phase function.
• The new parameter δ > 0 is introduced to control the
size of the support of u ∈ C0∞ (Ω).
2
1
δx
• Thanks to the large parameter λ, U = e 2 1 and its
inverse U −1 are nice bounded operator on the weighted
space of L2 and H 1 with parameter in (18).
Step 2: Scalling argument
For ε1 > 0, we set the new variables.
Ψε1 : R × Rn−1 → R × Rn−1 as (x1 , x0 ) 7→ (y1 , y 0 ) by
y1 = ε11 x1 , y 0 = x0 . In the final step of the proof the small
parameter ε1 > 0 will be fixed.
We study
1 e
I=I=
3
ε1
where
Ie =
Z
Z
=
e
Ω
Ω
e2λϕeδ (x1 ) |{Dx21 − R(x1 , x0 , Dx0 )}u|2 dx
eϕ
2λ
eδ (y1 )
e
e
Ω
Z
|{Dy21 − ε21 R(ε1 y1 , y 0 , Dy0 )}e
u|2 dy
|T 2 − ε21 R(ε1 y1 , y 0 , Dy0 )}e
v |2 dy
=||Pλ ve||2L2 (Ω)
e ,
and
Pλ =T 2 − ε21 R(ε1 y1 , y 0 , Dy0 ),
e + δy
e 1 ),
T =T (y1 , D1 ) = Dy1 + iλ(1
e =ε1 λ,
λ
δe = ε1 δ,
n
o
M
1
n
e
Ω =Ψε1 (Ω) ⊂ y ∈ R | |y1 | ≤
, |y 0 | ≤ M2 .
ε1
Step 3: Commutator argument between Pλ and Pλ∗
∗
2
∗
||Pλ ve||2L2 (Ω)
−
||P
v
e
||
=
h[P
,
P
v , veiL2 (Ω)
λ
e
λ
λ ]e
e
e
L2 (Ω)
e where
for ve ∈ C0∞ (Ω),
eδ(T
e T ∗ + T ∗ T ) − ε2 J1 + ε4 [R∗ , R]
[Pλ , Pλ∗ ] =4λ
1
1
∗ 2
∗
J1 =[(T ) , R] − [T , R ].
2
By following the argument by Hörmander [3], we can show
the estimate as below.
Lemma 5.1
³
´
eδe ||T ve||2 + ||T ∗ ve||2 + ||P ∗ ve||2
4λ
λ
D
E
∂r
3e
e
+ 4ε1 λ (1 + δy1 )
(ε1 y1 , y 0 , Dy0 )e
v , ve
∂x1
≤ ||Pλ ve|| +
2
where
(1)
|hJ1 ve, vei|
(2)
2
ε1 hJ1 ve, vei
¡
+ ε41 hJ2 ve, vei,
¢
≤ Cε1 ||T ve|| + ||T ve|| ||e
v ||(0,1)
∗
|hJ2 ve, vei| ≤ C||e
v ||2(0,1) ,
and ||e
v ||(0,1) = ||e
v ||L2 +
n
X
j=2
||Dyj ve||L2 .
Step 4: Weak pseudo convexity
Special case: {p, {p, x1 }} = 0 (Limiting Carleman weight)
The symbol r(x1 , x0 , ξ 0 ) is independent of x1 . We have
D
E
∂r
0
e
(1 + δy1 )
(ε1 y1 , y , Dy0 )e
v , ve = 0.
∂x1
Gereral case: {p, {p, x1 }} ≥ 0 (weak pseudo convexity)
The term as above becomes small in a sense by the
construction of the symbols and the sharp Gårding
inequality. Here we have used pseudo differential operators.
(We skip this part today.)
Step 5: Estimate for ||e
v || and ||Dy1 ve||
It is not so difficult to get the estimates
³
´
e2 ||e
λ
v ||2 ≤ 2 ||T ve||2 + ||T ∗ ve||2
e2 ||Dy ve||2 ≤ ||Pλ ve||2 + ||P ∗ ve||2
λ
λ
1
e2 δe2 ||e
v ||2 + ε61 ||e
v ||2(0,1) .
+ 4λ
Step 6: Estimate for ||Dyj ve|| (2 ≤ j ≤ n)
Instead of the pseudo convexity, we control ||Dyj ve|| in y 0
direction by the principal type condition. To make our idea
clear, we review the idea of Hörmander’s calculation in [3].
Original approach If we have a time, we give a explanation.
Our approach If we have a time, we give a explanation.
In the talk we had to skip this part. The original
Hörmander’s calculation in [3] needed to make |x0 | small
enough. So we introduce the new commutator argument in
our result.
Combining the 6 steps, we have obtained Theorem 4.1.
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