Carleman estimates for elliptic PDE and applications Sylvain - PowerPoint PPT Presentation

Cauchy Linear Weight General Weights Full Info Partial Info Intermediate In particular, the formula g D ( ξ ′ ) + sinh ( | ξ ′ | x 1 ) u ( x 1 , ξ ′ ) = cosh ( | ξ ′ | x 1 )ˆ g N ( ξ ′ ) ˆ ˆ | ξ ′ | � x 1 sinh ( | ξ ′ | ( x 1 − x )) f ( x , ξ ′ ) dx . ˆ + | ξ ′ | 0 may not belong to S ′ ( R d − 1 ) and therefore, the inverse Fourier transform may not exist. We only have : ∃ C > 0, s.t. ∀ x 1 ∈ ( 0 , 1 ) , � x 1 � � u ( x 1 , ξ ′ ) | ≤ C exp ( | ξ ′ | x 1 ) g D ( ξ ′ ) | + | ˆ g N ( ξ ′ ) | + | ˆ f ( x , ξ ′ ) | dx | ˆ | ˆ . 0 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 ( x 1 , ξ ′ ) | 2 exp ( − 2 | ξ ′ | x 1 ) d ξ ′ dx 1 R d − 1 | ˆ 0 � � � g D � 2 L 2 ( R d − 1 ) + � g N � 2 L 2 ( R d − 1 ) + � f � 2 ≤ C . L 2 (( 0 , 1 ) × R d − 1 ) The left hand-side defines a norm on u but weaker than any norm of the form L 2 ( 0 , 1 ; H − k ( R d − 1 )) . � This is a prototype setting in which the linear mapping u ∈ H 2 (Ω) �→ ( f , g D , g N ) ∈ L 2 (Ω) × H 3 / 2 (Γ) × H 1 / 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 Another question We briefly presented the case of Full informations Γ = ∂ Ω . Partial informations Γ ⊂ ∂ Ω . What is in between ? ∂ 11 u + ∆ ′ u = f , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,   for x ′ ∈ R d − 1 ,  u ( 0 , x ′ ) = g D ( x ′ ) ,  for x ′ ∈ R d − 1 , ∂ 1 u ( 0 , x ′ ) = g N ( x ′ ) ,  for x ′ ∈ R d − 1 .  u ( 1 , x ′ ) = g 1 ( x ′ ) ,  In this case, one can solve ∆ u = f in Ω with full Dirichlet data, hence g N is fully determined by g D , g 1 . 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, u − | ξ ′ | 2 ˆ u = ˆ  for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 , ∂ 11 ˆ f ,  for ξ ′ ∈ R d − 1 , u ( 0 , ξ ′ ) = ˆ g D ( ξ ′ ) ,  ˆ  for ξ ′ ∈ R d − 1 , u ( 0 , ξ ′ ) = ˆ g N ( ξ ′ ) , ∂ 1 ˆ  for ξ ′ ∈ R d − 1 .  u ( 1 , ξ ′ ) = ˆ g 1 ( ξ ′ ) , ˆ  � 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 u − | ξ ′ | 2 ˆ u = ˆ ∂ 11 ˆ f , for x 1 ∈ ( 0 , 1 ) , as � ( ∂ 1 − | ξ ′ | )ˆ u = ˆ v , for x 1 ∈ ( 0 , 1 ) , v = ˆ ( ∂ 1 + | ξ ′ | )ˆ for x 1 ∈ ( 0 , 1 ) , f , OR � ( ∂ 1 + | ξ ′ | )ˆ u = ˆ w , for x 1 ∈ ( 0 , 1 ) , w = ˆ ( ∂ 1 − | ξ ′ | ) ˆ for x 1 ∈ ( 0 , 1 ) . f , � 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 x 1 = { 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 ( x 1 ) = z ( 0 ) exp ( − ax 1 ) , OR z ( x 1 ) = z ( 1 ) exp ( a ( 1 − x 1 )) . 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 one. Information comes from the right. 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 x 1 ∈ ( 0 , 1 ) ,  v = ˆ  ( ∂ 1 + | ξ ′ | )ˆ for x 1 ∈ ( 0 , 1 ) , f ,    u ( 0 , ξ ′ ) = ˆ g D ( ξ ′ ) , ˆ , u ( 0 , ξ ′ ) = ˆ g N ( ξ ′ ) , ∂ 1 ˆ     u ( 1 , ξ ′ ) = ˆ g 1 ( ξ ′ ) ,  ˆ that we solve in two steps : v = ˆ � v + | ξ ′ | ˆ ∂ 1 ˆ for x 1 ∈ ( 0 , 1 ) , f v ( 0 , ξ ′ ) = ˆ g N ( ξ ′ ) − | ξ ′ | ˆ g D ( ξ ′ ) , ˆ then  u − | ξ ′ | ˆ ∂ 1 ˆ u = ˆ for x 1 ∈ ( 0 , 1 ) , v  u ( 0 , ξ ′ ) = ˆ g D ( ξ ′ ) , ˆ u ( 1 , ξ ′ ) = ˆ g 1 ( ξ ′ ) . ˆ  Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Full Info Partial Info Intermediate Solving v = ˆ � v + | ξ ′ | ˆ for x 1 ∈ ( 0 , 1 ) , ∂ 1 ˆ f v ( 0 , ξ ′ ) = ˆ g N ( ξ ′ ) − | ξ ′ | ˆ g D ( ξ ′ ) , ˆ we find � x 1 � ˆ v ( x 1 , ξ ′ ) = exp ( −| ξ ′ | x 1 )ˆ v ( 0 , ξ ′ )+ −| ξ ′ | ( x 1 − x ) f ( x , ξ ′ ) dx , � ˆ exp 0 Solving u − | ξ ′ | ˆ  for x 1 ∈ ( 0 , 1 ) , ∂ 1 ˆ u = ˆ v  u ( 0 , ξ ′ ) = ˆ g D ( ξ ′ ) , ˆ u ( 1 , ξ ′ ) = ˆ g 1 ( ξ ′ ) . ˆ  we find � 1 u ( x 1 , ξ ′ ) = e −| ξ ′ | ( 1 − x ) ˆ g 1 ( ξ ′ ) − −| ξ ′ | ( x − x 1 ) v ( x , ξ ′ ) dx . � � ˆ exp ˆ x 1 Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Full Info Partial Info Intermediate This leads � 1 u ( x 1 , ξ ′ ) = e −| ξ ′ | ( 1 − x ) ˆ g 1 ( ξ ′ ) − ˆ v ( 0 , ξ ′ ) −| ξ ′ | ( 2 x − x 1 ) � � ˆ exp dx x 1 � 1 �� 1 � ˆ x , ξ ′ ) exp ( −| ξ ′ | ( 2 x − x 1 − ˜ − f (˜ x )) dx d ˜ x , 0 max { x 1 , ˜ x } v ( 0 , ξ ′ ) = ˆ g N ( ξ ′ ) − | ξ ′ | ˆ g D ( ξ ′ ) . with ˆ � 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 ′ , � 1 � � C v ( 0 , ξ ′ ) −| ξ ′ | ( 2 x − x 1 ) v ( 0 , ξ ′ ) | , � � � � ≤ 1 + | ξ ′ | 3 / 2 | ˆ � ˆ exp dx � � � x 1 L 2 ( 0 , 1 ) and � 1 �� 1 � � � � � ˆ x , ξ ′ ) exp ( −| ξ ′ | ( 2 x − x 1 − ˜ f (˜ x )) dx d ˜ x � � � � 0 max { x 1 , ˜ x } � � L 2 ( 0 , 1 ) � 1 � � C | ˆ x , ξ ′ ) | exp ( −| ξ ′ || x 1 − ˜ � � ≤ f (˜ x | ) d ˜ x � � 1 + | ξ ′ | � 0 � L 2 ( 0 , 1 ) C � � � ˆ f ( x 1 , ξ ′ ) 1 x 1 ∈ ( 0 , 1 ) ∗ x 1 exp ( −| ξ ′ || x 1 | ) 1 x 1 ∈ ( − 2 , 2 ) ≤ � � 1 + | ξ ′ | � L 2 ( 0 , 1 ) C � � � ˆ f ( · , ξ ′ ) � exp ( −| ξ ′ || x 1 | ) 1 x 1 ∈ ( − 2 , 2 ) � � ≤ � � � L 1 ( − 2 , 2 ) 1 + | ξ ′ | � L 2 ( 0 , 1 ) C � � � ˆ f ( · , ξ ′ ) ≤ L 2 ( 0 , 1 ) , � � 1 + | ξ ′ | 2 � 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 ′ , � 2 ( 1 + | ξ ′ | 4 ) � u ( · , ξ ′ ) � L 2 ( 0 , 1 ) ≤ C ( 1 + | ξ ′ | ) 3 / 2 | ˆ g 1 ( ξ ′ ) | + C ( 1 + | ξ ′ | ) | ˆ v ( 0 , ξ ′ ) | 2 � ˆ 2 � � � ˆ f ( · , ξ ′ ) + C L 2 ( 0 , 1 ) . � � � Integrating in ξ ′ ∈ R d − 1 and using Parseval’s identity � u � 2 L 2 ( 0 , 1 ; H 2 ( R d − 1 )) ≤ C � f � 2 L 2 (( 0 , 1 ) × R d − 1 ) + C � g 1 � 2 H 3 / 2 ( R d − 1 ) � v ( 0 , ξ ′ ) | 2 d ξ ′ . R d − 1 ( 1 + | ξ ′ | ) | ˆ + C Using ∂ 11 u = f − ∆ ′ u ∈ L 2 (( 0 , 1 ) × R d − 1 ) , � u � 2 H 2 (( 0 , 1 ) × R d − 1 )) ≤ C � f � 2 L 2 (( 0 , 1 ) × R d − 1 ) + C � g 1 � 2 H 3 / 2 ( R d − 1 ) � v ( 0 , ξ ′ ) | 2 d ξ ′ . R d − 1 ( 1 + | ξ ′ | ) | ˆ + C Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Full Info Partial Info Intermediate v ( 0 , ξ ′ ) = ˆ g N ( ξ ′ ) − | ξ ′ | ˆ g D ( ξ ′ ) From the explicit form of ˆ � v ( 0 , ξ ′ ) | 2 d ξ ′ ≤ C � g D � 2 R d − 1 ( 1 + | ξ ′ | ) | ˆ H 3 / 2 ( R d − 1 ) + C � g N � 2 H 1 / 2 ( R d − 1 ) . We thus conclude � u � H 2 (( 0 , 1 ) × R d − 1 ) ≤ C � f � L 2 (( 0 , 1 ) × R d − 1 ) + C � g 1 � H 3 / 2 ( R d − 1 ) + C � g D � H 3 / 2 ( R d − 1 ) + C � g N � H 1 / 2 ( R d − 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 � u � H 2 (( 0 , 1 ) × R d − 1 ) ≤ C � f � L 2 (( 0 , 1 ) × R d − 1 ) + C � g D � H 3 / 2 ( R d − 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 u ( x 1 , ξ ′ ) =ˆ g 1 ( ξ ′ ) e −| ξ ′ | ( 1 − x 1 ) ˆ � 1 g N ( ξ ′ ) − | ξ ′ | ˆ g D ( ξ ′ )) −| ξ ′ | ( 2 x − x 1 ) � � − (ˆ exp dx x 1 � 1 �� 1 � ˆ x , ξ ′ ) exp ( −| ξ ′ | ( 2 x − x 1 − ˜ − f (˜ x )) dx d ˜ x , 0 max { x 1 , ˜ x } is thus “stable”. However, it does not provide easily the property  f = 0 in Ω  g D = 0 on Γ , ⇒ u = 0 in Ω . g N = 0 on Γ  � 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 Outline The Cauchy problem for the Laplace operator 1 The case of full informations The case of partial data Intermediate case Carleman estimates with a linear weight 2 Goal Proof of the Carleman estimate : Fourier techniques Proof of the Carleman estimate : Multiplier techniques More general geometric settings Application to the Calderón Problem More general Carleman Weights 3 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 Goal The goal of this section is to get a stable estimate for solutions of  ∆ u = f , in Ω ,  u = g D , on ∂ Ω , ∂ n u = g N , on Γ .  allowing to prove that  f = 0 in Ω  g D = 0 on Γ , ⇒ u = 0 in Ω , g N = 0 on Γ ,  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 ) × R d − 1 , Γ = { 0 } × R d − 1 , with homogeneous Dirichlet boundary conditions to simplify the presentation : ∂ 11 u + ∆ ′ u = f , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,   for x ′ ∈ R d − 1 , u ( 0 , x ′ ) = u ( 1 , x ′ ) = 0 , (2) for x ′ ∈ R d − 1 . ∂ 1 u ( 0 , x ′ ) = g N ( x ′ ) ,  A Carleman estimate ∃ C > 0, s.t. for all k ≥ 1, any solution u of (2) with f ∈ L 2 (Ω) and g N ∈ H 1 / 2 (Γ) satisfies � � � � � ue − kx 1 � ∇ ue − kx 1 L 2 (( 0 , 1 ) × R d − 1 ) + k � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) � � L 2 (( 0 , 1 ) × R d − 1 ) + Ck 1 / 2 � g N � L 2 ( R d − 1 ) . � fe − kx 1 ≤ 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 ∈ L 2 (Ω) and g N ∈ H 1 / 2 (Γ) satisfies � � � � � ue − kx 1 � ∇ ue − kx 1 k L 2 (( 0 , 1 ) × R d − 1 ) + � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) � � L 2 (( 0 , 1 ) × R d − 1 ) + Ck 1 / 2 � g N � L 2 ( R d − 1 ) . � fe − kx 1 ≤ 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 − kx 1 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 Some consequences Carleman estimate for a Laplace operator with potential Let q ∈ L ∞ (( 0 , 1 ) × R d − 1 ) . There exists a constant C > 0 such that for all k ≥ 1, any solution u ∈ L 2 (( 0 , 1 ) × R d − 1 ) of ∂ 11 u + ∆ ′ u + qu = f , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,   for x ′ ∈ R d − 1 , u ( 0 , x ′ ) = u ( 1 , x ′ ) = 0 , for x ′ ∈ R d − 1 , ∂ 1 u ( 0 , x ′ ) = g N ( x ′ ) ,  with source term f ∈ L 2 (( 0 , 1 ) × R d − 1 ) and Neumann data g N ∈ L 2 ( R d − 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 Proof Apply the Carleman estimate (3) to u with f replaced by f − qu : ∃ C > 0, ∀ k ≥ 1, � � � � � ue − kx 1 � ∇ ue − kx 1 k L 2 (( 0 , 1 ) × R d − 1 ) + � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) � � L 2 (( 0 , 1 ) × R d − 1 ) + Ck 1 / 2 � g N � L 2 ( R d − 1 ) . � ( f − qu ) e − kx 1 ≤ C � � � Using � � � � � ( f − qu ) e − kx 1 � fe − kx 1 L 2 (( 0 , 1 ) × R d − 1 ) ≤ � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) � � � ue − kx 1 + � q � L ∞ (( 0 , 1 ) × R d − 1 ) L 2 (( 0 , 1 ) × R d − 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 ≥ k q = 2 C � q � L ∞ (( 0 , 1 ) × R d − 1 ) , � � � � � ue − kx 1 � ∇ ue − kx 1 k L 2 (( 0 , 1 ) × R d − 1 ) + � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) � � L 2 (( 0 , 1 ) × R d − 1 ) + 2 Ck 1 / 2 � g N � L 2 ( R d − 1 ) . � fe − kx 1 ≤ 2 C � � � Now, ∃ C > 0 s.t. ∀ k ≥ 1, � � � � � ue − kx 1 � ∇ ue − kx 1 k L 2 (( 0 , 1 ) × R d − 1 ) + � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) � � L 2 (( 0 , 1 ) × R d − 1 ) + 2 Ck 1 / 2 � g N � L 2 ( R d − 1 ) , � fe − kx 1 ≤ 2 C � � � as the case k ∈ [ 1 , k q ] can be handled by immediate bounds on the estimate for k = k q . 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 ) × R d − 1 ) , and u ∈ L 2 (( 0 , 1 ) × R d − 1 ) be the solution of  ∂ 11 u + ∆ ′ u + qu = f , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,  for x ′ ∈ R d − 1 , u ( 0 , x ′ ) = 0 , for x ′ ∈ R d − 1 , ∂ 1 u ( 0 , x ′ ) = g N ( x ′ ) ,  with u ( 1 , x ′ ) ∈ H 3 / 2 ( R d − 1 ) and Neumann data g N = 0 and source term f ∈ L 2 (( 0 , 1 ) × R d − 1 ) satisfying, for some a ∈ ( 0 , 1 ) , for x 1 ∈ ( 0 , a ) , x ′ ∈ R d − 1 . f ( x 1 , x ′ ) = 0 Then u vanishes in ( 0 , a ) × R d − 1 . Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Proof If u ( 1 , x ′ ) = 0, apply the Carleman estimate to u : for all k ≥ 1, � � � � � ue − kx 1 � fe − kx 1 k L 2 (( 0 , 1 ) × R d − 1 ) ≤ C L 2 (( 0 , 1 ) × R d − 1 ) . � � � � � � But on one hand, � � � � L 2 (( 0 , a ) × R d − 1 ) ≥ ke − ka � u � L 2 (( 0 , a ) × R d � ue − kx 1 � ue − kx 1 k L 2 (( 0 , 1 ) × R d − 1 ) ≥ k � � � � � � On the other hand, as f vanishes in ( 0 , a ) × R d − 1 , � � L 2 (( 0 , 1 ) × R d − 1 ) ≤ e − ka � f � L 2 (( 0 , 1 ) × R d − 1 ) . � fe − kx 1 � � � Using the Carleman estimate then yields, for all k ≥ 1, k � u � L 2 (( 0 , a ) × R d − 1 ) ≤ C � f � L 2 (( 0 , 1 ) × R d − 1 ) . Taking the limit k → ∞ , u necessarily vanishes in ( 0 , a ) × R d − 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 ′ ) ∈ H 3 / 2 ( R d − 1 ) , u ∈ H 2 ( R d − 1 ) . We therefore set v ( x ) = η ( x 1 ) u ( x ) , in Ω , where η = η ( x 1 ) is a smooth cut-off function taking value 1 on ( 0 , a ) , vanishing in x 1 = 1. Then v solves ∂ 11 v + ∆ ′ v + qv = η f − [ ∂ 11 , η ] u , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,   for x ′ ∈ R d − 1 , v ( 0 , x ′ ) = v ( 1 , x ′ ) = 0 , for x ′ ∈ R d − 1 , ∂ 1 v ( 0 , x ′ ) = 0 ,  Thus, by the previous case, v ( x 1 , x ′ ) = 0 for x 1 ∈ ( 0 , a ) , i.e. u ( x 1 , x ′ ) = 0 for ( x 1 , x ′ ) ∈ ( 0 , a ) × R d − 1 . Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Recall the Carleman estimate Theorem ∃ C > 0, s.t. ∀ k ≥ 1, any solution u of  ∂ 11 u + ∆ ′ u = f , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,  for x ′ ∈ R d − 1 , u ( 0 , x ′ ) = u ( 1 , x ′ ) = 0 , for x ′ ∈ R d − 1 , ∂ 1 u ( 0 , x ′ ) = g N ( x ′ ) ,  with f ∈ L 2 (Ω) and g N ∈ H 1 / 2 ( R d − 1 ) satisfies � � � � � ue − kx 1 � ∇ ue − kx 1 k L 2 (( 0 , 1 ) × R d − 1 ) + � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) � � L 2 (( 0 , 1 ) × R d − 1 ) + Ck 1 / 2 � g N � L 2 ( R d − 1 ) . � fe − kx 1 ≤ 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 Proof by Fourier techniques Since we have to prove estimates on ue − kx 1 in terms of fe − kx 1 , it is convenient to work on the conjugate variables : � U ( x 1 , x ′ ) = u ( x 1 , x ′ ) e − kx 1 , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 . F ( x 1 , x ′ ) = f ( x 1 , x ′ ) e − kx 1 , The equation on u then rewrites in terms of U , as U ( x 1 , x ′ ) = e kx 1 u ( x 1 , x ′ ) : ∂ 11 U + 2 k ∂ 1 U + k 2 U + ∆ ′ U = F , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,   for x ′ ∈ R d − 1 , U ( 0 , x ′ ) = U ( 1 , x ′ ) = 0 , for x ′ ∈ R d − 1 , ∂ 1 U ( 0 , x ′ ) = g N ( x ′ ) ,  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 : U ( x 1 , ξ ′ ) = F x ′ → ξ ′ U ( x 1 , · ) , ˆ ˆ F ( x 1 , ξ ′ ) = F x ′ → ξ ′ F ( x 1 , · ) , g N ( ξ ′ ) = F x ′ → ξ ′ g N ( · ) . ˆ We obtain a family of ODE indexed by ξ ′ : U + k 2 ˆ U − | ξ ′ | 2 ˆ ∂ 11 ˆ U + 2 k ∂ 1 ˆ U = ˆ for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 ,  F ,  for ξ ′ ∈ R d − 1 , U ( 0 , ξ ′ ) = ˆ ˆ U ( 0 , ξ ′ ) = 0 , for ξ ′ ∈ R d − 1 . ∂ 1 ˆ U ( 0 , x ′ ) = ˆ g N ( 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 operator factorized as follows : ∂ 11 + 2 k ∂ 1 + k 2 − | ξ ′ | 2 = ( ∂ 1 + k + | ξ ′ | )( ∂ 1 + k − | ξ ′ ) . Therefore, we introduce the function V = ( ∂ 1 + k − | ξ | ) ˆ ˆ ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 . U , � Back to  ∂ 1 ˆ U + ( k − | ξ ′ | ) ˆ U = ˆ for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 , V   ∂ 1 ˆ V + ( k + | ξ ′ | ) ˆ V = ˆ for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 ,  F  for ξ ′ ∈ R d − 1 , U ( 0 , ξ ′ ) = ˆ ˆ U ( 1 , ξ ′ ) = 0   for ξ ′ ∈ R d − 1 . ˆ V ( 0 , ξ ′ ) = ˆ g N ( ξ ′ )   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 for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 , V   ∂ 1 ˆ V + ( k + | ξ ′ | ) ˆ V = ˆ for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 ,  F  for ξ ′ ∈ R d − 1 , U ( 0 , ξ ′ ) = ˆ ˆ U ( 1 , ξ ′ ) = 0   for ξ ′ ∈ R d − 1 . ˆ V ( 0 , ξ ′ ) = ˆ g N ( ξ ′ )   For fixed ξ ′ ∈ R d − 1 , we can then solve this system in two steps : 1 Compute ˆ V in terms of ˆ F and of ˆ g N : � ∂ 1 ˆ V + ( k + | ξ ′ | ) ˆ V = ˆ for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 , F for ξ ′ ∈ R d − 1 . V ( 0 , ξ ′ ) = ˆ ˆ g N ( ξ ′ ) 2 Compute ˆ U in terms of ˆ V : � ∂ 1 ˆ U + ( k − | ξ ′ | ) ˆ U = ˆ for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 , V for ξ ′ ∈ R d − 1 , U ( 0 , ξ ′ ) = ˆ ˆ U ( 1 , ξ ′ ) = 0 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 for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 , F for ξ ′ ∈ R d − 1 . ˆ V ( 0 , ξ ′ ) = ˆ g N ( ξ ′ ) yields : � x 1 V ( x 1 , ξ ′ ) = e − ( k + | ξ ′ | ) x 1 ˆ ˆ g N ( ξ ′ )+ exp ( − ( k + | ξ ′ | )( x 1 − x )) ˆ F ( x , ξ ′ ) dx . 0 The computation of ˆ U also is straightforward, as it solves : � ∂ 1 ˆ U + ( k − | ξ ′ | ) ˆ U = ˆ for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 , V for ξ ′ ∈ R d − 1 , U ( 0 , ξ ′ ) = ˆ ˆ U ( 1 , ξ ′ ) = 0 But now we have two possible formulae, depending whether we use the boundary condition at x 1 = 0 or at x 1 = 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 for ( x 1 , ξ ′ ) ∈ ( 0 , 1 ) × R d − 1 , V for ξ ′ ∈ R d − 1 , U ( 0 , ξ ′ ) = ˆ ˆ U ( 1 , ξ ′ ) = 0 Two formulae : � x 1 ˆ exp ( − ( k − | ξ ′ | )( x 1 − x )) ˆ U ( x 1 , ξ ′ ) V ( x , ξ ′ ) dx , = 0 � 1 ˆ exp ( − ( k − | ξ ′ | )( x 1 − x )) ˆ U ( x 1 , ξ ′ ) V ( x , ξ ′ ) dx . = − x 1 � 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 The low frequency case | ξ ′ | ≤ k Here, we use the first formula : � x 1 ˆ U ( x 1 , ξ ′ ) = ˆ g N ( ξ ′ ) exp ( − ( k − | ξ ′ | ) x 1 ) exp ( − 2 | ξ ′ | x ) dx 0 � x 1 �� x 1 � x + x 1 − 2 x ) dx ˆ x , ξ ′ ) e − k ( x 1 − ˜ x ) e | ξ ′ | (˜ + F (˜ d ˜ x . 0 x ˜ Direct estimates : � x 1 � � g N ( ξ ′ ) exp ( − ( k − | ξ ′ | ) x 1 ) exp ( − 2 | ξ ′ | x ) dx � � � ˆ � � 0 � L 2 ( 0 , 1 ) � � � � 1 1 C g N ( ξ ′ ) | g N ( ξ ′ ) | . ≤ C | ˆ ≤ k 1 / 2 | ˆ 1 + | ξ ′ | 1 + ( k − | ξ ′ | ) 1 / 2 Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon The low frequency case | ξ ′ | ≤ k On the other hand, writing � x 1 �� x 1 � x + x 1 − 2 x ) dx ˆ x , ξ ′ ) e − k ( x 1 − ˜ x ) e | ξ ′ | (˜ F (˜ d ˜ x 0 ˜ x � 1 ˆ x , ξ ′ ) H ( x 1 − ˜ x , ξ ′ ) d ˜ = F (˜ x 0 with � X e − 2 | ξ ′ | x dx , H ( X , ξ ′ ) = 1 X > 0 e − ( k −| ξ ′ | ) X 0 we obtain � x 1 �� x 1 � � � x + x 1 − 2 x ) dx ˆ e | ξ ′ | (˜ � x , ξ ′ ) e − k ( x 1 − ˜ x ) � F (˜ d ˜ x � � � � 0 x ˜ L 2 ( 0 , 1 ) � � � ˆ F ( · , ξ ′ ) � H ( · , ξ ′ ) � � ≤ C � � � L 1 ( 0 , 2 ) � L 2 ( 0 , 1 ) � 1 � � 1 � ≤ C � � � � � ˆ F ( · , ξ ′ ) � ˆ F ( · , ξ ′ ) ≤ C L 2 ( 0 , 1 ) . � � � � 1 + k − | ξ ′ | 1 + | ξ ′ | � k � L 2 ( 0 , 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 low frequency case | ξ ′ | ≤ k We thus get, for ξ ′ ∈ R d − 1 with | ξ ′ | ≤ k , C g N ( ξ ′ ) | + C � � � � � ˆ U ( x 1 , ξ ′ ) � ˆ F ( · , ξ ′ ) L 2 ( 0 , 1 ) ≤ k 1 / 2 | ˆ L 2 ( 0 , 1 ) , � � � � k � � i.e. ∀ ξ ′ ∈ R d − 1 with | ξ ′ | ≤ k 2 2 k 2 � � g N ( ξ ′ ) | 2 + C � � � ˆ U ( x 1 , ξ ′ ) � ˆ F ( · , ξ ′ ) L 2 ( 0 , 1 ) ≤ Ck | ˆ L 2 ( 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 The high frequency case | ξ ′ | ≥ k Here, we use the other formula : � 1 e − 2 | ξ ′ | x dx ˆ g N ( ξ ′ ) e − ( k −| ξ ′ | ) x 1 U ( x 1 , ξ ′ ) = − ˆ x 1 � 1 �� 1 − max { ˜ � x , x 1 } e − 2 | ξ ′ | x dx ˆ x ) e −| ξ ′ || x 1 − ˜ x , ξ ′ ) e − k ( x 1 − ˜ x | − F (˜ 0 0 For the first term : � 1 � � e − 2 | ξ ′ | x dx g N ( ξ ′ ) e − ( k −| ξ ′ | ) x 1 � � � − ˆ � � � x 1 L 2 ( 0 , 1 ) C � � g N ( ξ ′ ) e − ( k + | ξ ′ | ) x 1 ≤ � − ˆ � � 1 + | ξ ′ | � L 2 ( 0 , 1 ) C 1 C g N ( ξ ′ ) | g N ( ξ ′ ) | . ≤ 1 + | ξ ′ || ˆ ( k + | ξ ′ | ) 1 / 2 ≤ k 3 / 2 | ˆ Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon The high frequency case | ξ ′ | ≥ k The second term can be bounded as follows : � 1 �� 1 − max { ˜ � � � x , x 1 } � e − 2 | ξ ′ | x dx � ˆ x , ξ ′ ) e − k ( x 1 − ˜ x ) e −| ξ ′ || x 1 − ˜ x | � − F (˜ d ˜ x � � � � 0 0 � L 2 ( 0 , 1 ) � 1 �� 1 − max { ˜ � � � x , x 1 } � � e − 2 | ξ ′ | x dx | ˆ x , ξ ′ ) | e − k ( x 1 − ˜ x ) e −| ξ ′ || x 1 − ˜ x | ≤ F (˜ d ˜ x � � � � 0 0 � � L 2 ( 0 � 1 �� ∞ � � � e − 2 | ξ ′ | x dx | ˆ x , ξ ′ ) | e ( k −| ξ ′ | ) | x 1 − ˜ x | � � ≤ F (˜ d ˜ x � � � � 0 0 L 2 ( 0 , 1 ) ≤ C � F | ( x 1 , ξ ′ ) 1 x 1 ∈ ( 0 , 1 ) ∗ x 1 e ( k −| ξ ′ | ) | x 1 | � � | ˆ � � | ξ ′ | � L 2 ( 0 , 1 ) ≤ C � � � e ( k −| ξ ′ | ) | x 1 | � � � ˆ F ( · , ξ ′ ) � � � � | ξ ′ | � � L 2 ( 0 , 1 ) L 1 ( − 2 , 2 ) C L 2 ( 0 , 1 ) ≤ C � � � F ( · , ξ ′ ) � ˆ F ( · , ξ ′ ) � � ≤ L 2 ( 0 , 1 ) . � � � | ξ ′ | ( 1 − k + | 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 The high frequency case | ξ ′ | ≥ k We thus get, for ξ ′ ∈ R d − 1 with | ξ ′ | ≥ k , C g N ( ξ ′ ) | + C � � � � � ˆ U ( x 1 , ξ ′ ) � ˆ F ( · , ξ ′ ) L 2 ( 0 , 1 ) ≤ k 1 / 2 | ˆ L 2 ( 0 , 1 ) , � � � � � k � i.e. ∀ ξ ′ ∈ R d − 1 with | ξ ′ | ≥ k 2 2 k 2 � � g N ( ξ ′ ) | 2 + C � � � ˆ � ˆ U ( x 1 , ξ ′ ) F ( · , ξ ′ ) L 2 ( 0 , 1 ) ≤ Ck | ˆ L 2 ( 0 , 1 ) , � � � � � � with C independent of ξ ′ and k . This is the same estimate as the one obtained for ξ ′ ∈ R d − 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 Low and High-frequency together As the estimates obtained for | ξ ′ | ≤ k and for | ξ ′ | ≥ k are the same, ∃ C > 0, such that ∀ ξ ′ ∈ R d − 1 , 2 2 k 2 � � � � g N ( ξ ′ ) | 2 + C � ˆ U ( x 1 , ξ ′ ) � ˆ F ( · , ξ ′ ) L 2 ( 0 , 1 ) ≤ Ck | ˆ L 2 ( 0 , 1 ) . � � � � � � Integrating this estimate with respect to ξ ′ ∈ R d − 1 and using Parseval’s identity : k 2 � U � 2 L 2 (( 0 , 1 ) × R d − 1 ) ≤ Ck � g N � 2 L 2 ( R d − 1 ) + C � F � 2 L 2 (( 0 , 1 ) × R d − 1 ) . i.e. 2 2 k 2 � � � � � ue − kx 1 L 2 (( 0 , 1 ) × R d − 1 ) ≤ Ck � g N � 2 � fe − kx 1 L 2 ( R d − 1 ) + C L 2 (( 0 , 1 ) × R d − 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 − kx 1 in L 2 (Ω) , we multiply the equation � ∆ u = f , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 , for x ′ ∈ R d − 1 , u ( 0 , x ′ ) = u ( 1 , x ′ ) = 0 , by ue − 2 kx 1 : � � ( 0 , 1 ) × R d − 1 |∇ u | 2 e − 2 kx 1 dx 1 dx ′ + 2 k 2 ( 0 , 1 ) × R d − 1 | u | 2 e − 2 kx 1 dx 1 dx ′ − � ( 0 , 1 ) × R d − 1 fue − 2 kx 1 dx 1 dx ′ , = so that 2 2 � � L 2 (( 0 , 1 ) × R d − 1 ) ≤ k 2 � � � ∇ ue − kx 1 � ue − kx 1 � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) � � � � � fe − kx 1 � ue − kx 1 + L 2 (( 0 , 1 ) × R d − 1 ) . � � � � � L 2 (( 0 , 1 ) × R d − 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 2 2 k 2 � � � � � ue − kx 1 L 2 (( 0 , 1 ) × R d − 1 ) ≤ Ck � g N � 2 � fe − kx 1 L 2 ( R d − 1 ) + C L 2 (( 0 , 1 ) × R d − 1 ) . � � � � � � in 2 2 � � L 2 (( 0 , 1 ) × R d − 1 ) ≤ k 2 � � � ∇ ue − kx 1 � ue − kx 1 � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) � � � � � fe − kx 1 � ue − kx 1 + L 2 (( 0 , 1 ) × R d − 1 ) . � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) We therefore get 2 2 � � � � � ∇ ue − kx 1 L 2 (( 0 , 1 ) × R d − 1 ) ≤ Ck � g N � 2 � fe − kx 1 L 2 ( R d − 1 ) + C L 2 (( 0 , 1 ) × R d − 1 ) . � � � � � � Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Comments on this proof 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 − kx 1 is estimated, a weighted type regularity estimate give an estimate on ∇ ue − kx 1 . Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Proof by multiplier techniques U = ue − kx 1 solves the equation ∂ 11 U + 2 k ∂ 1 U + k 2 U + ∆ ′ U = F , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,   for x ′ ∈ R d − 1 , U ( 0 , x ′ ) = U ( 1 , x ′ ) = 0 , for x ′ ∈ R d − 1 , ∂ 1 U ( 0 , x ′ ) = g N ( x ′ ) ,  Multiplying the equation by ∂ 1 U we easily get � 2 k � ∂ 1 U � 2 ( 0 , 1 ) × R d − 1 F ∂ 1 U dx 1 dx ′ L 2 (( 0 , 1 ) × R d − 1 ) = + 1 R d − 1 | ∂ 1 U ( 0 , x ′ ) | 2 dx ′ − 1 � � R d − 1 | ∂ 1 U ( 1 , x ′ ) | 2 dx ′ , 2 2 so that L 2 (( 0 , 1 ) × R d − 1 ) ≤ C k � ∂ 1 U � 2 k � F � 2 L 2 (( 0 , 1 ) × R d − 1 ) + C � g N � 2 L 2 ( R d − 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 x 1 = 0 and x 1 = 1 � Poincaré’s inequality applies : � U � 2 L 2 (( 0 , 1 ) × R d − 1 ) ≤ C � ∂ 1 U � 2 L 2 (( 0 , 1 ) × R d − 1 ) . Thus, k 2 � U � 2 L 2 (( 0 , 1 ) × R d − 1 ) ≤ C � F � 2 L 2 (( 0 , 1 ) × R d − 1 ) + Ck � g N � 2 L 2 ( R d − 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  ∂ 11 U + 2 k ∂ 1 U + k 2 U + ∆ ′ U = F , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,  for x ′ ∈ R d − 1 , U ( 0 , x ′ ) = U ( 1 , x ′ ) = 0 , for x ′ ∈ R d − 1 , ∂ 1 U ( 0 , x ′ ) = g N ( x ′ ) ,  by U , we obtain � L 2 (( 0 , 1 ) × R d − 1 ) − k 2 � U � 2 � � �∇ U � 2 � � L 2 (( 0 , 1 ) × R d − 1 ) � ≤ � F � L 2 (( 0 , 1 ) × R d − 1 ) � U � L 2 (( 0 , 1 ) × R d − 1 ) , As U is already suitably estimated, L 2 (( 0 , 1 ) × R d − 1 ) ≤ Ck 2 � U � 2 �∇ U � 2 L 2 (( 0 , 1 ) × R d − 1 ) + C � F � 2 L 2 (( 0 , 1 ) × R d − 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 k 2 � U � 2 L 2 (( 0 , 1 ) × R d − 1 ) + �∇ U � 2 L 2 (( 0 , 1 ) × R d − 1 ) ≤ C � F � 2 L 2 (( 0 , 1 ) × R d − 1 ) + Ck � g N � 2 L 2 ( R d − 1 ) . Using then that u = Ue kx 1 , | u | e − kx 1 ≤ | U | , |∇ u | e − kx 1 ≤ |∇ U | + k | U | . Therefore, 2 2 k 2 � � � � � ue − kx 1 � ∇ ue − kx 1 L 2 (( 0 , 1 ) × R d − 1 ) + � � � � � � L 2 (( 0 , 1 ) × R d − 1 ) 2 � � � fe − kx 1 L 2 (( 0 , 1 ) × R d − 1 ) + Ck � g N � 2 ≤ C L 2 ( R d − 1 ) . � � � Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Comments on the second proof 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 L 2 . In particular, the first strategy is more precise and can handled : Non-homogeneous Dirichlet boundary conditions in H 3 / 2 ( ∂ Ω) and Neumann boundary data in H 1 / 2 (Γ) . L p 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 More general geometric settings Theorem Let Ω be a smooth ( C 2 ) bounded domain of R d . ∃ C > 0 s.t. ∀ k ∈ R d with | k | ≥ 1, any solution u ∈ L 2 (Ω) of � ∆ u = f , for x ∈ Ω for x ∈ ∂ Ω , u ( x ) = 0 , with source term f ∈ L 2 (Ω) satisfies 2 2 1 2 | k | 2 � � ue − k · x � � � ∇ ue − k · x � � ue − k · x � � L 2 (Ω) + L 2 (Ω) + � � � � � � | k | 2 � � � H 2 (Ω) 2 2 � ∂ n ue − k · x � � � � fe − k · x � ≤ C | k | L 2 (Γ k ) + C L 2 (Ω) , � � � � � � where Γ k = { x ∈ ∂ Ω , | k · n x < 0 } . Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Proof by multiplier techniques Setting U = ue − k · x and F = fe − k · x , U satisfies � ∆ U + 2 k · ∇ U + | k | 2 U = F for x ∈ Ω , U ( x ) = 0 , for x ∈ ∂ Ω . Multiplying the equation by U , we first derive L 2 (Ω) ≤ C | k | 2 � U � 2 �∇ U � 2 L 2 (Ω) + C � F � 2 L 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 Multiplying � ∆ U + 2 k · ∇ U + | k | 2 U = F for x ∈ Ω , U ( x ) = 0 , for x ∈ ∂ Ω . by k · ∇ U , we derive � � � | k · ∇ U | 2 dx + Fk · ∇ U dx = 2 ∂ n Uk · ∇ U d σ Ω Ω ∂ Ω − 1 � k · n x |∇ U | 2 d σ 2 ∂ Ω � | k · ∇ U | 2 dx + 1 � k · n x | ∂ n U | 2 d σ, = 2 2 Ω ∂ Ω where we used that, as U = 0 on ∂ Ω , ∇ U = ( ∂ n U ) n x on ∂ Ω . Therefore, ∃ C independent of k ∈ R d s.t. � � � | k · ∇ U | 2 dx ≤ C | F | 2 dx + C | k | | ∂ n U | 2 d σ. Ω Ω Γ k 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 � U � 2 | k · ∇ U | 2 dx . L 2 (Ω) ≤ C Ω Thus, ∀ k ∈ R d , | k | 2 � U � 2 L 2 (Ω) ≤ C � F � 2 L 2 (Ω) + C | k | � ∂ n U � 2 L 2 (Γ k ) . Based on the estimate of ∇ U in terms of U , we thus derive | k | 2 � U � 2 L 2 (Ω) + �∇ U � 2 L 2 (Ω) ≤ C � F � 2 L 2 (Ω) + C | k | � ∂ n U � 2 L 2 (Γ k ) . Recalling that ue − k · x = U and ∇ u e − k · x = ∇ U + kU , we immediately conclude the Carleman estimate. The H 2 -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 Comments on the proof 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 Γ k = { x ∈ ∂ Ω , | k · n x < 0 } depends on the direction of k . � 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 Application to the Calderón Problem 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 R d , and consider the elliptic problem � div ( σ ∇ u ) = 0 , for x ∈ Ω , u ( x ) = g d ( 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 Λ σ : g d �→ σ∂ n u , where u denotes the solution � div ( σ ∇ u ) = 0 , for x ∈ Ω , for x ∈ ∂ Ω . u ( x ) = g d ( x ) , Physically : Λ σ is the so-called Voltage-to-current map. g d is a voltage imposed on the boundary of the object. We can measure the current ( σ ∇ u ) · n x on the boundary ∂ Ω . Mathematically : Λ σ 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 Mathematical framework 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 ) = g d ( x ) , for x ∈ ∂ Ω . Assumption 1 : ∃ C ∗ > 0 , s . t . ∀ x ∈ Ω , 1 σ ∈ C 0 (Ω) , ≤ σ ( x ) ≤ C ∗ . C ∗ If g d ∈ H 1 / 2 ( ∂ Ω) yields u ∈ H 1 (Ω) and ( σ ∇ u ) · n x = σ∂ n u ∈ H − 1 / 2 ( ∂ Ω) . Λ σ : H 1 / 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 Λ σ 1 = Λ σ 2 , can we deduce σ 1 = σ 2 ? Stability : If Λ σ 1 − Λ σ 2 is small (in suitable norms), can we deduce that σ 1 − σ 2 is small (in suitable norms) ? Reconstruction : Given Λ σ , can we compute σ ? In the following, we will focus on the uniqueness problem, i.e. on the injectivity of the map Λ : σ �→ Λ σ . Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Injectivity of Λ : σ �→ Λ σ Main difficulty The map Λ is non-linear. In here, we shall further assume the following : σ and ∂ n σ 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 Liouville’s transform If u solves � div ( σ ∇ u ) = 0 , for x ∈ Ω , for x ∈ ∂ Ω , u ( x ) = g d ( x ) , Then v = σ 1 / 2 u in Ω , solves � ∆ v + qv = 0 , for x ∈ Ω , v ( x ) = h d ( x ) , for x ∈ ∂ Ω . with q = − ∆( σ 1 / 2 ) , in Ω σ 1 / 2 h d = σ 1 / 2 g d , ∂ n v = σ − 1 / 2 ( σ∂ n u ) + ∂ n ( σ 1 / 2 ) g d , 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) → ˜ ˜ Λ q , where Λ q : h d ∈ H 1 / 2 ( ∂ Ω) → ∂ n v ∈ H − 1 / 2 ( ∂ Ω) , ˜ where ∂ n v is given by � ∆ v + qv = 0 , for x ∈ Ω , v ( x ) = h d ( x ) , for x ∈ ∂ Ω . Claim Under the previous assumptions on σ , Λ σ 1 = Λ σ 2 is equivalent to Λ q 1 = ˜ ˜ Λ q 2 , where q 1 = − ∆( σ 1 / 2 q 2 = − ∆( σ 1 / 2 ) ) 1 2 , . σ 1 / 2 σ 1 / 2 1 2 Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Goal Λ : q ∈ L ∞ (Ω) → Λ q , where Λ q : h d ∈ H 1 / 2 ( ∂ Ω) → ∂ n v ∈ H − 1 / 2 ( ∂ Ω) , where ∂ n v is given by � ∆ v + qv = 0 , for x ∈ Ω , v ( x ) = h d ( x ) , for x ∈ ∂ Ω . Theorem Assume d ≥ 3. � q 1 , q 2 ∈ L ∞ (Ω) , ⇒ q 1 = q 2 . Λ q 1 = Λ q 2 Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Preliminary properties of Λ q Proposition Given q ∈ L ∞ (Ω) , the map Λ q is self-adjoint on H 1 / 2 ( ∂ Ω) . i.e. ∀ h 1 , h 2 ∈ H 1 / 2 ( ∂ Ω) , � Λ q h 1 , h 2 � H − 1 / 2 ( ∂ Ω) , H 1 / 2 ( ∂ Ω) = � h 1 , Λ q h 2 � H 1 / 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 Proof Let q ∈ L ∞ (Ω) and, for h 1 and h 2 in H 1 / 2 ( ∂ Ω) , set � ∆ v 1 + qv 1 = 0 , � ∆ v 2 + qv 2 = 0 , for x ∈ Ω , for x ∈ Ω , v 1 ( x ) = h 1 ( x ) , for x ∈ ∂ Ω , v 2 ( x ) = h 2 ( x ) , for x ∈ ∂ Ω . Thus, � Λ q h 1 , h 2 � H − 1 / 2 ( ∂ Ω) , H 1 / 2 ( ∂ Ω) = � ∂ n v 1 , v 2 � H − 1 / 2 ( ∂ Ω) , H 1 / 2 ( ∂ Ω) � � = ∆ v 1 v 2 dx + ∇ v 1 · ∇ v 2 dx Ω Ω � � = − qv 1 v 2 dx + ∇ v 1 · ∇ v 2 dx Ω Ω = � h 1 , Λ q h 2 � H 1 / 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 A polarization formula Proposition Let q 1 and q 2 in L ∞ (Ω) . Then for all h 1 , h 2 in H 1 / 2 ( ∂ Ω) , the solutions v 1 , v 2 of � ∆ v 1 + q 1 v 1 = 0 , � ∆ v 2 + q 2 v 2 = 0 , in Ω , in Ω , v 1 ( x ) = h 1 ( x ) , on ∂ Ω , v 2 ( x ) = h 2 ( x ) , on ∂ Ω , satisfy � � (Λ q 1 − Λ q 2 ) h 1 , h 2 � H − 1 / 2 ( ∂ Ω) , H 1 / 2 ( ∂ Ω) = ( q 2 − q 1 ) v 1 v 2 dx . Ω In particular, if Λ q 1 = Λ q 2 , for all v 1 and v 2 as above, � ( q 2 − q 1 ) v 1 v 2 dx = 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 Proof We compute � Λ q 1 h 1 , h 2 � H − 1 / 2 ( ∂ Ω) , H 1 / 2 ( ∂ Ω) = � ∂ n v 1 , v 2 � H − 1 / 2 ( ∂ Ω) , H 1 / 2 ( ∂ Ω) � � = ∆ v 1 v 2 dx + ∇ v 1 · ∇ v 2 dx Ω Ω � � = − ∇ v 1 · ∇ v 2 dx . q 1 v 1 v 2 dx + Ω Ω Similar computations yield � � � h 1 , Λ q 2 h 2 � H 1 / 2 ( ∂ Ω) , H − 1 / 2 ( ∂ Ω) = − q 2 v 1 v 2 dx + ∇ v 1 · ∇ v 2 dx . Ω Ω As Λ q 2 is self-adjoint, � h 1 , Λ q 2 h 2 � H 1 / 2 ( ∂ Ω) , H − 1 / 2 ( ∂ Ω) = � Λ q 2 h 1 , h 2 � H − 1 / 2 ( ∂ Ω) , H 1 / 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 Idea In particular, if Λ q 1 = Λ q 2 , for all v 1 and v 2 as before, � ( q 2 − q 1 ) v 1 v 2 dx = 0 . Ω Therefore, our next goal is to generate a dense set of functions of the form v 1 v 2 , where � ∆ v 1 + q 1 v 1 = 0 , � ∆ v 2 + q 2 v 2 = 0 , in Ω , in Ω , v 1 ( x ) = h 1 ( x ) , on ∂ Ω , v 2 ( x ) = h 2 ( x ) , on ∂ Ω , Therefore, our goal is to find function v 1 v 2 which approximates the Fourier basis x �→ exp ( − i ξ x ) . Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Looking for specific solutions Remark If ρ ∈ C d satisfies ρ · ρ = 0, then ∆( e ρ · x ) = 0 in R d . There are many such ρ ∈ C d ! If ρ = a + i b , with a , b ∈ R d , � | a | = | b | , ρ · ρ = 0 ⇔ a · b = 0 . NB : If ρ · ρ = 0 and ρ ∈ R d , 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 Complex Geometric Optics solutions Theorem Let q ∈ L ∞ (Ω) . For all ρ = a + i b ∈ C d with ρ · ρ = 0 and | a | ≥ 1, there exists a solution v ρ ∈ L 2 (Ω) of ∆ v ρ + qv ρ = 0 in Ω , that can be written as v ρ ( x ) = e ρ · x + e a · x r ρ ( x ) , with r ρ satisfying the estimate � r ρ � L 2 (Ω) ≤ C | a | . In other words, v ρ ( x ) ≃ e ρ · 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 Proof Remark that v ρ ( x ) = e ρ · x + e a · x r ρ ( x ) solves ∆ v ρ + qv ρ = 0 iff e − a · x (∆ + q )( e a · x r ρ ) = − qe ib · x . q = − qe ib · x . We set ˜ To be proved ∃ r ρ ∈ L 2 (Ω) solution of e − a · x (∆ + q )( e a · x r ρ ) = ˜ q in Ω , with � r ρ � L 2 (Ω) ≤ C | a | . Here, the difficult part is the estimate on r ρ . Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon A duality approach The function r ρ solves e − a · x (∆ + q )( e a · x r ρ ) = ˜ q iff for all w ∈ D (Ω) , � � e a · x (∆ + q )( e − a · x w ) � � dx = ˜ r ρ qw dx , Ω Ω or, by density, for all w ∈ H 2 0 (Ω) . Consequence The set { r ρ solving e − a · x (∆ + q )( e a · x r ρ ) = ˜ q } is an affine space of 0 (Ω) } ⊥ L 2 (Ω) . direction { e a · x (∆ + q )( e − a · x w ) , w ∈ H 2 � The function r ρ of minimal L 2 (Ω) norm : r ρ ∈ { e a · x (∆ + q )( e − a · x w ) , w ∈ H 2 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 { e a · x (∆ + q )( e − a · x w ) , w ∈ H 2 0 (Ω) } = { e a · x (∆ + q )( e − a · x w ) , w ∈ H 2 0 (Ω) } . Indeed, e a · x (∆ + q )( e − a · x w ) = ∆ w − 2 a · ∇ w + | a | 2 w + qw , so the Carleman estimate applied to e − ax w yields 1 | a | 2 � w � 2 � 2 | a | 2 � w � 2 � e a · x (∆ + q )( e − a · x w ) � � H 2 (Ω) ≤ C L 2 (Ω) + L 2 (Ω) . In particular, for all | a | ≥ 1, � e a · x (∆ + q )( e − a · x w ) � � L 2 (Ω) is equivalent to � w � H 2 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 r ρ of minimal L 2 norm writes, for some W ∈ H 2 0 (Ω) , r ρ = e a · x (∆ + q )( e − a · x W ) , and satisfies, for all w ∈ H 2 0 (Ω) , for all w ∈ D (Ω) , � � e a · x (∆ + q )( e − a · x w ) � � dx = ˜ r ρ qw dx , Ω Ω We take w = W : � � | e a · x (∆ + q )( e − a · x W ) | 2 dx = qW dx ≤ � ˜ ˜ q � L 2 � W � L 2 . Ω 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 got � | e a · x (∆ + q )( e − a · x W ) | 2 dx ≤ � ˜ q � L 2 � W � L 2 , Ω while the Carleman estimate implies, for all | a | ≥ 1 and w ∈ H 2 0 (Ω) | a | 2 � w � 2 � 2 � e a · x (∆ + q )( e − a · x w ) � � L 2 (Ω) ≤ C L 2 (Ω) . Consequently, � 1 / 2 �� | e a · x (∆ + q )( e − a · x W ) | 2 dx � r ρ � L 2 (Ω) = Ω ≤ C � ˜ q � L ∞ ≤ C � q � L ∞ . | a | | a | Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Back to the Calderón problem Recall if Λ q 1 = Λ q 2 , for all v 1 and v 2 with ∆ v 1 + q 1 v 1 = 0 , in Ω , ∆ v 2 + q 2 v 2 = 0 , in Ω , we have � ( q 2 − q 1 ) v 1 v 2 dx = 0 . Ω � For each ξ ∈ R d , we will chose ( v 1 , n ) n ∈ N ≃ ( e ρ 1 , n · x ) n ∈ N , ( v 2 , n ) n ∈ N ≃ ( e ρ 2 , n · x ) n ∈ N ,  ρ 1 , n · ρ 1 , n = 0 , lim n →∞ | ρ 1 , n | = ∞ ,  with ρ 2 , n · ρ 2 , n = 0 , lim n →∞ | ρ 2 , n | = ∞ , ρ 1 , n + ρ 2 , n = − i ξ.  Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Choosing ρ 1 , n , ρ 2 , n We want :  ρ 1 , n · ρ 1 , n = 0 , lim n →∞ | ρ 1 , n | = ∞ ,  ρ 2 , n · ρ 2 , n = 0 , lim n →∞ | ρ 2 , n | = ∞ , ρ 1 , n + ρ 2 , n = − i ξ.  As d ≥ 3, we start by choosing α, β ∈ R d such that | α | = | β | = 1 and α · β = β · ξ = α · ξ = 0 . We then set, for n large enough, � � � � γ n β − ξ γ n β + ξ ρ 1 , n = n α + i , ρ 2 , n = − n α − i , 2 2 where n = n 2 − | ξ | 2 γ 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 End of the proof of the uniqueness With the constructed ρ 1 , n , ρ 2 , n , we obtain v 1 , n ( x ) = e ρ 1 , n · x + r 1 , n e n α · x , v 2 , n ( x ) = e ρ 2 , n · x + r 2 , n e − n α · x . with � r 1 , n � L 2 (Ω) ≤ C � q 1 � L ∞ � r 2 , n � L 2 (Ω) ≤ C � q 2 � L ∞ , . | n | | n | Therefore, as Λ q 1 = Λ q 2 , � ( q 2 − q 1 ) e − i ξ x dx Ω � � � e − i ( γ n β + ξ/ 2 ) x r n , 1 ( x ) + e i ( γ n β − ξ/ 2 ) x r n , 2 ( x ) + r n , 1 r n , 2 = − ( q 2 − q 1 ) dx . Ω Taking the limit n → ∞ , � ( q 2 − q 1 ) e − i ξ x dx = 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 Therefore, ∀ ξ ∈ R d , � ( q 2 − q 1 ) e − i ξ x dx = 0 . Ω ⇒ q 1 = q 2 in Ω . Our result is proved : Λ q 1 = Λ q 2 ⇒ q 1 = q 2 . Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Goal Proof 1 Proof 2 Geometry Calderon Comments Here, we studied the Calderón problem for a scalar conductivity σ with the knowledge of the full Dirichlet to Neumann map. • With partial Dirichlet to Neumann map, the Calderón problem can be solved provided there exists suitably “Limiting Carleman Weights”. � [Kenig Sjöstrand Uhlmann 2007] • For anisotropic σ , the Calderón problem is still relevant, 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 = Λ σ , � D Ψ T (Ψ − 1 ( y )) × σ (Ψ − 1 ( y )) × D Ψ(Ψ − 1 ( y )) � where ˜ σ ( y ) = . | det ( D Ψ(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 Cloaking ? 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 � 1 + | x | � 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. Outline The Cauchy problem for the Laplace operator 1 The case of full informations The case of partial data Intermediate case Carleman estimates with a linear weight 2 Goal Proof of the Carleman estimate : Fourier techniques Proof of the Carleman estimate : Multiplier techniques More general geometric settings Application to the Calderón Problem More general Carleman Weights 3 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. General Weight functions 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 ( x 1 ) ? ∃ C > 0, s.t. ∀ s ≥ 1, ∀ u ∈ H 2 ∩ H 1 0 (Ω) , s # � ue s ϕ � L 2 (( 0 , 1 ) × R d − 1 ) ≤ C � ∆ ue s ϕ � L 2 (( 0 , 1 ) × R d − 1 ) + Cs ∗ � � ∂ n u ( 0 , · ) e s ϕ ( 0 ) � L 2 ( R d − 1 ) . � � � 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. Back to the case of a strip Ω = ( 0 , 1 ) × R d − 1 , Γ = { 0 } × R d − 1 . Our goal is to understand for which functions ϕ = ϕ ( x 1 ) one can get an estimate of the form ∃ C > 0, s.t. ∀ s ≥ 1, ∀ u ∈ H 2 ∩ H 1 0 (Ω) , s # � ue s ϕ � L 2 (( 0 , 1 ) × R d − 1 ) ≤ C � ∆ ue s ϕ � L 2 (( 0 , 1 ) × R d − 1 ) + Cs ∗ � � ∂ n u ( 0 , · ) e s ϕ ( 0 ) � L 2 ( R d − 1 ) . � � � 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  ∂ 11 u + ∆ ′ u = f , for ( x 1 , x ′ ) ∈ ( 0 , 1 ) × R d − 1 ,  for x ′ ∈ R d − 1 , u ( 0 , x ′ ) = u ( 1 , x ′ ) = 0 , for x ′ ∈ R d − 1 , ∂ 1 u ( 0 , x ′ ) = g N ( x ′ ) ,  and we set U ( x 1 , x ′ ) = u ( x 1 , x ′ ) e s ϕ ( x 1 ) , F ( x 1 , x ′ ) = f ( x 1 , x ′ ) e s ϕ ( x 1 ) , G n ( x ′ ) = g n ( x ′ ) e s ϕ ( 0 ) . The equations now read ∂ 11 U − 2 s ∂ 1 ϕ∂ 1 U + ( s 2 | ∂ 1 ϕ | 2 − s ∂ 11 ϕ ) U + ∆ ′ U = F ,  in Ω ,  U ( 0 , x ′ ) = U ( 1 , x ′ ) = 0 , in R d − 1 , ∂ 1 U ( 0 , x ′ ) = G N ( x ′ ) , in R d − 1 .  Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont. Factorization of the operator After partial Fourier transform, U + ( s 2 | ∂ 1 ϕ | 2 − s ∂ 11 ϕ ) U − | ξ ′ | 2 ˆ ∂ 11 ˆ U − 2 s ∂ 1 ϕ∂ 1 ˆ U = ˆ  F , in ( 0 , 1 ) ×  U ( 0 , ξ ′ ) = ˆ ˆ U ( 0 , ξ ′ ) = 0 , in R d − 1 , ∂ 1 ˆ U ( 0 , ξ ′ ) = ˆ G N ( ξ ′ ) , in R d − 1 ,  We then recognize that ∂ 11 − 2 s ∂ 1 ϕ∂ 1 + ( s 2 | ∂ 1 ϕ | 2 − s ∂ 11 ϕ ) − | ξ ′ | 2 = ( ∂ 1 − s ∂ 1 ϕ ) 2 − | ξ ′ | 2 = ( ∂ 1 − s ∂ 1 ϕ − | ξ ′ | )( ∂ 1 − s ∂ 1 ϕ + | ξ ′ | ) = ( ∂ 1 − s ∂ 1 ϕ + | ξ ′ | )( ∂ 1 − s ∂ 1 ϕ − | ξ ′ | ) . Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications

Cauchy Linear Weight General Weights Strip Multiplier(1) General Unique Cont. Which conditions on ϕ ? To see what are the needed conditions on ϕ , note that we have to consider the two ODE with operators ( ∂ 1 − s ∂ 1 ϕ − | ξ ′ | ) , ( ∂ 1 − s ∂ 1 ϕ + | ξ ′ | ) . Claim The sign of − s ∂ 1 ϕ − | ξ ′ | and of − s ∂ 1 ϕ + | 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. Conditions on ϕ ? We are considering operators of the form : ( ∂ 1 − s ∂ 1 ϕ − | ξ ′ | ) , ( ∂ 1 − s ∂ 1 ϕ + | ξ ′ | ) . Therefore, for all ξ ′ ∈ R d − 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 ∂ 1 ϕ } > 0 , x 1 ∈ [ 0 , 1 ] ∂ 1 ϕ ( x 1 ) < 0 . max i . e . 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 U + ( s 2 | ∂ 1 ϕ | 2 − s ∂ 11 ϕ ) U − | ξ ′ | 2 ˆ ∂ 11 ˆ U − 2 s ∂ 1 ϕ∂ 1 ˆ U = ˆ  F , in ( 0 , 1 ) ,  U ( 0 , ξ ′ ) = ˆ ˆ U ( 0 , ξ ′ ) = 0 , ∂ 1 ˆ U ( 0 , ξ ′ ) = ˆ G N ( ξ ′ ) ,  we introduce ˆ V ( x 1 , ξ ′ ) = ( ∂ 1 − s ∂ 1 ϕ − | ξ ′ | ) ˆ U ( x 1 , ξ ′ ) :  ∂ 1 ˆ U + ( − s ∂ 1 ϕ − | ξ ′ | ) ˆ U = ˆ V in ( 0 , 1 )   ∂ 1 ˆ V + ( − s ∂ 1 ϕ + | ξ ′ | ) ˆ V = ˆ  in ( 0 , 1 ) F  U ( 0 , ξ ′ ) = ˆ ˆ U ( 1 , ξ ′ ) = 0 ,   V ( 0 , ξ ′ ) = ˆ ˆ G N ( ξ ′ ) .   � No problem to solve ˆ V . What happens for ˆ U ? � ∂ 1 ˆ U + ( − s ∂ 1 ϕ − | ξ ′ | ) ˆ U = ˆ V in ( 0 , 1 ) U ( 0 , ξ ′ ) = ˆ ˆ U ( 1 , ξ ′ ) = 0 , Sylvain Ervedoza Mai 2017 Carleman estimates for elliptic PDE and applications