Carleman estimates for elliptic PDE and applications Sylvain - - PowerPoint PPT Presentation
Cauchy Linear Weight General Weights Carleman estimates for elliptic PDE and applications Sylvain Ervedoza Institut de Mathmatiques de Toulouse & CNRS Monastir - Mai 2017 Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE
Cauchy Linear Weight General Weights
Sylvain Ervedoza
Institut de Mathématiques de Toulouse & CNRS
Monastir - Mai 2017
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights
Carleman estimates for elliptic PDE and applications Cauchy Problem for the Laplace operator. Carleman estimates with a linear weight. Applications to the Calderón problem. Carleman estimates with a general weight : the role of strict pseudo-convexity. Applications to the unique continuation property for the Laplace operator.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
1
The Cauchy problem for the Laplace operator The case of full informations The case of partial data Intermediate case
2
Carleman estimates with a linear weight Goal Proof of the Carleman estimate : Fourier techniques Proof of the Carleman estimate : Multiplier techniques More general geometric settings Application to the Calderón Problem
3
More general Carleman Weights The case of a strip The case of a strip with a multiplier technique The general case More on unique continuation
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
The Cauchy Problem for the Laplace operator We look for u solution of ∆u = f , in Ω, u = gD,
∂nu = gN,
Known data of the problem : The geometric setting, the domain Ω and a part Γ of ∂Ω where the Dirichlet and Neumann conditions are known. f source term in Ω. gD Dirichlet data on Γ. gN Neumann data on Γ. Physically (Electronic) : u is a potential, ∇u is a current.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
The Cauchy problem, Case Γ = ∂Ω We look for u solution of ∆u = f , in Ω, u = gD,
∂nu = gN,
When Γ = ∂Ω, the information on the boundary is redundant : Using classical elliptic theory, one can solve ∆u = f , in Ω, u = gD,
OR ∆u = f , in Ω, ∂nu = gN,
yielding u ∈ H2(Ω) for f ∈ L2(Ω) and gD ∈ H3/2(∂Ω) f ∈ L2(Ω) and gN ∈ H1/2(∂Ω), in the case
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
The Cauchy problem, Case Γ = ∂Ω We look for u solution of ∆u = f , in Ω, u = gD,
∂nu = gN,
Consequences (When Γ = ∂Ω) : (f , gD) completely determines u, thus gN. (f , gN) completely determines u, thus gD. A correct functional setting is given by f ∈ L2(Ω), gD ∈ H3/2(∂Ω), gN ∈ H1/2(∂Ω), u ∈ H2(Ω).
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
The Cauchy Problem for the Laplace operator We look for u solution of ∆u = f , in Ω, u = gD,
∂nu = gN,
Known data of the problem : The geometric setting, the domain Ω and a part Γ of ∂Ω where the Dirichlet and Neumann conditions are known. f source term in Ω. gD Dirichlet data on Γ. gN Neumann data on Γ.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
The Cauchy Problem for the Laplace operator We look for u solution of ∆u = f , in Ω, u = gD,
∂nu = gN,
To simplify the study, we will restrict ourselves to the following simplified geometric setting : Ω = (0, 1) × Rd−1 is a vertical strip (d ≥ 1). Γ = {0} × Rd−1 is one side of the strip.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
We look for u solution of ∆u = f , in Ω, u = gD,
∂nu = gN,
i.e. ∂11u + ∆′u = f , for (x1, x′) ∈ (0, 1) × Rd−1, u(0, x′) = gD(x′), for x′ ∈ Rd−1, ∂1u(0, x′) = gN(x′), for x′ ∈ Rd−1, where ∆′ = ∂22 + · · · + ∂dd is the Laplace operator in the variable x′ = (x2, · · · , xd) ∈ Rd−1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
We introduce ˆ u(x1, ξ′) = Fx′→ξ′u(x1, ·), ˆ f (x1, ξ′) = Fx′→ξ′f (x1, ·), ˆ gD(ξ′) = Fx′→ξ′gD(·), ˆ gN(ξ′) = Fx′→ξ′gN(·), where Fx′→ξ′ is the Fourier transform in the variable x′, defined for function v ∈ S (Rd−1) of x′ by ∀ξ′ ∈ Rd−1, (Fx′→ξ′v)(ξ′) =
Fx′→ξ′ enjoys the same properties as the usual Fourier transform : It is an isometry of L2(Rd−1). Its action on derivatives is easily understood : Fx′→ξ′(∆x′v)(ξ′) = −|ξ′|2(Fx′→ξ′v)(ξ′).
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
Therefore, the Cauchy problem ∂11u + ∆′u = f , for (x1, x′) ∈ (0, 1) × Rd−1, u(0, x′) = gD(x′), for x′ ∈ Rd−1, ∂1u(0, x′) = gN(x′), for x′ ∈ Rd−1, now reads ∂11 ˆ u − |ξ′|2 ˆ u = ˆ f , for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ u(0, ξ′) = ˆ gD(ξ′), for ξ′ ∈ Rd−1, ∂1 ˆ u(0, ξ′) = ˆ gN(ξ′), for ξ′ ∈ Rd−1, Advantages : We are back to analyze a family of ODE indexed by the frequency parameter ξ′ ∈ Rd−1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
At ξ′ ∈ Rd−1 fixed, one can solve ∂11 ˆ u − |ξ′|2 ˆ u = ˆ f , for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ u(0, ξ′) = ˆ gD(ξ′), for ξ′ ∈ Rd−1, ∂1 ˆ u(0, ξ′) = ˆ gN(ξ′), for ξ′ ∈ Rd−1, using Duhamel formula : ˆ u(x1, ξ′) = cosh(|ξ′|x1)ˆ gD(ξ′) + sinh(|ξ′|x1) |ξ′| ˆ gN(ξ′) + x1 sinh(|ξ′|(x1 − x)) |ξ′| ˆ f (x, ξ′) dx. As the Fourier transform is invertible and this identity holds ∀ξ′, the Cauchy problem is solved. We even have a fomula to give the solution !
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
At ξ′ ∈ Rd−1 fixed, one can solve ∂11 ˆ u − |ξ′|2 ˆ u = ˆ f , for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ u(0, ξ′) = ˆ gD(ξ′), for ξ′ ∈ Rd−1, ∂1 ˆ u(0, ξ′) = ˆ gN(ξ′), for ξ′ ∈ Rd−1, using Duhamel formula : ˆ u(x1, ξ′) = cosh(|ξ′|x1)ˆ gD(ξ′) + sinh(|ξ′|x1) |ξ′| ˆ gN(ξ′) + x1 sinh(|ξ′|(x1 − x)) |ξ′| ˆ f (x, ξ′) dx. As the Fourier transform is invertible and this identity holds ∀ξ′, the Cauchy problem is solved. We even have a fomula to give the solution ! This last sentence is partially wrong !
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
What we have proved : If we know that there exists u solving ∆u = f , in Ω, u = gD,
∂nu = gN,
then it is unique, and it is given by the formula ˆ u(x1, ξ′) = cosh(|ξ′|x1)ˆ gD(ξ′) + sinh(|ξ′|x1) |ξ′| ˆ gN(ξ′) + x1 sinh(|ξ′|(x1 − x)) |ξ′| ˆ f (x, ξ′) dx. However, we never said that such u exist !
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
In particular, the formula ˆ u(x1, ξ′) = cosh(|ξ′|x1)ˆ gD(ξ′) + sinh(|ξ′|x1) |ξ′| ˆ gN(ξ′) + x1 sinh(|ξ′|(x1 − x)) |ξ′| ˆ f (x, ξ′) dx. may not belong to S′(Rd−1) and therefore, the inverse Fourier transform may not exist. We only have : ∃C > 0, s.t. ∀x1 ∈ (0, 1), |ˆ u(x1, ξ′)| ≤ C exp(|ξ′|x1)
gD(ξ′)| + |ˆ gN(ξ′)| + x1 |ˆ f (x, ξ′)| dx
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
We thus obtain 1
u(x1, ξ′)|2 exp(−2|ξ′|x1) dξ′dx1 ≤ C
L2(Rd−1) + gN2 L2(Rd−1) + f 2 L2((0,1)×Rd−1)
The left hand-side defines a norm on u but weaker than any norm
This is a prototype setting in which the linear mapping u ∈ H2(Ω) → (f , gD, gN) ∈ L2(Ω) × H3/2(Γ) × H1/2(Γ) is well-defined, injective, but its inverse is not continuous for reasonable topologies. Therefore, the Cauchy problem is ill-posed in the Hadamard sense.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
We briefly presented the case of Full informations Γ = ∂Ω. Partial informations Γ ⊂ ∂Ω. What is in between ? ∂11u + ∆′u = f , for (x1, x′) ∈ (0, 1) × Rd−1, u(0, x′) = gD(x′), for x′ ∈ Rd−1, ∂1u(0, x′) = gN(x′), for x′ ∈ Rd−1, u(1, x′) = g1(x′), for x′ ∈ Rd−1. In this case, one can solve ∆u = f in Ω with full Dirichlet data, hence gN is fully determined by gD, g1. Question Can we give a better formula than the one in the case of partial informations ?
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
Same strategy as before : After Fourier transform, ∂11 ˆ u − |ξ′|2 ˆ u = ˆ f , for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ u(0, ξ′) = ˆ gD(ξ′), for ξ′ ∈ Rd−1, ∂1 ˆ u(0, ξ′) = ˆ gN(ξ′), for ξ′ ∈ Rd−1, ˆ u(1, ξ′) = ˆ g1(ξ′), for ξ′ ∈ Rd−1. Again, family of ODE of order 2 with 3 boundary conditions !.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
We decompose the operator ∂11 − |ξ′|2 = (∂1 − |ξ′|)(∂1 + |ξ′|) = (∂1 + |ξ′|)(∂1 − |ξ′|). Therefore, one can rewrite ∂11 ˆ u − |ξ′|2 ˆ u = ˆ f , for x1 ∈ (0, 1), as (∂1 − |ξ′|)ˆ u = ˆ v, for x1 ∈ (0, 1), (∂1 + |ξ′|)ˆ v = ˆ f , for x1 ∈ (0, 1), OR (∂1 + |ξ′|)ˆ u = ˆ w, for x1 ∈ (0, 1), (∂1 − |ξ′|) ˆ w = ˆ f , for x1 ∈ (0, 1). Is there one decomposition better than the other ? Which one ?
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
How to chose the correct decomposition of the operator ? With the boundary conditions : We have more informations (2 boundary conditions) on the left point x1 = {0} than on the right end point (1 boundary condition). On the other hand, if one considers the ODE (∂1 + a)z = 0 in (0, 1), then we can get the two following formulas, z(x1) = z(0) exp(−ax1), OR z(x1) = z(1) exp(a(1 − x1)). In particular, If a > 0, the first formula is better (more stable) than the second one. Information comes from the left. If a < 0, the first formula is worst (less stable) than the second
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
We therefore choose to write (∂1 − |ξ′|)ˆ u = ˆ v, for x1 ∈ (0, 1), (∂1 + |ξ′|)ˆ v = ˆ f , for x1 ∈ (0, 1), ˆ u(0, ξ′) = ˆ gD(ξ′), , ∂1 ˆ u(0, ξ′) = ˆ gN(ξ′), ˆ u(1, ξ′) = ˆ g1(ξ′), that we solve in two steps :
v + |ξ′|ˆ v = ˆ f for x1 ∈ (0, 1), ˆ v(0, ξ′) = ˆ gN(ξ′) − |ξ′|ˆ gD(ξ′), then ∂1 ˆ u − |ξ′|ˆ u = ˆ v for x1 ∈ (0, 1), ˆ u(0, ξ′) = ˆ gD(ξ′), ˆ u(1, ξ′) = ˆ g1(ξ′).
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
Solving
v + |ξ′|ˆ v = ˆ f for x1 ∈ (0, 1), ˆ v(0, ξ′) = ˆ gN(ξ′) − |ξ′|ˆ gD(ξ′), we find ˆ v(x1, ξ′) = exp(−|ξ′|x1)ˆ v(0, ξ′)+ x1 exp
ˆ f (x, ξ′) dx, Solving ∂1 ˆ u − |ξ′|ˆ u = ˆ v for x1 ∈ (0, 1), ˆ u(0, ξ′) = ˆ gD(ξ′), ˆ u(1, ξ′) = ˆ g1(ξ′). we find ˆ u(x1, ξ′) = e−|ξ′|(1−x)ˆ g1(ξ′) − 1
x1
exp
v(x, ξ′) dx.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
This leads ˆ u(x1, ξ′) =e−|ξ′|(1−x)ˆ g1(ξ′) − ˆ v(0, ξ′) 1
x1
exp
− 1 ˆ f (˜ x, ξ′) 1
max{x1,˜ x}
exp(−|ξ′|(2x − x1 − ˜ x)) dx
x, with ˆ v(0, ξ′) = ˆ gN(ξ′) − |ξ′|ˆ gD(ξ′). Explicit formula with only decaying exponentials. We can provide estimates from the above formula in reasonable norms.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
For some C independent of ξ′,
v(0, ξ′) 1
x1
exp
≤ C 1 + |ξ′|3/2 |ˆ v(0, ξ′)|, and
ˆ f (˜ x, ξ′) 1
max{x1,˜ x}
exp(−|ξ′|(2x − x1 − ˜ x)) dx
x
≤ C 1 + |ξ′|
|ˆ f (˜ x, ξ′)| exp(−|ξ′||x1 − ˜ x|) d ˜ x
≤ C 1 + |ξ′|
f (x1, ξ′)1x1∈(0,1) ∗x1 exp(−|ξ′||x1|)1x1∈(−2,2)
≤ C 1 + |ξ′|
f (·, ξ′)
≤ C 1 + |ξ′|2
f (·, ξ′)
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
For C independent of ξ′, (1+|ξ′|4)
u(·, ξ′)
L2(0,1) ≤ C(1+|ξ′|)3/2|ˆ
g1(ξ′)|+C(1+|ξ′|)|ˆ v(0, ξ′)|2 + C
f (·, ξ′)
L2(0,1) .
Integrating in ξ′ ∈ Rd−1 and using Parseval’s identity u2
L2(0,1;H2(Rd−1)) ≤ C f 2 L2((0,1)×Rd−1) + C g12 H3/2(Rd−1)
+ C
v(0, ξ′)|2 dξ′. Using ∂11u = f − ∆′u ∈ L2((0, 1) × Rd−1), u2
H2((0,1)×Rd−1)) ≤ C f 2 L2((0,1)×Rd−1) + C g12 H3/2(Rd−1)
+ C
v(0, ξ′)|2 dξ′.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
From the explicit form of ˆ v(0, ξ′) = ˆ gN(ξ′) − |ξ′|ˆ gD(ξ′)
v(0, ξ′)|2 dξ′ ≤ C gD2
H3/2(Rd−1)+C gN2 H1/2(Rd−1) .
We thus conclude uH2((0,1)×Rd−1) ≤ C f L2((0,1)×Rd−1) + C g1H3/2(Rd−1) + C gDH3/2(Rd−1) + C gNH1/2(Rd−1) . (1)
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
Note that we also have, from the well-posedness of the elliptic problem with Dirichlet boundary conditions that uH2((0,1)×Rd−1) ≤ C f L2((0,1)×Rd−1) + C gDH3/2(Rd−1) , which seems stronger than what we proved. In fact, this estimate can also be proved along the same lines of the one above. But more importantly, the strategy developed to prove the estimate (1) can be adapted to prove Carleman estimates.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Full Info Partial Info Intermediate
The formula ˆ u(x1, ξ′) =ˆ g1(ξ′)e−|ξ′|(1−x1) − (ˆ gN(ξ′) − |ξ′|ˆ gD(ξ′)) 1
x1
exp
− 1 ˆ f (˜ x, ξ′) 1
max{x1,˜ x}
exp(−|ξ′|(2x − x1 − ˜ x)) dx
x, is thus “stable”. However, it does not provide easily the property f = 0 in Ω gD = 0 on Γ, gN = 0 on Γ ⇒ u = 0 in Ω. Carleman estimates.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
1
The Cauchy problem for the Laplace operator The case of full informations The case of partial data Intermediate case
2
Carleman estimates with a linear weight Goal Proof of the Carleman estimate : Fourier techniques Proof of the Carleman estimate : Multiplier techniques More general geometric settings Application to the Calderón Problem
3
More general Carleman Weights The case of a strip The case of a strip with a multiplier technique The general case More on unique continuation
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
The goal of this section is to get a stable estimate for solutions of ∆u = f , in Ω, u = gD,
∂nu = gN,
allowing to prove that f = 0 in Ω gD = 0 on Γ, gN = 0 on Γ, ⇒ u = 0 in Ω, even when Γ = ∂Ω. The strategy then consists in mixing the previous computations, by considering norms which allow a linear exponential growth.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We will first focus on the case of a strip Ω = (0, 1) × Rd−1, Γ = {0} × Rd−1, with homogeneous Dirichlet boundary conditions to simplify the presentation : ∂11u + ∆′u = f , for (x1, x′) ∈ (0, 1) × Rd−1, u(0, x′) = u(1, x′) = 0, for x′ ∈ Rd−1, ∂1u(0, x′) = gN(x′), for x′ ∈ Rd−1. (2) A Carleman estimate ∃C > 0, s.t. for all k ≥ 1, any solution u of (2) with f ∈ L2(Ω) and gN ∈ H1/2(Γ) satisfies k
≤ C
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Theorem ∃C > 0, s.t. ∀k ≥ 1, any solution u of (2) with f ∈ L2(Ω) and gN ∈ H1/2(Γ) satisfies k
≤ C
(3) Estimate (3) is a Carleman estimate with linear weight : Weighted norms appear containing exponential terms. Free parameter k ≥ 1, which can be made arbitrarily large. C does not depend on k. The weight e−kx1 is larger on the side on which the measurements are done.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Carleman estimate for a Laplace operator with potential Let q ∈ L∞((0, 1) × Rd−1). There exists a constant C > 0 such that for all k ≥ 1, any solution u ∈ L2((0, 1) × Rd−1) of ∂11u + ∆′u + qu = f , for (x1, x′) ∈ (0, 1) × Rd−1, u(0, x′) = u(1, x′) = 0, for x′ ∈ Rd−1, ∂1u(0, x′) = gN(x′), for x′ ∈ Rd−1, with source term f ∈ L2((0, 1) × Rd−1) and Neumann data gN ∈ L2(Rd−1) satisfies (3).
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Apply the Carleman estimate (3) to u with f replaced by f − qu : ∃C > 0, ∀k ≥ 1, k
≤ C
Using
+ qL∞((0,1)×Rd−1)
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Hence, taking k ≥ kq = 2C qL∞((0,1)×Rd−1), k
≤ 2C
Now, ∃C > 0 s.t. ∀k ≥ 1, k
≤ 2C
as the case k ∈ [1, kq] can be handled by immediate bounds on the estimate for k = kq.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Corollary Let q ∈ L∞((0, 1) × Rd−1), and u ∈ L2((0, 1) × Rd−1) be the solution of ∂11u + ∆′u + qu = f , for (x1, x′) ∈ (0, 1) × Rd−1, u(0, x′) = 0, for x′ ∈ Rd−1, ∂1u(0, x′) = gN(x′), for x′ ∈ Rd−1, with u(1, x′) ∈ H3/2(Rd−1) and Neumann data gN = 0 and source term f ∈ L2((0, 1) × Rd−1) satisfying, for some a ∈ (0, 1), f (x1, x′) = 0 for x1 ∈ (0, a), x′ ∈ Rd−1. Then u vanishes in (0, a) × Rd−1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
If u(1, x′) = 0, apply the Carleman estimate to u : for all k ≥ 1, k
But on one hand, k
On the other hand, as f vanishes in (0, a) × Rd−1,
Using the Carleman estimate then yields, for all k ≥ 1, k uL2((0,a)×Rd−1) ≤ C f L2((0,1)×Rd−1) . Taking the limit k → ∞, u necessarily vanishes in (0, a) × Rd−1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
If u(1, x′) ∈ H3/2(Rd−1), u ∈ H2(Rd−1). We therefore set v(x) = η(x1)u(x), in Ω, where η = η(x1) is a smooth cut-off function taking value 1 on (0, a), vanishing in x1 = 1. Then v solves ∂11v + ∆′v + qv = ηf − [∂11, η]u, for (x1, x′) ∈ (0, 1) × Rd−1, v(0, x′) = v(1, x′) = 0, for x′ ∈ Rd−1, ∂1v(0, x′) = 0, for x′ ∈ Rd−1, Thus, by the previous case, v(x1, x′) = 0 for x1 ∈ (0, a), i.e. u(x1, x′) = 0 for (x1, x′) ∈ (0, a) × Rd−1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Theorem ∃C > 0, s.t. ∀k ≥ 1, any solution u of ∂11u + ∆′u = f , for (x1, x′) ∈ (0, 1) × Rd−1, u(0, x′) = u(1, x′) = 0, for x′ ∈ Rd−1, ∂1u(0, x′) = gN(x′), for x′ ∈ Rd−1, with f ∈ L2(Ω) and gN ∈ H1/2(Rd−1) satisfies k
≤ C
Two proofs : By Fourier techniques. By multiplier techniques.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Since we have to prove estimates on ue−kx1 in terms of fe−kx1, it is convenient to work on the conjugate variables : U(x1, x′) = u(x1, x′)e−kx1, F(x1, x′) = f (x1, x′)e−kx1, for (x1, x′) ∈ (0, 1) × Rd−1. The equation on u then rewrites in terms of U, as U(x1, x′) = ekx1u(x1, x′) : ∂11U + 2k∂1U + k2U + ∆′U = F, for (x1, x′) ∈ (0, 1) × Rd−1, U(0, x′) = U(1, x′) = 0, for x′ ∈ Rd−1, ∂1U(0, x′) = gN(x′), for x′ ∈ Rd−1,
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Similarly as before, we take the partial Fourier transform in the x′-variable : ˆ U(x1, ξ′) = Fx′→ξ′U(x1, ·), ˆ F(x1, ξ′) = Fx′→ξ′F(x1, ·), ˆ gN(ξ′) = Fx′→ξ′gN(·). We obtain a family of ODE indexed by ξ′ : ∂11 ˆ U + 2k∂1 ˆ U + k2 ˆ U − |ξ′|2 ˆ U = ˆ F, for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ U(0, ξ′) = ˆ U(0, ξ′) = 0, for ξ′ ∈ Rd−1, ∂1 ˆ U(0, x′) = ˆ gN(ξ′), for ξ′ ∈ Rd−1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
The operator factorized as follows : ∂11 + 2k∂1 + k2 − |ξ′|2 = (∂1 + k + |ξ′|)(∂1 + k − |ξ′). Therefore, we introduce the function ˆ V = (∂1 + k − |ξ|) ˆ U, (x1, ξ′) ∈ (0, 1) × Rd−1. Back to ∂1 ˆ U + (k − |ξ′|) ˆ U = ˆ V for (x1, ξ′) ∈ (0, 1) × Rd−1, ∂1 ˆ V + (k + |ξ′|) ˆ V = ˆ F for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0 for ξ′ ∈ Rd−1, ˆ V (0, ξ′) = ˆ gN(ξ′) for ξ′ ∈ Rd−1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
∂1 ˆ U + (k − |ξ′|) ˆ U = ˆ V for (x1, ξ′) ∈ (0, 1) × Rd−1, ∂1 ˆ V + (k + |ξ′|) ˆ V = ˆ F for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0 for ξ′ ∈ Rd−1, ˆ V (0, ξ′) = ˆ gN(ξ′) for ξ′ ∈ Rd−1. For fixed ξ′ ∈ Rd−1, we can then solve this system in two steps :
1 Compute ˆ
V in terms of ˆ F and of ˆ gN : ∂1 ˆ V + (k + |ξ′|) ˆ V = ˆ F for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ V (0, ξ′) = ˆ gN(ξ′) for ξ′ ∈ Rd−1.
2 Compute ˆ
U in terms of ˆ V : ∂1 ˆ U + (k − |ξ′|) ˆ U = ˆ V for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0 for ξ′ ∈ Rd−1,
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
The computation of ˆ V is straightforward : ∂1 ˆ V + (k + |ξ′|) ˆ V = ˆ F for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ V (0, ξ′) = ˆ gN(ξ′) for ξ′ ∈ Rd−1. yields : ˆ V (x1, ξ′) = e−(k+|ξ′|)x1 ˆ gN(ξ′)+ x1 exp(−(k+|ξ′|)(x1−x)) ˆ F(x, ξ′) dx. The computation of ˆ U also is straightforward, as it solves : ∂1 ˆ U + (k − |ξ′|) ˆ U = ˆ V for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0 for ξ′ ∈ Rd−1, But now we have two possible formulae, depending whether we use the boundary condition at x1 = 0 or at x1 = 1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
∂1 ˆ U + (k − |ξ′|) ˆ U = ˆ V for (x1, ξ′) ∈ (0, 1) × Rd−1, ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0 for ξ′ ∈ Rd−1, Two formulae : ˆ U(x1, ξ′) = x1 exp(−(k − |ξ′|)(x1 − x)) ˆ V (x, ξ′) dx, ˆ U(x1, ξ′) = − 1
x1
exp(−(k − |ξ′|)(x1 − x)) ˆ V (x, ξ′) dx. We have to do a choice : The first formula for |ξ′| ≤ k, Low frequency case. The second formula for |ξ′| ≥ k, High frequency case.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Here, we use the first formula : ˆ U(x1, ξ′) = ˆ gN(ξ′) exp(−(k − |ξ′|)x1) x1 exp(−2|ξ′|x) dx + x1 ˆ F(˜ x, ξ′)e−k(x1−˜
x)
x1
˜ x
e|ξ′|(˜
x+x1−2x) dx
x. Direct estimates :
gN(ξ′) exp(−(k − |ξ′|)x1) x1 exp(−2|ξ′|x) dx
≤ C|ˆ gN(ξ′)|
1 + |ξ′| 1 1 + (k − |ξ′|)1/2
C k1/2 |ˆ gN(ξ′)|.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
On the other hand, writing x1 ˆ F(˜ x, ξ′)e−k(x1−˜
x)
x1
˜ x
e|ξ′|(˜
x+x1−2x) dx
x = 1 ˆ F(˜ x, ξ′)H(x1 − ˜ x, ξ′) d ˜ x with H(X, ξ′) = 1X>0e−(k−|ξ′|)X X e−2|ξ′|x dx, we obtain
ˆ F(˜ x, ξ′)e−k(x1−˜
x)
x1
˜ x
e|ξ′|(˜
x+x1−2x) dx
x
≤ C
F(·, ξ′)
≤ C
F(·, ξ′)
1 + k − |ξ′| 1 1 + |ξ′|
k
F(·, ξ′)
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We thus get, for ξ′ ∈ Rd−1 with |ξ′| ≤ k,
U(x1, ξ′)
C k1/2 |ˆ gN(ξ′)| + C k
F(·, ξ′)
i.e. ∀ξ′ ∈ Rd−1 with |ξ′| ≤ k k2
U(x1, ξ′)
L2(0,1) ≤ Ck|ˆ
gN(ξ′)|2 + C
F(·, ξ′)
L2(0,1) ,
with C independent of ξ′ and k.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Here, we use the other formula : ˆ U(x1, ξ′) = − ˆ gN(ξ′)e−(k−|ξ′|)x1 1
x1
e−2|ξ′|x dx − 1 ˆ F(˜ x, ξ′)e−k(x1−˜
x)e−|ξ′||x1−˜ x|
1−max{˜
x,x1}
e−2|ξ′|x dx
gN(ξ′)e−(k−|ξ′|)x1 1
x1
e−2|ξ′|x dx
≤ C 1 + |ξ′|
gN(ξ′)e−(k+|ξ′|)x1
≤ C 1 + |ξ′||ˆ gN(ξ′)| 1 (k + |ξ′|)1/2 ≤ C k3/2 |ˆ gN(ξ′)|.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
The second term can be bounded as follows :
1 ˆ F(˜ x, ξ′)e−k(x1−˜
x)e−|ξ′||x1−˜ x|
1−max{˜
x,x1}
e−2|ξ′|x dx
x
≤
| ˆ F(˜ x, ξ′)|e−k(x1−˜
x)e−|ξ′||x1−˜ x|
1−max{˜
x,x1}
e−2|ξ′|x dx
x
≤
| ˆ F(˜ x, ξ′)|e(k−|ξ′|)|x1−˜
x|
∞ e−2|ξ′|x dx
x
≤ C |ξ′|
F|(x1, ξ′)1x1∈(0,1) ∗x1 e(k−|ξ′|)|x1|
≤ C |ξ′|
F(·, ξ′)
≤ C |ξ′|(1 − k + |ξ′|)
k
F(·, ξ′)
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We thus get, for ξ′ ∈ Rd−1 with |ξ′| ≥ k,
U(x1, ξ′)
C k1/2 |ˆ gN(ξ′)| + C k
F(·, ξ′)
i.e. ∀ξ′ ∈ Rd−1 with |ξ′| ≥ k k2
U(x1, ξ′)
L2(0,1) ≤ Ck|ˆ
gN(ξ′)|2 + C
F(·, ξ′)
L2(0,1) ,
with C independent of ξ′ and k. This is the same estimate as the one obtained for ξ′ ∈ Rd−1 with |ξ′| ≤ k.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
As the estimates obtained for |ξ′| ≤ k and for |ξ′| ≥ k are the same, ∃C > 0, such that ∀ξ′ ∈ Rd−1, k2
U(x1, ξ′)
L2(0,1) ≤ Ck|ˆ
gN(ξ′)|2 + C
F(·, ξ′)
L2(0,1) .
Integrating this estimate with respect to ξ′ ∈ Rd−1 and using Parseval’s identity : k2 U2
L2((0,1)×Rd−1) ≤ Ck gN2 L2(Rd−1) + C F2 L2((0,1)×Rd−1) .
i.e. k2
L2((0,1)×Rd−1) ≤ Ck gN2 L2(Rd−1)+C
L2((0,1)×Rd−1) .
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
In order to estimate ∇ue−kx1 in L2(Ω), we multiply the equation ∆u = f , for (x1, x′) ∈ (0, 1) × Rd−1, u(0, x′) = u(1, x′) = 0, for x′ ∈ Rd−1, by ue−2kx1 : −
=
so that
L2((0,1)×Rd−1) ≤ k2
L2((0,1)×Rd−1)
+
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We can thus use the proved estimate k2
L2((0,1)×Rd−1) ≤ Ck gN2 L2(Rd−1)+C
L2((0,1)×Rd−1) .
in
L2((0,1)×Rd−1) ≤ k2
L2((0,1)×Rd−1)
+
We therefore get
L2((0,1)×Rd−1) ≤ Ck gN2 L2(Rd−1)+C
L2((0,1)×Rd−1) .
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
To sum up : This proof yields a formula to get u in terms of the known quantities. For low frequency |ξ′| ≤ k, the information comes from the left. For high frequency |ξ′| ≥ k, the information comes from the left and from the right. Once ue−kx1 is estimated, a weighted type regularity estimate give an estimate on ∇ue−kx1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
U = ue−kx1 solves the equation ∂11U + 2k∂1U + k2U + ∆′U = F, for (x1, x′) ∈ (0, 1) × Rd−1, U(0, x′) = U(1, x′) = 0, for x′ ∈ Rd−1, ∂1U(0, x′) = gN(x′), for x′ ∈ Rd−1, Multiplying the equation by ∂1U we easily get 2k ∂1U2
L2((0,1)×Rd−1) =
+ 1 2
2
so that k ∂1U2
L2((0,1)×Rd−1) ≤ C
k F2
L2((0,1)×Rd−1) + C gN2 L2(Rd−1) .
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
But U satisfies homogeneous boundary conditions at x1 = 0 and x1 = 1 Poincaré’s inequality applies : U2
L2((0,1)×Rd−1) ≤ C ∂1U2 L2((0,1)×Rd−1) .
Thus, k2 U2
L2((0,1)×Rd−1) ≤ C F2 L2((0,1)×Rd−1) + Ck gN2 L2(Rd−1) .
Remains to estimate ∇U.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Multiplying ∂11U + 2k∂1U + k2U + ∆′U = F, for (x1, x′) ∈ (0, 1) × Rd−1, U(0, x′) = U(1, x′) = 0, for x′ ∈ Rd−1, ∂1U(0, x′) = gN(x′), for x′ ∈ Rd−1, by U, we obtain
L2((0,1)×Rd−1) − k2 U2 L2((0,1)×Rd−1)
As U is already suitably estimated, ∇U2
L2((0,1)×Rd−1) ≤ Ck2 U2 L2((0,1)×Rd−1) + C F2 L2((0,1)×Rd−1) .
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We thus have k2 U2
L2((0,1)×Rd−1) + ∇U2 L2((0,1)×Rd−1)
≤ C F2
L2((0,1)×Rd−1) + Ck gN2 L2(Rd−1) .
Using then that u = Uekx1, |u|e−kx1 ≤ |U|, |∇u|e−kx1 ≤ |∇U| + k|U|. Therefore, k2
L2((0,1)×Rd−1) +
L2((0,1)×Rd−1)
≤ C
L2((0,1)×Rd−1) + Ck gN2 L2(Rd−1) .
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
This second proof is much shorter than the first one. However, It is not very precise on the boundary terms. It seems to strongly use the Hilbertian property of L2. In particular, the first strategy is more precise and can handled : Non-homogeneous Dirichlet boundary conditions in H3/2(∂Ω) and Neumann boundary data in H1/2(Γ). Lp settings. Still, the second strategy generalizes easily in more general geometric setting....
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Theorem Let Ω be a smooth (C 2) bounded domain of Rd. ∃C > 0 s.t. ∀k ∈ Rd with |k| ≥ 1, any solution u ∈ L2(Ω) of ∆u = f , for x ∈ Ω u(x) = 0, for x ∈ ∂Ω, with source term f ∈ L2(Ω) satisfies |k|2
L2(Ω) +
L2(Ω) +
1 |k|2
H2(Ω)
≤ C|k|
L2(Γk) + C
L2(Ω) ,
where Γk = {x ∈ ∂Ω, | k · nx < 0}.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Setting U = ue−k·x and F = fe−k·x, U satisfies ∆U + 2k · ∇U + |k|2U = F for x ∈ Ω, U(x) = 0, for x ∈ ∂Ω. Multiplying the equation by U, we first derive ∇U2
L2(Ω) ≤ C|k|2 U2 L2(Ω) + C F2 L2(Ω) .
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Multiplying ∆U + 2k · ∇U + |k|2U = F for x ∈ Ω, U(x) = 0, for x ∈ ∂Ω. by k · ∇U, we derive
Fk · ∇U dx = 2
|k · ∇U|2 dx +
∂nUk · ∇U dσ − 1 2
k · nx|∇U|2 dσ = 2
|k · ∇U|2 dx + 1 2
k · nx|∂nU|2 dσ, where we used that, as U = 0 on ∂Ω, ∇U = (∂nU)nx on ∂Ω. Therefore, ∃C independent of k ∈ Rd s.t.
|k · ∇U|2 dx ≤ C
|F|2 dx + C |k|
|∂nU|2 dσ.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
But Ω is bounded, so by Poincaré’s inequality |k|2 U2
L2(Ω) ≤ C
|k · ∇U|2 dx. Thus, ∀k ∈ Rd, |k|2 U2
L2(Ω) ≤ C F2 L2(Ω) + C |k| ∂nU2 L2(Γk) .
Based on the estimate of ∇U in terms of U, we thus derive |k|2 U2
L2(Ω) + ∇U2 L2(Ω) ≤ C F2 L2(Ω) + C |k| ∂nU2 L2(Γk) .
Recalling that ue−k·x = U and ∇u e−k·x = ∇U + kU, we immediately conclude the Carleman estimate. The H2-norm of ue−k·x comes from the equation satisfied by U.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Proof based on the multiplier technique developed in the case of the strip. Mainly the same proof as in the case of the strip. A similar proof based on a Fourier decomposition technique would be much more intricate. Note that the observation set Γk = {x ∈ ∂Ω, | k · nx < 0} depends
Geometry of the domain and the observation part Γ of the boundary and of the wave functions are linked.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Calderón problem (or Electrical Impedance Tomography) This corresponds to a medical imaging technique which consists in the recovery of the conductivity of a tissue (or a material) by applying currents on the surface on the body and measuring the electrical potentials on the surface of the body. Let Ω be a bounded domain of Rd, and consider the elliptic problem div (σ∇u) = 0, for x ∈ Ω, u(x) = gd(x), for x ∈ ∂Ω. Here, σ = σ(x) is a scalar function modeling the conductivity of the material, σ is unknown.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Mathematical Formulation Recover σ knowing the map Λσ : gd → σ∂nu, where u denotes the solution div (σ∇u) = 0, for x ∈ Ω, u(x) = gd(x), for x ∈ ∂Ω. Physically : Λσ is the so-called Voltage-to-current map. gd is a voltage imposed on the boundary of the object. We can measure the current (σ∇u) · nx on the boundary ∂Ω. Mathematically : Λσ is the Dirichlet to Neumann map.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Calderón Problem Can we determine the conductivity of the material σ from the knowledge on the Voltage-to-Current map ? div (σ∇u) = 0, for x ∈ Ω, u(x) = gd(x), for x ∈ ∂Ω. Assumption 1 : σ ∈ C 0(Ω), ∃C∗ > 0, s.t. ∀x ∈ Ω, 1 C∗ ≤ σ(x) ≤ C∗. If gd ∈ H1/2(∂Ω) yields u ∈ H1(Ω) and (σ∇u) · nx = σ∂nu ∈ H−1/2(∂Ω). Λσ : H1/2(∂Ω) → H−1/2(∂Ω).
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
The Calderón problem is an inverse problem. Subproblems : Uniqueness : If Λσ1 = Λσ2, can we deduce σ1 = σ2 ? Stability : If Λσ1 − Λσ2 is small (in suitable norms), can we deduce that σ1 − σ2 is small (in suitable norms) ? Reconstruction : Given Λσ, can we compute σ ? In the following, we will focus on the uniqueness problem, i.e. on the injectivity of the map Λ : σ → Λσ.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Main difficulty The map Λ is non-linear. In here, we shall further assume the following : σ and ∂nσ are known on the boundary. √σ belongs to C 2(Ω). Under these assumptions, we can transform the Calderón problem in the recovery of a potential.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
If u solves div (σ∇u) = 0, for x ∈ Ω, u(x) = gd(x), for x ∈ ∂Ω, Then v = σ1/2u in Ω, solves ∆v + qv = 0, for x ∈ Ω, v(x) = hd(x), for x ∈ ∂Ω. with q= −∆(σ1/2) σ1/2 , in Ω hd = σ1/2gd, ∂nv = σ−1/2(σ∂nu) + ∂n(σ1/2)gd, on ∂Ω.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We therefore introduce the map ˜ Λ : q ∈ L∞(Ω) → ˜ Λq, where ˜ Λq : hd ∈ H1/2(∂Ω) → ∂nv ∈ H−1/2(∂Ω), where ∂nv is given by ∆v + qv = 0, for x ∈ Ω, v(x) = hd(x), for x ∈ ∂Ω. Claim Under the previous assumptions on σ, Λσ1 = Λσ2 is equivalent to ˜ Λq1 = ˜ Λq2, where q1 = −∆(σ1/2
1
) σ1/2
1
, q2 = −∆(σ1/2
2
) σ1/2
2
.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Λ : q ∈ L∞(Ω) → Λq, where Λq : hd ∈ H1/2(∂Ω) → ∂nv ∈ H−1/2(∂Ω), where ∂nv is given by ∆v + qv = 0, for x ∈ Ω, v(x) = hd(x), for x ∈ ∂Ω. Theorem Assume d ≥ 3. q1, q2 ∈ L∞(Ω), Λq1 = Λq2 ⇒ q1 = q2.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Proposition Given q ∈ L∞(Ω), the map Λq is self-adjoint on H1/2(∂Ω). i.e. ∀h1, h2 ∈ H1/2(∂Ω), Λqh1, h2H−1/2(∂Ω),H1/2(∂Ω) = h1, Λqh2H1/2(∂Ω),H−1/2(∂Ω),
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Let q ∈ L∞(Ω) and, for h1 and h2 in H1/2(∂Ω), set ∆v1 + qv1 = 0, for x ∈ Ω, v1(x) = h1(x), for x ∈ ∂Ω, ∆v2 + qv2 = 0, for x ∈ Ω, v2(x) = h2(x), for x ∈ ∂Ω. Thus, Λqh1, h2H−1/2(∂Ω),H1/2(∂Ω) = ∂nv1, v2H−1/2(∂Ω),H1/2(∂Ω) =
∆v1v2 dx +
∇v1 · ∇v2 dx = −
qv1v2 dx +
∇v1 · ∇v2 dx = h1, Λqh2H1/2(∂Ω),H−1/2(∂Ω).
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Proposition Let q1 and q2 in L∞(Ω). Then for all h1, h2 in H1/2(∂Ω), the solutions v1, v2 of ∆v1 + q1v1 = 0, in Ω, v1(x) = h1(x),
∆v2 + q2v2 = 0, in Ω, v2(x) = h2(x),
satisfy (Λq1 − Λq2)h1, h2H−1/2(∂Ω),H1/2(∂Ω) =
(q2 − q1)v1v2 dx. In particular, if Λq1 = Λq2, for all v1 and v2 as above,
(q2 − q1)v1v2 dx = 0.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We compute Λq1h1, h2H−1/2(∂Ω),H1/2(∂Ω) = ∂nv1, v2H−1/2(∂Ω),H1/2(∂Ω) =
∆v1v2 dx +
∇v1 · ∇v2 dx = −
q1v1v2 dx +
∇v1 · ∇v2 dx. Similar computations yield h1, Λq2h2H1/2(∂Ω),H−1/2(∂Ω) = −
q2v1v2 dx +
∇v1 · ∇v2 dx. As Λq2 is self-adjoint, h1, Λq2h2H1/2(∂Ω),H−1/2(∂Ω) = Λq2h1, h2H−1/2(∂Ω),H1/2(∂Ω). Subtraction of the two above identities gives the result.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
In particular, if Λq1 = Λq2, for all v1 and v2 as before,
(q2 − q1)v1v2 dx = 0. Therefore, our next goal is to generate a dense set of functions of the form v1v2, where ∆v1 + q1v1 = 0, in Ω, v1(x) = h1(x),
∆v2 + q2v2 = 0, in Ω, v2(x) = h2(x),
Therefore, our goal is to find function v1v2 which approximates the Fourier basis x → exp(−iξx).
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Remark If ρ ∈ Cd satisfies ρ · ρ = 0, then ∆(eρ·x) = 0 in Rd. There are many such ρ ∈ Cd ! If ρ = a + ib, with a, b ∈ Rd, ρ · ρ = 0 ⇔ |a| = |b|, a · b = 0. NB : If ρ · ρ = 0 and ρ ∈ Rd, then ρ = 0 of course !
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Theorem Let q ∈ L∞(Ω). For all ρ = a + ib ∈ Cd with ρ · ρ = 0 and |a| ≥ 1, there exists a solution vρ ∈ L2(Ω) of ∆vρ + qvρ = 0 in Ω, that can be written as vρ(x) = eρ·x + ea·xrρ(x), with rρ satisfying the estimate rρL2(Ω) ≤ C |a|. In other words, vρ(x) ≃ eρ·x when |a| → ∞.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Remark that vρ(x) = eρ·x + ea·xrρ(x) solves ∆vρ + qvρ = 0 iff e−a·x(∆ + q)(ea·xrρ) = −qeib·x. We set ˜ q = −qeib·x. To be proved ∃rρ ∈ L2(Ω) solution of e−a·x(∆ + q)(ea·xrρ) = ˜ q in Ω, with rρL2(Ω) ≤ C |a|. Here, the difficult part is the estimate on rρ.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
The function rρ solves e−a·x(∆ + q)(ea·xrρ) = ˜ q iff for all w ∈ D(Ω),
rρ
˜ qw dx,
0(Ω).
Consequence The set {rρ solving e−a·x(∆ + q)(ea·xrρ) = ˜ q} is an affine space of direction {ea·x(∆ + q)(e−a·xw), w ∈ H2
0(Ω)}⊥L2(Ω).
The function rρ of minimal L2(Ω) norm : rρ ∈ {ea·x(∆ + q)(e−a·xw), w ∈ H2
0(Ω)}.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We claim that {ea·x(∆ + q)(e−a·xw), w ∈ H2
0(Ω)}
= {ea·x(∆ + q)(e−a·xw), w ∈ H2
0(Ω)}.
Indeed, ea·x(∆ + q)(e−a·xw) = ∆w − 2a · ∇w + |a|2w + qw, so the Carleman estimate applied to e−axw yields |a|2 w2
L2(Ω) +
1 |a|2 w2
H2(Ω) ≤ C
L2(Ω) .
In particular, for all |a| ≥ 1,
0(Ω) .
hence the closedness of the above vector space.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Therefore, the solution rρ of minimal L2 norm writes, for some W ∈ H2
0(Ω),
rρ = ea·x(∆ + q)(e−a·xW ), and satisfies, for all w ∈ H2
0(Ω), for all w ∈ D(Ω),
rρ
˜ qw dx, We take w = W :
|ea·x(∆ + q)(e−a·xW )|2 dx =
˜ qW dx≤ ˜ qL2 W L2.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We got
|ea·x(∆ + q)(e−a·xW )|2 dx ≤ ˜ qL2 W L2 , while the Carleman estimate implies, for all |a| ≥ 1 and w ∈ H2
0(Ω)
|a|2 w2
L2(Ω) ≤ C
L2(Ω) .
Consequently, rρL2(Ω) =
|ea·x(∆ + q)(e−a·xW )|2 dx 1/2 ≤ C ˜ qL∞ |a| ≤ C qL∞ |a| .
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Recall if Λq1 = Λq2, for all v1 and v2 with ∆v1 + q1v1 = 0, in Ω, ∆v2 + q2v2 = 0, in Ω, we have
(q2 − q1)v1v2 dx = 0. For each ξ ∈ Rd, we will chose (v1,n)n∈N ≃ (eρ1,n·x)n∈N, (v2,n)n∈N ≃ (eρ2,n·x)n∈N, with ρ1,n · ρ1,n = 0, limn→∞ |ρ1,n| = ∞, ρ2,n · ρ2,n = 0, limn→∞ |ρ2,n| = ∞, ρ1,n + ρ2,n = −iξ.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
We want : ρ1,n · ρ1,n = 0, limn→∞ |ρ1,n| = ∞, ρ2,n · ρ2,n = 0, limn→∞ |ρ2,n| = ∞, ρ1,n + ρ2,n = −iξ. As d ≥ 3, we start by choosing α, β ∈ Rd such that |α| = |β| = 1 and α · β = β · ξ = α · ξ = 0. We then set, for n large enough, ρ1,n = nα + i
2
ρ2,n = −nα − i
2
where γ2
n = n2 − |ξ|2
4 .
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
With the constructed ρ1,n, ρ2,n, we obtain v1,n(x) = eρ1,n·x + r1,nenα·x, v2,n(x) = eρ2,n·x + r2,ne−nα·x. with r1,nL2(Ω) ≤ C q1L∞ |n| , r2,nL2(Ω) ≤ C q2L∞ |n| . Therefore, as Λq1 = Λq2,
(q2 − q1)e−iξx dx = −
(q2−q1)
Taking the limit n → ∞,
(q2 − q1)e−iξx dx = 0.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Therefore, ∀ξ ∈ Rd,
(q2 − q1)e−iξx dx = 0. ⇒ q1 = q2 in Ω. Our result is proved : Λq1 = Λq2 ⇒ q1 = q2.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Here, we studied the Calderón problem for a scalar conductivity σ with the knowledge of the full Dirichlet to Neumann map.
can be solved provided there exists suitably “Limiting Carleman Weights”. [Kenig Sjöstrand Uhlmann 2007]
corresponding to anisotropic materials. But there are counterexamples in this case ! ! [Tartar].
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
In the case of anisotropic conductivities, Tartar proposed a simple construction to show that uniqueness for the Calderón problem cannot hold. Indeed, given any C 2 diffeomorphism Ψ on Ω with Ψ = Id on the boundary ∂Ω, then Λ˜
σ = Λσ,
where ˜ σ(y) = DΨT(Ψ−1(y)) × σ(Ψ−1(y)) × DΨ(Ψ−1(y)) |det (DΨ(Ψ−1(y)))|
This shows in particular that one cannot distinguish between σ and ˜ σ when allowing anisotropic conductivities.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon
Actually, this counterexample is the basis of several recent works on invisibility (cloaking), see [Uhlmann 2009]. The idea is to construct diffeomorphisms that approximate the singular transformation between B(0, 2) \ {0} and B(0, 2) \ B(0, 1) given by Ψ(x) = x |x|
2
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
1
The Cauchy problem for the Laplace operator The case of full informations The case of partial data Intermediate case
2
Carleman estimates with a linear weight Goal Proof of the Carleman estimate : Fourier techniques Proof of the Carleman estimate : Multiplier techniques More general geometric settings Application to the Calderón Problem
3
More general Carleman Weights The case of a strip The case of a strip with a multiplier technique The general case More on unique continuation
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
The goal of this section is to make precise what weight functions can be used in Carleman estimates. In a strip : what weight functions ϕ(x1) ? ∃C > 0, s.t. ∀s ≥ 1, ∀u ∈ H2 ∩ H1
0(Ω),
s# uesϕL2((0,1)×Rd−1) ≤ C ∆uesϕL2((0,1)×Rd−1) + Cs∗
In general geometries : Specific issue : Can we made the observation set arbitrarily small ?
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Ω = (0, 1) × Rd−1, Γ = {0} × Rd−1. Our goal is to understand for which functions ϕ = ϕ(x1) one can get an estimate of the form ∃C > 0, s.t. ∀s ≥ 1, ∀u ∈ H2 ∩ H1
0(Ω),
s# uesϕL2((0,1)×Rd−1) ≤ C ∆uesϕL2((0,1)×Rd−1) + Cs∗
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
We start from ∂11u + ∆′u = f , for (x1, x′) ∈ (0, 1) × Rd−1, u(0, x′) = u(1, x′) = 0, for x′ ∈ Rd−1, ∂1u(0, x′) = gN(x′), for x′ ∈ Rd−1, and we set U(x1, x′) = u(x1, x′)esϕ(x1), F(x1, x′) = f (x1, x′)esϕ(x1), Gn(x′) = gn(x′)esϕ(0). The equations now read ∂11U − 2s∂1ϕ∂1U + (s2|∂1ϕ|2 − s∂11ϕ)U + ∆′U = F, in Ω, U(0, x′) = U(1, x′) = 0, in Rd−1, ∂1U(0, x′) = GN(x′), in Rd−1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
After partial Fourier transform, ∂11 ˆ U − 2s∂1ϕ∂1 ˆ U + (s2|∂1ϕ|2 − s∂11ϕ)U − |ξ′|2 ˆ U = ˆ F, in (0, 1) × ˆ U(0, ξ′) = ˆ U(0, ξ′) = 0, in Rd−1, ∂1 ˆ U(0, ξ′) = ˆ GN(ξ′), in Rd−1, We then recognize that ∂11 − 2s∂1ϕ∂1 + (s2|∂1ϕ|2 − s∂11ϕ) − |ξ′|2 = (∂1 − s∂1ϕ)2 − |ξ′|2 = (∂1 − s∂1ϕ − |ξ′|)(∂1 − s∂1ϕ + |ξ′|) = (∂1 − s∂1ϕ + |ξ′|)(∂1 − s∂1ϕ − |ξ′|).
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
To see what are the needed conditions on ϕ, note that we have to consider the two ODE with operators (∂1−s∂1ϕ − |ξ′|), (∂1−s∂1ϕ + |ξ′|). Claim The sign of −s∂1ϕ − |ξ′| and of −s∂1ϕ + |ξ′| indicates how the information propagates.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
We are considering operators of the form : (∂1−s∂1ϕ − |ξ′|), (∂1−s∂1ϕ + |ξ′|). Therefore, for all ξ′ ∈ Rd−1, we want to be able to drive the information from the left to the right. ⇒ To handle the case ξ′ = 0, we need inf{−s∂1ϕ} > 0, i.e. max
x1∈[0,1] ∂1ϕ(x1) < 0.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Thus, to solve ∂11 ˆ U − 2s∂1ϕ∂1 ˆ U + (s2|∂1ϕ|2 − s∂11ϕ)U − |ξ′|2 ˆ U = ˆ F, in (0, 1), ˆ U(0, ξ′) = ˆ U(0, ξ′) = 0, ∂1 ˆ U(0, ξ′) = ˆ GN(ξ′), we introduce ˆ V (x1, ξ′) = (∂1−s∂1ϕ − |ξ′|) ˆ U(x1, ξ′) : ∂1 ˆ U + (−s∂1ϕ − |ξ′|) ˆ U = ˆ V in (0, 1) ∂1 ˆ V + (−s∂1ϕ + |ξ′|) ˆ V = ˆ F in (0, 1) ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0, ˆ V (0, ξ′) = ˆ GN(ξ′). No problem to solve ˆ V . What happens for ˆ U ? ∂1 ˆ U + (−s∂1ϕ − |ξ′|) ˆ U = ˆ V in (0, 1) ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0,
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
(∂1+a)z = f in (0, 1), z(0) = z(1) = 0, Then, using a primitive A of a, i.e. A′ = a, we have two formulas : z(x1) = x1 eA(˜
x)−A(x1)f (˜
x)d ˜ x, = − 1
x1
eA(˜
x)−A(x1)f (˜
x)d ˜ x. Recall the goal : Have bounds without exponential factors. If inf[0,1] a > 0, information propagates from the left. If sup[0,1] a < 0, information propagates from the right. What happens otherwise, i.e. if a cancels in [0, 1] ?
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
We focus on the strict cases, in which a changes sign. We have A′ = a and for each x1 ∈ (0, 1), either ∀x < x1, A(x) < A(x1) or ∀x > x1, A(x) < A(x1). We assumed that a vanishes in some point x∗ ∈ [0, 1] and changes
A′(x∗) = 0 and A attains a strict local maximum in x∗. There cannot exist other strict local maximizers ˜ x of A. Otherwise, ∃xm between ˜ x and x∗ which is a local minimizer, and the above condition is violated at x1 = xm. The condition is equivalent to ∀x < x1 < x∗, A(x) < A(x1)(< A(x∗)), ∀x > x1 > x∗, A(x) < A(x1)(< A(x∗)),
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Recall A′ = a and A′(x∗) = 0. The condition reads ∀x < x1 < x∗, A(x) < A(x1)(< A(x∗)), ∀x > x1 > x∗, A(x) < A(x1)(< A(x∗)), Therefore, one should have ∀x1 < x∗, a(x1) > a(x∗) = 0, ∀x1 > x∗, a(x1) < a(x∗) = 0.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
We consider ∂1 ˆ U + (−s∂1ϕ − |ξ′|) ˆ U = ˆ V in (0, 1) ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0, corresponding to a(x1, ξ′) = −s∂1ϕ(x1) − |ξ′|. For being suitable for Carleman estimates, we require, for all ξ′ ∈ Rd−1, that if ∃x∗,ξ such that a(x∗,ξ, ξ′) = 0, then ∀x1 < x∗,ξ, a(x1, ξ′) > a(x∗,ξ, ξ′) = 0, ∀x1 > x∗,ξ, a(x1, ξ′) < a(x∗,ξ, ξ′) = 0. Claim This latter condition is equivalent to a convexity condition on ϕ.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
a(x1, ξ′) = −s∂1ϕ(x1) − |ξ′|. Condition ∀(x∗, ξ′) ∈ Rd−1 with a(x∗, ξ′) = 0, ∀x1 < x∗, a(x1, ξ′) > a(x∗, ξ′) = 0, ∀x1 > x∗, a(x1, ξ′) < a(x∗, ξ′) = 0. Obviously, this condition is satisfied if −s∂1ϕ is strictly decreasing, i.e. if inf[0,1] ∂11ϕ > 0. Reciprocally, this condition implies inf[0,1] ∂11ϕ > 0 : ∀x∗ ∈ (0, 1), ∃ξ′ ∈ Rd−1 s.t. a(x∗, ξ′) = 0. Thus, ∀x1 < x∗, a(x1, ξ′) > 0, i.e. ∂1ϕ(x1) < ∂1ϕ(x∗).
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Ω = (0, 1) × Rd−1, Γ = {0} × Rd−1. Conditions on ϕ = ϕ(x1) ϕ = ϕ(x1) is smooth, max
[0,1]{∂1ϕ} < 0 and min [0,1]{∂11ϕ} > 0.
Theorem ∃C > 0, s.t. ∀s ≥ 1, ∀u ∈ H2 ∩ H1
0(Ω),
s3/2 uesϕL2((0,1)×Rd−1) + s1/2 ∇uesϕL2((0,1)×Rd−1) ≤ C ∆uesϕL2((0,1)×Rd−1) + Cs1/2
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
With a linear weight, we got : ∀k ≥ 1, k
≤ C
With a strictly convex weight ϕ, ∀s ≥ 1, s3/2 uesϕL2((0,1)×Rd−1) + s1/2 ∇uesϕL2((0,1)×Rd−1) ≤ C ∆uesϕL2((0,1)×Rd−1) + Cs1/2
Not the same powers of the parameter ! Note that a linear weight is not strictly convex.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
In both cases, we first solve ∂1 ˆ V + (−s∂1ϕ + |ξ′|) ˆ V = ˆ F in (0, 1) ˆ V (0, ξ′) = ˆ GN(ξ′). so that there exists a C > 0 independent of s and ξ′ such that s
V (·, ξ′)
GN(ξ′)| + C
F(·, ξ′)
Only uses max{∂1ϕ} < 0. So it applies also when sϕ(x1) = −sx1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
We then have to solve ∂1 ˆ U + (−s∂1ϕ − |ξ′|) ˆ U = ˆ V in (0, 1) ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0, In the context of linear weights, this equation is ∂1 ˆ U + (k − |ξ′|) ˆ U = ˆ V in (0, 1) ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0. In particular, when |ξ′| = k, we cannot have better than
U(·, ξ′)
V (·, ξ′)
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
For strictly convex weights, estimating ˆ U solving ∂1 ˆ U + (−s∂1ϕ − |ξ′|) ˆ U = ˆ V in (0, 1) ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0, yields : ∃C > 0 independent of s and ξ′ s.t. s1/2
U(·, ξ′)
V (·, ξ′)
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
To prove the estimate on ˆ U, multiply ∂1 ˆ U + (−s∂1ϕ − |ξ′|) ˆ U = ˆ V in (0, 1) ˆ U(0, ξ′) = ˆ U(1, ξ′) = 0, by (−s∂1ϕ − |ξ′|) ˆ U, integrate and take the real part : 1 | ˆ U(x1, ξ′)|2(s∂11ϕ + (−s∂1ϕ − |ξ′|)2) dx1 ≤ ˆ V (·, ξ′)L2(0,1)(−s∂1ϕ − |ξ′|) ˆ UL2(0,1). Consequently, s1/2
U(·, ξ′)
V (·, ξ′)
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Ω = (0, 1) × Rd−1, Γ = {0} × Rd−1. Conditions on ϕ = ϕ(x1) ϕ = ϕ(x1) is smooth, max
[0,1]{∂1ϕ} < 0 and min [0,1]{∂11ϕ} > 0.
Theorem ∃C > 0, s.t. ∀s ≥ 1, ∀u ∈ H2 ∩ H1
0(Ω),
s3/2 uesϕL2((0,1)×Rd−1) + s1/2 ∇uesϕL2((0,1)×Rd−1) ≤ C ∆uesϕL2((0,1)×Rd−1) + Cs1/2
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
U(x1, x′) = u(x1, x′)esϕ(x1), F(x1, x′) = ∆u(x1, x′)esϕ(x1), Gn(x′) = gn(x′)esϕ(0). The equations now read ∂11U − 2s∂1ϕ∂1U + (s2|∂1ϕ|2 − s∂11ϕ)U + ∆′U = F, in Ω, U(0, x′) = U(1, x′) = 0, in Rd−1, ∂1U(0, x′) = GN(x′), in Rd−1.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Multiply the equation by s∂1ϕ∂1U :
Fs∂1ϕ∂1U =
= s 2
x1=0
+ s 2
−
s 2∂11ϕ|∂1U|2 + 2s2(∂1ϕ)2|∂1U|2 + 3s3 2 ∂11ϕ(∂1ϕ)2|U|2
s2 2 ((∂11ϕ)2 + ∂1ϕ∂111ϕ)|U|2. We recall that max
[0,1]{∂1ϕ} < 0 and min [0,1]{∂11ϕ} > 0.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Therefore,
s 2∂11ϕ|∂1U|2 + 2s2(∂1ϕ)2|∂1U|2 + 3s3 2 ∂11ϕ(∂1ϕ)2|U|2
|Fs∂1U| + Cs
|∂1U|2 + s 2
∂11ϕ|∇′U|2 + Cs2
|U|2. We need to estimate s
Multiply the equation by s∂11ϕU :
Fs∂11ϕU =
= −s
∂11ϕ|∇U|2 + s 2
∂(4)
1 ϕ|U|2 + s2
∂1(∂1ϕ∂11ϕ)|U|2 + s3
∂11ϕ(∂1ϕ)2|U|2 − s2
(∂11ϕ)2|U|2.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Combining the two estimates, we obtain
s 2∂11ϕ|∂1U|2 + s2(∂1ϕ)2|∂1U|2 + s3∂11ϕ(∂1ϕ)2|U|2 ≤ C
|F|2 + Cs
|∂1U|2 + Cs2
|U|2. Taking s large enough,
≤ C
|F|2 + Cs
|∂1U|2. We have an estimate on U, hence on ∇U from the previous estimates.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
A direct approach consists in multiplying the equation of U by 2s∂1ϕ∂1U + s∂11ϕU. But the two above terms do not play the same role : s∂1ϕ∂1U is a multiplier chosen for analyzing the propagation in the x1 variable. The other term, s∂11ϕU, rather corresponds to an energy method to compensate the “bad” term
the multiplier s∂1ϕ∂1U. In particular, s∂11ϕU is a weaker order term in U than s∂1ϕ∂1U.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Carleman estimate Let Ω be a bounded domain. Assume that ϕ ∈ C 4(Ω), ∃α, β > 0, s.t. inf
x∈Ω
|∇ϕ(x)| ≥ α, and −|∇ϕ|2∆ϕ + 2D2ϕ(∇ϕ, ∇ϕ) ≥ β|∇ϕ|2. inf
x∈Ω ξ∈Rd, |ξ|=1
∃C > 0 s.t. for all s ≥ 1 and u ∈ H2 ∩ H1
0(Ω),
s3 esϕu2
L2(Ω) + s esϕ∇u2 L2(Ω)
≤ C
L2(Ω) + s esϕ∂nu2 L2(Γϕ)
where Γϕ = {x ∈ ∂Ω, ∂nϕ(x) > 0}.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
On the conditions on the weight functions : If ϕ = ϕ(x1), the two conditions are satisfied iff min[0,1]{|∂1ϕ|} > 0 and min[0,1]{∂11ϕ} > 0. If we further impose Γϕ to correspond to x1 = 0 in the case of a strip, one should have ∂1ϕ < 0. The regularity of ϕ can be lowered to C 2(Ω). The conditions on the weight functions used here are slightly too strong, see later. We have the following result : Lemma Let Γ ⊂ ∂Ω with Γ = ∅. ∃ϕ satisfying the conditions of the Theorem s.t. Γϕ ⊂ Γ.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Set U = u exp(sϕ) and F = ∆uesϕ. Then U solves ∆U − 2s∇ϕ · ∇U + s2|∇ϕ|2U − s∆ϕ U = F in Ω, U = 0
We thus write P1U = ∆U + s2|∇ϕ|2U, P2U = −2s∆ϕ U − 2s∇ϕ · ∇U, RU = −s∆ϕ U. so that P1U + P2U = F + RU. P1, P2 roughly correspond to the self-adjoint and skew-adjoint parts of the operator.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Taking the L2 norm of both sides of the identity P1U + P2U = F + RU,
|P1U|2 +
|P2U|2 + 2
P1U P2U ≤ 2
|F|2 + 2
|RU|2. Main idea Compute
After tedious computations, one gets :
P1U P2U = s3
−s
∆2ϕ |U|2+s
∆ϕ|∇U|2 + 2s
D2ϕ(∇U, ∇U)−s
∂nϕ|∂nU|2.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Denote by Ii,j the cross product between the i-th term of P1 and the j-th term of P2. I11 = −2s
∆ϕ U ∆U = 2s
∆ϕ|∇U|2 − s
∆2ϕ |U|2 I12 = −2s
∆U ∇ϕ · ∇U = 2s
∇U · ∇(∇ϕ · ∇U) − 2s
∂nϕ|∂nU|2 = 2s
D2ϕ(∇U, ∇U) − s
∆ϕ |∇U|2 − s
∂nϕ|∂nU|2 I21 = −2s3
|∇ϕ|2∆ϕ|U|2 I22 = −2s3
|∇ϕ|2∇ϕ · ∇U U = −s3
|∇ϕ|2∇ϕ · ∇(|U|2) = s3
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
We got
P1U P2U = s3
−s
∆2ϕ |U|2+s
∆ϕ|∇U|2 + 2s
D2ϕ(∇U, ∇U)−s
∂nϕ|∂nU|2. From our assumptions on ϕ, we obtain, for s ≥ s0 large enough, that s3
|U|2 + s
|∇U|2 ≤ C
P1U P2U + Cs
|∂nU|2. Thus, s3
|U|2 + s
|∇U|2 ≤ C
|F|2 + Cs2
|U|2 + Cs
|∂nU|2. Hence taking s0 larger if necessary, for all s ≥ s0, s3
|U|2 + s
|∇U|2 ≤ C
|F|2 + Cs
|∂nU|2.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Lemma Let Γ ⊂ ∂Ω with Γ = ∅. ∃ϕ satisfying the ∃α, β > 0, s.t. inf
x∈Ω
|∇ϕ(x)| ≥ α, and −|∇ϕ|2∆ϕ + 2D2ϕ(∇ϕ, ∇ϕ) ≥ β|∇ϕ|2. inf
x∈Ω ξ∈Rd, |ξ|=1
s.t. Γϕ ⊂ Γ. In particular, Γ can be arbitrarily small.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
We first construct a smooth function ψ in C 4(Ω) such that inf
x∈Ω
{|∇ψ(x)|} > 0, and ∀x ∈ ∂Ω \ Γ, ∂nψ(x) < 0. Theorem
([Fursikov Imanuvilov ’96], see also [Tucsnak Weiss, App. III])
There exists such ψ. We then look for ϕ of the form ϕ(x) = eλψ(x) for some λ > 1 chosen later large enough.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Explicit computations yield ∇ϕ = λ∇ψϕ ∆ϕ = λ2|∇ψ|2ϕ + λ∆ψϕ D2ϕ = λ2∇ψT∇ψϕ + λD2ψϕ. By construction, Γϕ ⊂ Γ. Taking λ large enough − |∇ϕ|2∆ϕ + 2D2ϕ(∇ϕ, ∇ϕ) = λ4|∇ψ|4ϕ3 − λ3D2ψ(∇, ∇ψ)ϕ3 + λ3∆ψ|∇ψ|2ϕ3> 0. ∆ϕ|ξ|2 + 2D2ϕ(ξ, ξ) = λ2|∇ψ|2|ξ|2ϕ+λ∆ψ|ξ|2ϕ+2λ2(∇ψ · ξ)2ϕ+λD2ψ(ξ, ξ)ϕ> 0.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
The difficult point is Theorem
([Fursikov Imanuvilov ’96], see also [Tucsnak Weiss, App. III])
There exists ψ such that inf
x∈Ω
{|∇ψ(x)|} > 0, and ∀x ∈ ∂Ω \ Γ, ∂nψ(x) < 0. Once such ψ is constructed, we can chose ϕ = f (ψ) for any function f sufficiently convex, for instance of the form f (y) = eλy for sufficiently large λ. In other words, f increases the convexity in the direction of ∇ψ compared to the curvature of the level sets {ψ = c}. This process is called a “convexification” process.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
We imposed : inf
x∈Ω
|∇ϕ(x)| ≥ α, and −|∇ϕ|2∆ϕ + 2D2ϕ(∇ϕ, ∇ϕ) ≥ β|∇ϕ|2. inf
x∈Ω ξ∈Rd, |ξ|=1
No critical point in the domain. A kind of convexity condition. One can weaken this convexity condition into a weaker convexity condition.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Carleman estimates for the Laplace operator holds for weight functions ϕ such that inf
x∈Ω
|∇ϕ(x)| ≥ α, and ∃β > 0, s.t ∀x ∈ Ω, ∀ξ ∈ Rd with ∇ϕ(x) · ξ = 0 and |∇ϕ(x)| = |ξ|, D2ϕx(∇ϕ(x), ∇ϕ(x)) + D2ϕx(ξ, ξ) ≥ β|∇ϕ(x)|2. This is the strict pseudo-convexity condition, see [Hörmander]. NB : A linear weight is degenerate, and belongs to the family of “Limiting Carleman Weights”.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Write the conjugated operator Ps = esϕ∆(e−sϕ ·) as Ps = As + Bs, with A∗
s = As, B∗ s = −Bs
Then if PsU = F, we have F2 = AsU2 + BsU2 +
=
s + B∗ s Bs + [As, Bs])U + Boundary Terms.
In terms of pseudo-differential operator and taking the symbols of the operators, we will be able to derive estimates on U if the symbols of A2
s + B∗ s Bs + [As, Bs] is positive. In terms of symbols,
this means as(x, ξ) = 0 and bs(x, ξ) = 0 ⇒ {a, b}(x, ξ) > 0.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
In our case, As = −∆ − s2|∇ϕ|2, Bs = 2s∇ϕ · ∇, so the symbols
a(x, ξ) = |ξ|2 − |∇ϕ(x)|2 b(x, ξ) = 2iξ · ∇ϕ(x) {a, b}(x, ξ) = D2ϕx(∇ϕ(x), ∇ϕ(x)) + D2ϕx(ξ, ξ). Therefore, as(x, ξ) = 0 and bs(x, ξ) = 0 ⇒ {a, b}(x, ξ) > 0 is equivalent to ∃β > 0, s.t ∀x ∈ Ω, ∀ξ ∈ Rd with ∇ϕ(x) · ξ = 0 and |∇ϕ(x)| = |ξ|, D2ϕx(∇ϕ(x), ∇ϕ(x)) + D2ϕx(ξ, ξ) ≥ β|∇ϕ(x)|2. See [Le Rousseau Lebeau 2010] for more details.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Theorem Let q ∈ L∞(Ω) and W ∈ (L∞(Ω))d, and let Γ be a non-empty
0(Ω) of
∆u + qu + W · ∇u = 0 for x ∈ Ω, u = 0 for x ∈ ∂Ω, which further satisfies ∂nu = 0 on Γ vanishes identically on Ω : u = 0 in Ω. This property is called the unique continuation property through Γ. When q is constant, this shows that any eigenvector of −∆ + W · ∇ cannot identically vanish on Γ. NB : Here, we handled the zero order term q and the first order term W · ∇.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Choose a suitable Carleman weight function ϕ so that Γϕ ⊂ Γ, and then apply the Carleman estimate to u : s3 esϕu2
L2(Ω) + s esϕ∇u2 L2(Ω) ≤ C esϕ∆u2 L2(Ω)
≤ C
L∞ esϕu2 L2(Ω) + W 2 L∞ esϕ∇u2 L2(Ω)
Taking s large enough, we easily obtain u = 0 in Ω.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
We can prove that for all u ∈ H1
0(Ω) with source term in L2(Ω) and
ϕ an admissible Carleman weight, s3 esϕu2
L2(Ω) + 1
s esϕu2
H2(Ω)
≤ C
L2(Ω) + s esϕ∂nu2 L2(Γϕ)
Following, by interpolation, for all θ ∈ (0, 1) we get s3−4θ esϕu2
H2θ(Ω) ≤ C
L2(Ω) + s esϕ∂nu2 L2(Γϕ)
Arguing as above allow then to derive unique continuation properties if q ∈ L2d/3+(Ω). Using Lp Carleman inequalities, [Jerison Kenig ’85] improves the result to q ∈ Ld/2+(Ω), which is sharp, see e.g. [Koch Tataru ’02].
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications
Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont.
Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications