Rotated weights in global Carleman estimates applied to an inverse problem for the wave equation

Rotated weights in global Carleman estimates applied to an inverse problem for the wave equation. A. Doubova†, A. Osses‡§ †Departamento E.D.A.N., Universidad de Sevilla, Tarfia s/n, E-41012 Sevilla Spain ([email protected]). Partially supported by D.G.E.S., Grant BFM2003–06446. ‡Departamento de Ingener´ıa Matem´atica, Facultad de Ciencias de F´ısicas y Matem´aticas, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile y Centro de Modelamiento Matem´atico, UMI 2807 CNRS-Uchile ([email protected]). This work has been partially supported by grants FONDECYT 1030808–7030059, ECOS-CONICYT C01E02–C04E08 and INRIA-CONICYT. Abstract. In this paper we establish geometrical conditions in order to solve an inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data from a single time-dependent Neumann boundary measurement on a suitable part of the boundary. We prove the uniqueness and the stability results for this problem when a Neumann measurement only located on a part of the boundary satisfying a rotated exit condition. The strategy consist in introducing an angle type dependence in the weight functions used to obtain global Carleman estimates for the wave equation and combination of several of these estimates and then apply it to the inverse problem. Submitted to: Inverse Problems §To whom correspondence should be addressed. Rotated weights in global Carleman estimates 2 1. Introduction The direct methods used to obtain global Carleman estimates for hyperbolic type equations (see, for instance [6, 9, 20, 21, 23]) implicitly involve a multiplier technique, which is a common tool in controllability theory (see [8, 15, 16]). Indeed, after splitting the corresponding partial differential operator, in the deduction of Carleman inequalities, it appears, among other things, an important terms multiplied by the gradient of the solution of the corresponding equation following some direction ∇Φ, where Φ is a specific weight function adapted to the nature of the equation. In the case of a global Carleman estimate for the wave equation, the direction ∇Φ is a radial field of type (x−x0), where x0is some exterior point to the domain. Our first main idea in this work is to modify this weight function Φ in such a way that the gradient ∇Φ is essentially a rotation of the original field (x−x0), but with a magnitude radially depending in space. Using that modified weight function we are able to obtain a modified global Carleman estimates. This idea is directly taken from the rotated multipliers used in exact controllability of hyperbolic type equations [18]. Our second main contribution is that in the Carleman estimates that we prove here, we can take a point x0not only inside the domain (see for instance [10, 20]), but also outside of it. To this end we use a combination of several Carleman estimates. This idea was already used in [5] in the context of Carleman inequality for the heat equation with discontinuous diffusion coefficients. It was shown (see for example [10, 22]) that global Carleman estimates can be applied in order to study the stability (also called lipschitz stability) for some specific kind of inverse problem for the wave equation. The problem is to locally retrieve a time-independent bounded potential in the wave equation by knowing only a single time-dependent over-determining measurement on the whole boundary (or some part of it) of the domain where the wave equation evolves. If the boundary condition is Dirichlet or Neumann, then the complementary (over-determining) measurement is Neumann or Dirichlet respectively. The case of Dirichlet boundary conditions and Neumann overdetermining measurements has been treated in [20, 22] and the other complementary case in [10]. The stability and uniqueness results were obtained (see [10, 20]) if the overdetermining measurements are taken on the part of the boundary which satisfies a geometrical hypothesis saying that it is sufficient that the measure region includes, so called “exit” boundary region with respect to some exterior point x0(see (8) for x0=x1, Γ0= Γ1and b= 0). In this paper we extend the previous geometrical hypothesis. More precisely, we will see that in order to solve the inverse problem (in the case of Dirichlet data and Neumann measurement), it suffices to measure on a rotated exit part of the boundary. This region can be in fact described by an angle parameter θ∈(−π/2, π/2). The case θ= 0 corresponds to the standard exit condition (“exit” boundary region, see Figure 1 left). If θapproaches π/2, the corresponding rotated part of the boundary is Rotated weights in global Carleman estimates 3 Figure 1. From left to right: “exit”,“outgoing” and “incoming” boundary regions in a domain with respect to some exterior point x0marked here with the symbol ×. Of course the two last cases are exactly the opposite if you invert the sense you go round the boundary. If you consider the wave equation with Dirichlet boundary conditions in this domain and if you take a single time-dependent Neumann measurement on some of these regions, then the inverse problem of recovering a time-independent potential in the equation can be locally solved. the one on which the distance to the point x0is increasing if we follow the boundary in the counterclock sense (“outgoing” boundary region, see Figure 1 center). If θis near −π/2, the rotated part is the one in which the distance from the point x0is decreasing if we follow the boundary in the same sense (“incoming” boundary region, see Figure 1 right). There are of course an infinite number of intermediate cases. Our generalization is indeed not optimal. In fact, a reasonable conjecture is that the measurements should be placed in order to capture the geometrical optic rays coming from the support of the unknown potential (see [2]). Although, we are far away to prove it with the techniques presented here. As mentioned before, the essential point in our proofs is a global Carleman estimate with modified weights. The main steps in the obtention of this inequality are taken from [20, 9]. Nevertheless, all our results related to both global Carleman estimates and stability of the inverse problem are stated here in two dimensional domains. In fact, we are able to construct the modified weight function only in two dimensions. The analysis in the case of higher dimensions could be the subject of another work. Axel??? On the other hand, in order to obtain this Carleman inequality we combine several Carleman inequalities with different (but similar) weight functions. This argument allows us to take the mentioned point x0also inside the domain and it does not depend on the dimension. Let us remark that extending our result to the Scrh¨odinger equation may be interesting by itself but the final stability result is not as strong as for the wave equation. In fact, there are also geometrical exit type conditions for the analogous inverse problem in the case of the Scrh¨odinger equation with Dirichlet data and Neumann measurements (see [3]). Nevertheless, in the case of the Scrh¨odinger operator, it is strongly possible Rotated weights in global Carleman estimates 4 that the geometrical optic condition, and particularly the exit type condition, are far from be necessary to solve the inverse problem considered here. Firstly, this is a know fact in control theory that geometrical optics condition is not necessary for the internal exact controllability of the Scrh¨odinger equation [7] or plates [13]. Secondly, there are some recent results in other similar kind of inverse problems for the Scrh¨odinger equation, showing that it is sufficient to measure the whole Dirichlet to Neumann map on a arbitrarily small strategical part of the boundary in order to have uniqueness (see [1, 14]). 2. Main results Let Ω be a bounded connected open subset of R2with boundary ∂Ω of class C2. We will denote by ν=ν(x) the unit outward normal vector to Ω at the point x∈∂Ω. Let UM be a M-bounded set in L∞(Ω), where Mis some positive constant, that is the following set: UM={q∈L∞(Ω) : ||q||L∞(Ω) ≤M}.(1) Let γbe a nonempty open subset of ∂Ω and T > 0. In this paper we will deal with the following inverse problem: Given u0,u1,hand ξin appropriate spaces, we look for sufficient conditions under γand Tsuch that we can find q∈ UMsuch that the solution uof the wave equation      ∂ttu−∆u+q(x)u= 0 in Ω×(0, T), u=hon ∂Ω×(0, T), u(x, 0) = u0, ∂tu(x, 0) = u1in Ω (2) satisfies the additional condition ∂u ∂ν =ξon γ×(0, T).(3) Notice that the dependence of uon qis nonlinear. Let us denote by u(q) and u(q) the solutions of (2) associated to qand to some known potential qrespectively. If we set f=q−qand y=u(q)−u(q), then qu(q)−qu(q) = qy −fu(q). Therefore, we obtain the following equivalent inverse problem for the function y: Given q∈ UM,R=−u(q)and ξ=∂u(q) ∂ν in appropriate spaces, we look for sufficient conditions for γand Tsuch that we can find a time independent function fsuch that the solution yof the wave equation      ∂tty−∆y+q(x)y=f(x)R(x, t)in Ω×(0, T), y= 0 on ∂Ω×(0, T), y(x, 0) = ∂ty(x, 0) = 0 in Ω (4) satisfies the additional condition ∂y ∂ν =ξ−ξon γ×(0, T).(5) Rotated weights in global Carleman estimates 5 Let x0and x1be two points in R2such that (x0≡x1in R2\Ω, |x0−x1| ≤ ²in Ω with ² > 0 sufficintly small. (6) The main novelty of this work is that we will consider that γis sufficiently large in such a way that γ= Γ0∪Γ1,where (7) Γi⊇ {x∈∂Ω:(x−xi)·(aI −bA)ν(x)>0}, i = 0,1 (8) with xi,i= 0,1 verifying (6) and for some a > 0 and b∈Rsuch that a2+b2= 1.(9) Here Iis the identity matrix and A=Ã0−1 1 0 !. Notice that (aI −bA)ν(x) corresponds to a clockwise rotation of the normal field in an angle (see Figure 2) θ= tan−1(b/a)∈(−π/2, π/2),(10) where a= cos θand b= sin θ. (11) Let us introduce the following radial φiand angular φ⊥ idependent scalar fields: φi(x) = |x−xi|2, i = 0,1,(12) φ⊥ i(x) = arg(x−xi), i = 0,1,(13) the last one being the argument of the vector x−xi, that is to say, the angle of the vector x−ximeasured counter-clockwise with respect to the horizontal axis (see Figure 2 for the case x0≡x16∈ Ω). We define the following constants only depending on xi,θand Ω: θi= inf x∈Ωφ⊥ i(x), θi= sup x∈Ω φ⊥ i(x), δθi=θi−θi, i = 0,1,(14) θmax = max(θ0, θ1), θmin = min(θ0, θ1), δθ =θmax −θmin (15) and Rmin = min µinf x∈Ωφ1/2 0(x),inf x∈Ωφ1/2 1(x)¶,(16) Rmax = max µsup x∈Ω φ1/2 0(x),sup x∈Ω φ1/2 1(x)¶,(17) Dθ=¡(Rmax)2exp(2|ρ|δθ)−(Rmin)2¢1/2.(18) Let us consider β∈(0, β2) with β2= min(β0 2, β1 2), where βi 2are two positive constants that will be introduced later on the paper and that will depend on xi,a, Rotated weights in global Carleman estimates 6 ν 0 φ⊥ x δθ Ω φ1/2 θb a x Figure 2. Main notations in the case x0=x16∈ Ω. b,θiand θi,i= 0,1 (see Lemma 4.1 for an exact expression). Given β∈(0, β2), let us introduce the following principal weight functions involved in this paper: Φi(x, t) = a φiexp ¡2ρ φ⊥ i¢−β t2, x ∈Ω, t ∈R, i = 0,1,(19) where we have introduced the parameter ρ=b a, ρ ∈(−∞,+∞).(20) We also introduce the following parameters that will be used all along the paper in order to simplify the expressions: α=β β2 ,0< α < 1,(21) m= max {1,2−4|ρ|} ,1≤m≤2,(22) Fi= exp(2|ρ|δθi)−m a2 4, Fi≥1 2, i = 0,1,(23) Fmin = min(F0, F1),(24) where δθiis given by (14). Obviously, when x0≡x16∈ Ω all these constants and all the both weight functions introduced before are the same for i= 0 and i= 1. Our first main result is on the stability for the inverse problem (2)–(3): Theorem 2.1 Suppose that γsatisfies (8) for some x0, x1∈R2verifying (6),a > 0 and b∈Rwith a2+b2= 1. Let u(q)and u(q)be the respective solutions of (2) associated to q∈ UMand q∈L∞(Ω) and with the measurements ξgiven by (3) and ξ=∂u(q) ∂ν . There exists a time T=T(Ω, θ, x0, x1)>0such that if T > T,u(q)∈H1(0, T;L∞(Ω)) and |u0| ≥ α0>0a.e. in Ω, then there exists a positive constant Cdepending on Ω,T, x0,x1,a,b,M,||u(q)||H1(L∞),||q||L∞,α0,u0,u1and hsuch that ||q−q||L2(Ω) ≤C° °ξ−ξ° °H1(0,T;L2(γ)) ∀q∈ UM.(25) Rotated weights in global Carleman estimates 7 Furthermore, Tis given by T=ÃFmin +pF2 min +m a2 m!1/2Dθ √α,(26) where Fmin,m,Dθand αare respectively defined in (24),(22),(18) and (21). Notice that this stability result is of a local nature, since for a given q∈L∞(Ω) the stability holds in an L∞-ball of radius Mand center q. The stability constant does not depend only on the radius of the ball but also on its center. An immediate consequence of the Theorem 2.1 is the following uniqueness result: Corollary 2.2 Under the hypothesis of Theorem 2.1, if u(q)and u(q)are the solutions of (2) associated to q∈L∞(Ω) and q∈L∞(Ω) respectively, such that ∂u(q) ∂ν =∂u(q) ∂ν on γ×(0, T), then q=qand u(q) = u(q). The proof of the Theorem 2.1, in view of the previous discussions, is a direct consequence of the following result concerning the linearized inverse problem (4)–(5), which is our second stability main result: Theorem 2.3 Assume that the same hypothesis under γ,x0,x1, a,band Tof Theorem 2.1hold. Let be given f∈L2(Ω),R∈H1(0, T;L∞(Ω)) with ||∂tR||L2(L∞)≤ M1, where M1is a positive constant and with |R(x, 0)| ≥ α0>0a.e. in Ω. Then, for each bounded subset UMof L∞(Ω) there exists a positive constant Cdepending on Ω, T,x0,x1,a,b,M,M1and α0such that if yis the solution of (4) we have ||f||L2(Ω) ≤C° ° ° ° ∂y ∂ν ° ° ° °H1(0,T;L2(γ)) ∀q∈ UM.(27) The proof of this theorem will be given in the Section 5. The essential ingredient in the proof is a global Carleman estimate for the wave equation, with the constants depending on qonly via M, which we will deduce in the Section 3 (see Theorem 3.1). We will also use the ideas introduced in [19] at later used, for example, in [3, 10, 22]. Remark 2.1 In our main results (Theorems 2.1 and 2.3) the respective sours terms q and fcan only depend on xand not on t. These restrictive hypothesis are imposed by the arguments used in the Section 5. The stability result for the sours terms depending on x and t(even when the overdetermining measurements are taken in the whole boundary) is, to our knowledge, an open problem. Remark 2.2 Notice that the constant Cin (27) depends on q∈ UMonly via M, therefore the result of Theorem 2.3 can be applied directly to the nonlinear inverse problem, in order to deduce the Theorem 2.1. Rotated weights in global Carleman estimates 8 Remark 2.3 Axel Hecer una remark sobre el hecho de que Γ0y Γ1son muy similares... probablemente un dibujo . Remark 2.4 If we take a= 1 and b= 0 in (19), this corresponds to the case in which no rotation is made, we recover the standard expression for the weight function Φi(x, t) = |x−xi|2−βt2,i= 0,1 which gives the standard “exit” boundary region corresponding to θ= 0 (see Figure 1 left). In addition, if x0≡x16∈ Ω, we have more simple already known situation (see [12]) in which ϕ0=ϕ1,θ= 0 and then Φ0= Φ1. In this case, we also recover the standard inversion time. Indeed, if a= 1 and b= 0, then Fmin = 1/2, ρ= 0, β2= 1 and the left factor in the right hand side of (26) is equal to 1, we have T=D √β, D = (R2 max −R2 min)1/2. Remark 2.5 If a→0 and b→ ±1 then θ→ ±π 2. This corresponds to a maximal rotation (see Figure 1 center and right). Notice that in this case the inversion time T→+∞and behaves asymptotically as T≈Rmaxr2 αexp µ2 cos θδθ¶as θ→ ±π 2. This fact recalls the one encountered in [18] where the control time for the wave equation with a rotated boundary control was of order 2Rmax cos θas θ→ ±π 2. Remark 2.6 The expression (26) for the inversion time Tdoes not come from the deduction of the Carleman inequality in Section 3 nor from the hypothesis in the technical Lemma of Section 4. It appears in the condition (161) of Section 5 in the last step of the analysis. Finally, the following uniqueness result is also an immediate consequence of Theorem 2.3 Corollary 2.4 Under the hypothesis of Theorem 2.3, if yis the solution of (4) such that ∂y ∂ν = 0 on γ×(0, T), then f= 0 and y= 0. The rest of the paper is organized as follows. In Section 3 we prove the global Carleman estimate which we use in the proof of Theorem 2.3. Section 4 deals with the proof of some technical results. Finally, in Section 5 we give the proof of Theorem 2.3. Rotated weights in global Carleman estimates 9 3. Global Carleman inequality with a rotated condition In this section we will deduce a global Carleman inequality we will need for the proof of Theorem 2.3. In the sequel, we will take the following notations Q= Ω ×(−T, T), Σ = ∂Ω×(−T, T) and Σi= Γi×(−T, T), where Γiis defined in (8), depending on the constants a > 0, b∈Rwith a2+b2= 1 and xiverifying (6), i= 0,1. Let us notice that we have chosen the radial and angular fields φiand φ⊥ i,i= 0,1 respectively introduced in (12) and (13) in such a way that they satisfy for all x∈Ω the following: ∇φi= 2(x−xi),(28) ∇φ⊥ i=1 |x−xi|2Ã0−1 1 0 !(x−xi) = 1 φA(x−xi).(29) Then, we can write that the weight functions Φidefined in (19) satisfies for all x∈Ω ∇Φi=¡a∇φi+ 2b φi∇φ⊥ i¢exp ¡2ρφ⊥ i¢ = 2(aI +bA)(x−xi) exp ¡2ρφ⊥ i¢, i = 0,1.(30) Let λsufficiently large positive constant which will be fixed later on. For i= 0,1 we introduce weight functions ϕi(x, t) = exp(λΦi(x, t)) = exp ¡λ¡aφi(x) exp ¡2ρφ⊥ i(x)¢−βt2¢¢.(31) Notice that the following calculations are quite simple ∂tϕi=λϕi∂tΦi=−2λϕiβt, (32) ∂ttϕi=λϕi∂ttΦi+λ∂tϕi∂tΦi=λϕi∂ttΦi+λ2ϕi|∂tΦi|2,(33) ∇ϕi=λϕi∇Φi,(34) ∆ϕi=λϕi∆Φi+λ2ϕi|∇Φi|2.(35) Let us set Z={v:v∈C2(Ω ×[−T, T]), v = 0 on Σ , v(x, −T) = v(x, T) = 0, ∂tv(x, −T) = ∂tv(x, T) = 0 in Ω}. We have the following global Carleman estimate: Theorem 3.1 Assume that x0, x1∈R2verifying (6),a > 0,b∈Rwith a2+b2= 1 and γ= Γ0∪Γ1is such that (8) holds. Then for all M > 0, there exist a positive constants λ,µand Conly depending on a,b,Ω,x0and x1, such that for any q∈L∞(Q)with ||q||L∞(Q)≤M, for any λ≥λany s≥s= (1 + T2) exp(µλT2)M2/3and any v∈Zwe have ZZQ (e2sϕ0+e2sϕ1)(sλ|∂tv|2+sλ|∇v|2+s3λ3|v|2)dx dt ≤CZZQ (e2sϕ0+e2sϕ1)|∂ttv−∆v+q(x, t)v|2dx dt +Csλ ZZγ×(0,T ) (e2sϕ0+e2sϕ1)¯¯¯¯ ∂v ∂ν ¯¯¯¯ 2 dσdt. (36) Rotated weights in global Carleman estimates 16 Then, from (44) taking into account (66) and (67) we obtain ||P1w||2 2+||P2w||2 2+sλ ZZQ ϕ[4∂ttΦ−2K(∂ttΦ−∆Φ)]|∂tw|2dx dt +sλ ZZQ ϕ[4H(Φ)∇w·∇w+K(∂ttΦ−∆Φ)|∇w|2]dx dt + 4s3λ4 ZZQ ϕ3(|∂tΦ|2−|∇Φ|2)2|w|2dx dt + 4s3λ3 ZZQ ϕ3£|∂tΦ|2∂ttΦ + H(Φ)∇Φ·∇Φ¤|w|2dx dt + 2Ks3λ3 ZZQ ϕ3(|∂tΦ|2−|∇Φ|2)(∂ttΦ−∆Φ)|w|2dx dt −2sλ ZZΣ ϕ(∇Φ·ν)¯¯¯¯ ∂w ∂ν ¯¯¯¯ 2 dσ dt ≤2 11 X i=1 |Xi|+||gs||2 2.(68) On the other hand, since |∂tΦ|=| − 2βt| ≤ CT and from (30) we can write that |∇Φ| ≤ C(a, b, Ω), then from (51) we get |X4| ≤ CT2sλ4 ZZQ ϕ(|∂tΦ|2−|∇Φ|2)2|w|2dx dt =CT2sλ4 ZZQ (ϕ2ϕ)ϕ−2(|∂tΦ|2−|∇Φ|2)2|w|2dx dt ≤Cs2λ4 ZZQ ϕ3(|∂tΦ|2−|∇Φ|2)2|w|2dx dt (69) and |X9| ≤ Csλ4 ZZQ ϕ(|∂tΦ|2−|∇Φ|2)2|w|2dx dt ≤Cs2λ4 ZZQ ϕ3(|∂tΦ|2−|∇Φ|2)2|w|2dx dt (70) for all s≥s1=µ1(a, b, Ω)(1 + T2)e2λT 2with µ1(a, b, Ω) a positive constant. In (69) and (70), we have used that (|∂tΦ|2−|∇Φ|2)≤C(|∂tΦ|2−|∇Φ|2)2 and ϕ−2= exp (−2λΦ) ≤exp ¡2λ¡βT2−aφ exp(2ρφ⊥)¢¢≤s(71) for s≥µ1(a, b, Ω) e2λT2. Thus, from (68), (69) and (70) we get ||P1w||2 2+||P2w||2 2+sλ ZZQ ϕ[4∂ttΦ−2K(∂ttΦ−∆Φ)]|∂tw|2dx dt Rotated weights in global Carleman estimates 17 +sλ ZZQ ϕ[4H(Φ)∇w·∇w+ 2K(∂ttΦ−∆Φ)|∇w|2]dx dt + 4s3λ4 ZZQ ϕ3(|∂tΦ|2−|∇Φ|2)2|w|2dx dt + 4s3λ3 ZZQ ϕ3£|∂tΦ|2∂ttΦ + H(Φ)∇Φ·∇Φ¤|w|2dx dt + 2Ks3λ3 ZZQ ϕ3(|∂tΦ|2−|∇Φ|2)(∂ttΦ−∆Φ)|w|2dx dt −2sλ ZZΣ ϕ(∇Φ·ν)¯¯¯¯ ∂w ∂ν ¯¯¯¯ 2 dσ dt ≤ 11 X i=1,i6=4,9|Xi|+||gs||2 2(72) for all s≥s2= max(s1, C) where s2is of the form s2=µ2(a, b, Ω)(1 + T2)e2λT 2with µ2(a, b, Ω) a positive constant. We will choose in the next section the constants βand Kin Lemma 4.1 such that in the left hand side of (72) all the term on |∂tw|2,|∇w|2and |w|2have a good (positive) sign. More precisely, we must have the following: 4∂ttΦ−2K(∂ttΦ−∆Φ) >0,(73) 4H(Φ)ξ·ξ+ 2K(∂ttΦ−∆Φ)|ξ|2>0 for all ξ∈R2\{0}(74) and 4λ(|∂tΦ|2−|∇Φ|2)2+ 4 £|∂tΦ|2∂ttΦ + H(Φ)∇Φ·∇Φ¤ + 2K(|∂tΦ|2−|∇Φ|2)(∂ttΦ−∆Φ) >0.(75) Let us use Lemma 4.1 for i= 0. According to it, if β∈(0, β0 2) and K≡K0is such that (109) holds for i= 0, then for λ≥λ0with λ0=λ0(a, b, Ω, x0) sufficiently large positive constant (73)–(75) are fulfilled. Moreover, we have uniform positiveness. Consequently, from (72) we deduce that sλ ZZQ ϕ(|∂tw|2+|∇w|2)dx dt +s3λ3 ZZQ ϕ3|w|2dx dt ≤CÃ||gs||2 2+sλ ZZΣ0 ϕ(∇Φ·ν)¯¯¯¯ ∂w ∂ν ¯¯¯¯ 2 dσ dt + 11 X i=1,i6=4,9|Xi|!(76) for all λ≥λ0and s≥s2. Now, we will see that all the term Xi,i= 1, . . . , 11, i6= 4,9 in the right hand side of (76) can be controlled by the terms in the left hand side. More precisely, using (19) and (30) we obtain |∂tΦ|=|−2βt| ≤ CT, (77) |∂ttΦ|=|−2β| ≤ C, (78) ∆Φ = 4 ae2ρφ⊥,(79) Rotated weights in global Carleman estimates 18 ||∂tΦ|2−|∇Φ|2|=|4β2t2−4φe4ρφ⊥| ≤ C(a, b, Ω, x0)T2(80) |∂ttΦ−∆Φ|=¯¯¯¯ 2µβ+2 ae2ρφ⊥¶¯¯¯¯≤C(a, b, Ω, x0).(81) Moreover, using the computation from Lemma 4.1 for i= 0, thanks to (124), (125) and (138) we get H(Φ)∇Φ·∇Φ = 81 + b2 aφe6ρφ⊥≤C(a, b, Ω, x0),(82) ¯¯¯¯ ∂2Φ ∂xk∂xj¯¯¯¯≤C(a, b, Ω, x0).(83) Recall that here we use the notations Φ = Φ0,φ=φ0,... Thus, from (47), (49), (50), (52)–(58), (60) and (61) and taking into account (77)–(81), we deduce the following: |X1| ≤ CT2sλ3 ZZQ ϕ|w|2dx dt =CT2sλ3 ZZQ (ϕ2ϕ)ϕ−2|w|2dx dt ≤Cs2λ3 ZZQ ϕ3|w|2dx dt, (84) |X2| ≤ CT2sλ3 ZZQ ϕ|w|2dx dt ≤Cs2λ3 ZZQ ϕ3|w|2dx dt, (85) |X3| ≤ Csλ3 ZZQ ϕ|w|2dx dt ≤Cs2λ3 ZZQ ϕ3|w|2dx dt, (86) |X5| ≤ Csλ2 ZZQ ϕ|w|2dx dt ≤Cs2λ2 ZZQ ϕ3|w|2dx dt, (87) |X6| ≤ Csλ2 ZZQ ϕ|w|2dx dt ≤Cs2λ2 ZZQ ϕ3|w|2dx dt, (88) |X7| ≤ Csλ3 ZZQ ϕ|w|2dx dt ≤Cs2λ3 ZZQ ϕ3|w|2dx dt, (89) |X8| ≤ CT2sλ3 ZZQ ϕ|w|2dx dt ≤Cs2λ3 ZZQ ϕ3|w|2dx dt, (90) |X10| ≤ Csλ3 ZZQ ϕ|w|2dx dt ≤Cs3λ3 ZZQ ϕ3|w|2dx dt, (91) |X11| ≤ Csλ2 ZZQ ϕ|w|2dx dt ≤Cs2λ2 ZZQ ϕ3|w|2dx dt. (92) for all s≥s3=µ3(a, b, Ω)(1 + T2)e2λT 2with µ3(a, b, Ω) a positive constant. On the other hand, from (41) and due to (71) we get ||gs||2 2≤ ||esϕg−q(x, t)w||2 2+Csλ ZZQ ϕ|w|2dx dt ≤ ||esϕg−q(x, t)w||2 2+Cs2λ ZZQ ϕ3|w|2dx dt. (93) Therefore, using (84)–(92) and (93) in (76) we obtain sλ ZZQ (|∂tw|2+|∇w|2)dx dt +s3λ3 ZZQ|w|2dx dt Rotated weights in global Carleman estimates 19 ≤CÃ||esϕg−q(x, t)w||2 2+sλ ZZΣ0 (∇Φ·ν)¯¯¯¯ ∂w ∂ν ¯¯¯¯ 2 dσ dt!(94) for all λ≥λ0and s≥s4= max(s2, s3, C), s4=µ4(a, b, Ω)(1+T2)e2λT2with µ4(a, b, Ω) a positive constant. In (94), we also have used that the function ϕgiven by (31) is such that ϕ > 0 and ϕ≤C(a, b, Ω, x0) in Ω ×(−T, T). Let us deduce from (94) that the inequality (37) holds for all s≥s0, where s0is as in the Theorem 3.2. Recall that w=esϕv. Then, using (32) and (34) we have ∂tw=esϕ(∂tv+s∂tϕv) = esϕ(∂tv−2sλβtv), ∇w=esϕ(∇v+s∇ϕv) = esϕ(∇v+sλϕ∇Φv). Then, we can write that e2sϕ∂tv=∂tw+ 2sλβtesϕv≡∂tw+ 2sλβtw, (95) esϕ∇v=∇w−sλϕ∇Φesϕv≡ ∇w−sλϕ∇Φw. (96) Consequently, we find the following: sλ ZZQ e2sϕ|∂tv|2dx dt ≤Csλ ZZQ|∂tw|2dx dt +Cs3λ3T2 ZZQ|w|2dx dt (97) and sλ ZZQ esϕ|∇v|2dx dt ≤Csλ ZZQ|∇w|2dx dt +Cs3λ3 ZZQ ϕ2|∇Φ|2|w|2dx dt ≤Csλ ZZQ|∇w|2dx dt +Cs3λ3 ZZQ|w|2dx dt. (98) In (98), we have used that ϕ≤C(a, b, Ω, x0) in Ω ×(−T, T),(99) |∇Φ| ≤ C(a, b, Ω, x0) in Ω ×(−T, T) (100) since (31) and (30) hold. On the other hand, on Σ we have ∂w ∂ν =esϕ µ∂v ∂ν +s∂ϕ ∂ν v¶≡esϕ ∂v ∂ν (101) since v= 0 on Σ. Then, taking into account (97), (98) and (101) in (94) we deduce that there exists a positive constant Conly depending on a,b, Ω, and x0such that ZZQ e2sϕ(sλ|∂tv|2+sλ|∇v|2+s3λ3|v|2)dx dt ≤CÃ||esϕg||2 2+||q||2 L∞(Q) ZZQ e2sϕ|v|2dx dt +sλ ZZΣ0 e2sϕ(∇Φ·ν)¯¯¯¯ ∂v ∂ν ¯¯¯¯ 2 dσ dt! ≤Cµ||esϕg||2 2+M2 ZZQ e2sϕ|v|2dx dt +sλ ZZΣ0 e2sϕ(∇Φ·ν)¯¯¯¯ ∂v ∂ν ¯¯¯¯ 2 dσ dt!(102) Rotated weights in global Carleman estimates 20 Finally, taking s≥s0with s0= (1 + T2)eµ0λT 2M2/3, where µ0is a positive constants only depending on a,b, Ω and x0we get the estimate (37). This ends the proof of Theorem 3.2. 3.2. Proof of Theorem 3.1 In order to obtain (36) we will use global Carleman inequality (37) and the arguments from [5]. Let x0, x1be two points in R2such that (6) holds. Let B0=B(x0, ε/3) ⊂⊂ Ω and B1=B(x1, ε/3) ⊂⊂ Ω be two fixed small balls of the radius ε/3 centered in x0and x1 respectively. Let us consider ϕi,i= 0,1 two wight functions given in (31). We notice that if xi∈Ω, then infxi∈Ωφi(x) = 0, that is to say φican vanish only in Bi,i= 0,1. Therefore, we can write two Carleman estimate similar to (37). More precisely, there exist a positive constants λi,si,µiand Conly depending on a,b, Ω and xi, such that for any λ≥λiwe have sλ ZZQ e2sϕi(|∂tv|2+|∇v|2)dx dt +s3λ3 ZZQ e2sϕi|v|2dx dt ≤CµZZQ e2sϕi|∂ttv−∆v+q(x, t)v|2dx dt +sλ ZZΓi×(0,T ) e2sϕi¯¯¯¯ ∂v ∂ν ¯¯¯¯ 2 dσdt +s3λ3 ZZBi×(0,T ) e2sϕi|v|2dx dt!.(103) for all s≥si= (1 + T2) exp(µiλT2)M2/3,i= 1,2 and all v∈Z. In the sequel we will see that from (103) and using the properties of the weight functions ϕ0and ϕ1we are able to get (36). More precisely, let us first justify the following properties of the wight functions: for each C > 0, there exists s7>0 such that e2sϕ1≥2Ce2sϕ0in B0, (104) e2sϕ0≥2Ce2sϕ1in B1, (105) for all s≥s7=σ7(a, b, Ω, x0, x1)T2. Indeed, using the definitions of ϕi(31), it easy to deduce that for all λ≥1, there exists a positive constant κ0such that ϕ1−ϕ0≥κ0ϕ1in B0, Then we obtain that for each C > 0, there exists s6>0 such that e2sϕ1 e2sϕ0=e2s(ϕ1−ϕ0)≥e2sκ0ϕ1≥e2sκ0min ϕ1≥2Cin B0 for all s≥s5=σ5(a, b, Ω, x0, x1)T2. This is exactly the inequality (104) for s≥s5. Using similar arguments and the fact that ϕ0−ϕ1≥κ1ϕ0in B1, Rotated weights in global Carleman estimates 21 for some positive constant κ1, we deduce that, for each C > 0 there exists s6such that e2sϕ0 e2sϕ1≥2Cin B1 for all s≥s6=σ6(a, b, Ω, x0, x1)T2. Thus, (105) holds for s≥s6.Taking now s7= max(s5, s6) we get (104) and (105) for s≥s7. Now, adding two Carleman inequality (103), we obtain sλ ZZQ (e2sϕ0+e2sϕ1)(|∂tv|2+|∇v|2)dx dt +s3λ3 ZZQ (e2sϕ0+e2sϕ1)|v|2dx dt ≤CµZZQ (e2sϕ0+e2sϕ1)|∂ttv−∆v+q(x, t)v|2dx dt¶ +Csλ ZZΓi×(0,T ) (e2sϕ0+e2sϕ1)¯¯¯¯ ∂v ∂ν ¯¯¯¯ 2 dσdt +Cµs3λ3 ZZB0×(0,T ) e2sϕ0|v|2dx dt +s3λ3 ZZB1×(0,T ) e2sϕ1|v|2dx dt¶.(106) for i= 0,1, for all s≥s8= max(s0, s1, s7) and λ≥λ= max(λ0, λ1). On the other hand, taking into account (104) and (105), we get Cs3λ3 ZZB0×(0,T ) e2sϕ0|v|2dx dt +Cs3λ3 ZZB1×(0,T ) e2sϕ1|v|2dx dt ≤s3λ3 2ZZB0×(0,T ) e2sϕ1|v|2dx dt +s3λ3 2ZZB1×(0,T ) e2sϕ0|v|2dx dt ≤s3λ3 2ZZQ (e2sϕ0+e2sϕ1)|v|2dx dt (107) for all s≥s9=σ9(a, b, Ω, x0, x1)T2. Combining (107) and (106) we find (36) for all s≥s= max(s8, s9). Moreover sis of the form s= (1 + T2) exp(µλT2)M2/3, with µa positive constant only depending on a,b, Ω, x0and x1.This ends the proof of Theorem 3.1. 4. A technical lemma In this section we precise the conditions over βand Kin order to fulfill the inequalities (73) to (75), which are necessary for the Carleman estimate of Theorem 3.2. Lemma 4.1 Assume that xi6∈ Ωand let the functions Φibe given by (19) i= 0,1, a > 0and b∈Rwith a2+b2= 1. Let us define for i= 0,1 θi +=(θiif b≥0 θiif b < 0θi −=(θiif b≥0 θiif b < 0(108) Then there exists a positive constant βi 2depending on a,b,θi +and θi −such that for all β∈(0, βi 2)and for all Kiin the nonempty interval 2β a β a + 2e2ρθi − < Ki<a2m e2ρθi − β a + 2e2ρθi + , i = 0,1 (109) Rotated weights in global Carleman estimates 22 (where ρand mare defined in (20) and (22) respectively) there exists λi(a, b, Ω, xi)>0 and Nj=Nj(a, b, Ω, xi),j= 1,2,3such that we have the following inequalities for all i= 0,1,λ≥λi,x∈Ωand t∈R: 4∂ttΦi−2K(∂ttΦi−∆Φi)≥N1>0,(110) 4H(Φi)ξ·ξ+ 2K(∂ttΦi−∆Φi)|ξ|2≥N2>0∀ξ∈R2\{0},(111) 4λ(|∂tΦi|2−|∇Φi|2)2+ 4 £|∂tΦi|2∂ttΦi+H(Φi)∇Φi·∇Φi¤+ 2K(|∂tΦi|2−|∇Φi|2)(∂ttΦi−∆Φi)≥N3>0.(112) Remark 4.1 Notice that for a= 1 and b= 0 (this corresponds to the case θ= 0 where no rotation is made) the inequality (109) becomes 2β β+ 2 < Ki<2 β+ 2 , i = 0,1 (113) which is nonempty for all β∈(0,1) and βi 2= 1 in this case. Proof of Lemma 4.1: In the sequel we will assume that i= 0,1. Let us first see that the interval in (109) is nonempty. We must have 2β a β a + 2 e2ρθi − <a2m e2ρθi − β a + 2 e2ρθi + .(114) The inequality (114) is equivalent to the following one (recall that a > 0): 2β2a+ 4β Fie2ρθi −−2a m e4ρθi −<0,(115) where Fwas defined in (23). The equation 2 β2a+ 4β Fie2ρθi −−2a m e4ρθi −= 0 has two solutions: βi 1=−Fi−p(Fi)2+m a2 ae2ρθi −(116) and βi 2=−Fi+p(Fi)2+m a2 ae2ρθi −.(117) Clearly βi 1<0 and βi 2>0. This proves that for β∈(0, βi 2) we have (114). In order to simplify the expressions that follows in this proof, let us introduce the notation xi=x−xiand xi= (xi1, xi2).(118) Notice that using the hypothesis that xi6∈ Ω we have that Φi(x)≥inf x∈Ω|xi|2>0,∀x∈Ω.(119) Let us now verify (111). Using (30) we can compute for x∈Ω the first derivatives ∂Φi ∂x1 = 2e2ρφ⊥ i(axi1−bxi1),(120) ∂Φi ∂x2 = 2e2ρφ⊥ i(axi2+bxi1),(121) Rotated weights in global Carleman estimates 23 and the second derivatives which are well defined thanks to (119) ∂2Φi ∂x2 1 = 2e2ρφ⊥ iµa−2ρ(axi1x2−bx2 i2) φi¶,(122) ∂2Φi ∂x2 2 = 2e2ρφ⊥ iµa+ 2ρ(axi1xi2+bx2 i1) φi¶,(123) ∂2Φi ∂x2∂x1 = 2e2ρφ⊥ iµ−b+ 2ρ(ax2 i1−bxi2xi1) φi¶,(124) ∂2Φi ∂x1∂x2 = 2e2ρφ⊥ iµb−2ρ(ax2 i2+bxi1xi2) φi¶.(125) The reader can easily verify that the last two derivatives (124) and (125) are of course equal, but it is convenient to write them in this form in order to have the following expression for the second derivative matrix or Hessian matrix H(Φi) = 2e2ρφ⊥ iÃ"a−b b a #+2b aφi D(Φi)!,(126) with D(Φi) = "−axi1xi2+bx2 i2ax2 i1−bxi1xi2 −ax2 i2−bxi1xi2axi1xi2+bx2 i1#.(127) Let us first see that H(Φi)ξ·ξ≥a m e2ρφ⊥ i|ξ|2for all ξ∈R2,ξ= (ξ1, ξ2), ξ6= 0, (128) where mis given in (22). First it is clear that "a−b b a #"ξ1 ξ2#·"ξ1 ξ2#=a|ξ|2.(129) Using now (127) and making the computations we get D(Φi)"ξ1 ξ2#·"ξ1 ξ2#=a(ξ1xi1+ξ2xi2)(ξ2xi1−ξ1xi2) + b(ξ2xi1−ξ1xi2)2. Using this equality and (126) in (129) we deduce that H(Φi)ξ·ξ= 2 e2ρφ⊥ iµa|ξ|2+2b2(ξ2xi1−ξ1xi2)2 aφi +2b(ξ2xi1−ξ1xi2)(ξ1xi1+ξ2xi2) φ¶.(130) There are at least two ways for finding an lower bound of the expression (130). Indeed, taking in account that ¯¯¯¯ 2b√a(ξ2xi1−ξ1xi2)(ξ1xi1+ξ2xi2) √aφi¯¯¯¯≤2b2(ξ2xi1−ξ1xi2)2 aφi +a 2|ξ|2 Rotated weights in global Carleman estimates 24 and ¯¯¯¯ 2b(ξ2xi1−ξ1xi2)(ξ1xi1+ξ2xi2) φi¯¯¯¯≤2|b||ξ|2, these bounds are respectively H(Φi)ξ·ξ≥a e2ρφ⊥ i|ξ|2 and H(Φi)ξ·ξ≥(2a−4|b|)e2ρφ⊥ i|ξ|2. Taking the best of the two lower bounds and using the definition of the positive constant mgiven in (22) we obtain (128). On the other hand, we have ∂tΦi=−2βt, (131) ∂ttΦi= 2β, (132) ∆Φi=4 ae2ρφ⊥ i,(133) ∂ttΦi−∆Φi=−2β−4 ae2ρφ⊥ i,(134) |∂tΦi|2−|∇Φi|2= 4β2t2−4φie4ρφ⊥ i.(135) In order to verify (111), taking into account (128) and (134), it is sufficient to have the following: 4a me2ρφ⊥ i−2Kµ2β+4 ae2ρφ⊥ i¶>0.(136) Since K < a2m e2ρθi − β a + 2 e2ρθi + = inf x∈Ω a2m e2ρφ⊥ i β a + 2 e2ρφ⊥ i , the inequality (136) holds and then (111) is also true. Now, we verify the condition (110). Using (132) and (134), (110) holds if −4β+ 2Kiµβ+2 ae2ρφ⊥ i¶>0.(137) By (109), Kiis such that Ki>2β a β a + 2 e2ρθi − = sup x∈Ω 2β a β a + 2 e2ρφ⊥ i . So, we have (137). This proves that (110) holds. Let us now check the condition (112). Taking ξ≡ ∇Φiwith ∇Φigiven by (120)- (121) we obtain |ξ|= 2 φie2ρφ⊥ i, ξ1xi1+ξ2xi2= 2a φie2ρφ⊥ i, ξ2xi1−ξ1xi2= 2b φie2ρφ⊥ i, Rotated weights in global Carleman estimates 25 thus from (130) we easily obtain H(Φi)∇Φi·∇Φi= 8φie6ρφ⊥ iµa+2b4 a+ 2ab2¶= 81 + b2 aφie6ρφ⊥ i,(138) where we have used that a2+b2= 1. Taking into account (131), (132), (134), (135) and (138), in order to prove the inequality (112) it suffices to show that for x∈Ω and t∈Rwe have: 4λ³β2t2−φie4ρφ⊥ i´2−2β3t2+ 21 + b2 aφie6ρφ⊥ i −K³β2t2−φie4ρφ⊥ i´µβ+2 ae2ρφ⊥ i¶>0. By adding and substracting 2βφie4ρφ⊥ iand after introducing the auxiliary function Y(x, t) = β2t2−φie4ρφ⊥ i, we can rewrite the previous condition as 4λ Y 2−c1(x)Y+c2(x)φi(x)>0,(139) where c1(x) = (2+K)β+2K ae2ρφ⊥ i, c2(x) = 2 µ1 + b2 ae2ρφ⊥ i−β¶e4ρφ⊥ i.(140) Notice that (2 + Ki)β+2Ki ae2ρθi −≤c1(x)≤(2 + Ki)β+2K ae2ρθi +,∀x∈Ω (141) and if 0< β < βi 3with βi 3=1 + b2 ae2ρθi − and using the fact that xi6∈ Ω, we would have that there exist a constant Csuch that c2(x)φi(x)≥C > 0,∀x∈Ω.(142) This is true for β∈(0, βi 2) since in fact βi 2≤βi 3. To see this, notice that a≤1 and from the definitions of mand Fiin (22) and (23) we have m≤2 and Fi≤1 2, thus a2m≤2≤2(1 + b2)Fi+ (1 + b2)2,(143) that is (Fi)2+a2m≤(Fi)2+ 2(1 + b2)Fi+ (1 + b2)2= (Fi+ (1 + b2))2(144) so we obtain βi 2=−Fi+p(Fi)2+m a2 ae2ρθi −≤1 + b2 ae2ρθi −≡βi 3.(145) In conclusion, if β∈(0, βi 2) and xi6∈ Ω, the coefficients c1(x) and c2(x)φi(x) in (139) are respectively uniformly bounded and uniformly positive for x∈Ω, therefore, it is sufficient to take λ≥λiwith λi=λi(a, b, Ω, xi) large enough in order to obtain (139) and consequently the desired condition (112). This ends the proof of this lemma. Rotated weights in global Carleman estimates 32 +Cs Z0 −TZΩ (e2sϕ0+e2sϕ1)|∂tw+sλw|2dx dt ≤CZ0 −TZΩ (e2sϕ0+e2sϕ1)(s|∂tw|2+sλ|∇w|2+s3λ2|w|2)dx dt. (188) since (30) and (32) hold. Now, taking into account (183), (33) and (35) we can write that me he parado here |I5| ≤ Csλ2Z0 −TZΩ (esϕ0+esϕ1)2|w(∂tw+sλw)|dx dt ≤Csλ Z0 −TZΩ (e2sϕ0+e2sϕ1)|∂tw|2dx dt +C(s2λ3+sλ3)Z0 −TZΩ (e2sϕ0+e2sϕ1)|w|2dx dt ≤CZ0 −TZΩ (e2sϕ0+e2sϕ1)(sλ|∂tw|2+s2λ3|w|2)dx dt. (189) By (184), (32) and (34) we deduce that |I6| ≤ CZ0 −TZΩ (e2sϕ0+e2sϕ1)(s2λ2|w∂tw|+s3λ3|w|2)dx dt ≤CZ0 −TZΩ (e2sϕ0+e2sϕ1)(sλ|∂tw|2+s3λ3|w|2)dx dt. (190) Then, from (178) and (185)–(190) we get Z0 −TZΩ (Lz)∂tz dx dt ≤CZ0 −TZΩ (e2sϕ0+e2sϕ1)(sλ|∂tw|2+sλ|∇w|2+s3λ3|w|2)dx dt +CZ0 −TZΩ (e2sϕ0+e2sϕ1)χ2|f(x)|2|∂tR(x, t)|2dx dt +CZ0 −TZΩ (e2sϕ0+e2sϕ1)|2∂tχ∂tψ+∂ttχψ|2dx dt ≤CZZQ (e2sϕ0+e2sϕ1)(sλ|∂tw|2+sλ|∇w|2+s3λ3|w|2)dx dt +CZ0 −TZΩ (e2sϕ0+e2sϕ1)χ2|f(x)|2|∂tR(x, t)|2dx dt +CZ0 −TZΩ (e2sϕ0+e2sϕ1)|2∂tχ∂tψ+∂ttχψ|2dx dt (191) with Ca positive constant only depending on a,b, Ω, x0,x1and T. Now, using Carleman inequality (172) in (191), we deduce that the following holds: Z0 −TZΩ (Lz)∂tz dx dt ≤CZZQ (e2sϕ0+e2sϕ1)χ2|f(x)|2|∂tR(x, t)|2dx dt Rotated weights in global Carleman estimates 33 +CZZQ (e2sϕ0+e2sϕ1)|2∂tχ∂tψ+∂ttχψ|2dx dt +Csλ ZZγ×(0,T ) (e2sϕ0+e2sϕ1)¯¯¯¯ ∂ψ ∂ν ¯¯¯¯ 2 dσdt for all s≥sand λ≥λand with a positive constant C=C(a, b, Ω, x0, x1, T). Here, we also have used that ¯¯¯¯ ∂w ∂ν ¯¯¯¯ =¯¯¯¯ ∂ψ ∂ν χ¯¯¯¯ , since w=χψ with χgiven by (163). The last inequality is exactly the estimate (169). Step 4: We will conclude the proof of Theorem 2.3. Let us consider the second term of the right hand side of (169). Using that ∂tχ=∂ttχ= 0 in [−T, −T+δ/2] ∪[−T+ δ, T −δ]∪[T−δ/2, T] and taking into account (162) and (152) we have ZZQ (e2sϕ0+e2sϕ1)|2∂tχ∂tψ+∂ttχψ|2dx dt ≤CZ−T+δ −T+δ/2ZΩ (e2sϕ0+e2sϕ1)(|∂tψ|2+|ψ|2)dx dt +CZT−δ/2 T−δZΩ (e2sϕ0+e2sϕ1)(|∂tψ|2+|ψ|2)dx dt ≤Ce2sexp(λ(c−ε)) ³||∂tψ||2 L2(L2)+||ψ||2 L2(L2)´ ≤Ce2sexp(λ(c−ε)) ³||f(·)R(·,0)||2 L2+||f∂tR||2 L2(L2)´ ≤e2sexp(λ(c−ε))||f||2 L2³||R(·,0)||2 L∞+||∂tR||2 L2(L∞)´,(192) where Cis a positive constant depending on Ω, Tand M. Let us now consider the first term on the right hand side of (169). Since R∈H1(−T, T;L∞(Ω)) and |R(x, 0)| ≥ α0>0, then we define g0∈L2(−T, T) by g0(t) = sup x∈Ω |∂tR(x, t)| |R(x, 0)|. Then, taking into account (31) we have ZZQ (e2sϕ0+e2sϕ1)χ2|f(x)|2|∂tR(x, t)|2dx dt ≤CZZQ (e2sϕ0+e2sϕ1)|f(x)|2|R(x, 0)|2|g0(t)|2dx dt ≤CZΩ e2sϕ0(x,0)|f(x)|2|R(x, 0)|2G0(x)dx +CZΩ e2sϕ1(x,0)|f(x)|2|R(x, 0)|2G1(x)dx, (193) where Gi(x) = ZT −T|g0(t)|2e2sϕi(x,0)(e−λβt2−1) dt, i = 0,1. Rotated weights in global Carleman estimates 34 Thanks the Lebesgue’s theorem we deduce that for ssufficiently large Gi→0 in Ω, i = 0,1.(194) On the other hand, since |R(x, 0)| ≥ α0>0 holds and using (158) we obtain 1 2ZΩ (esϕ0(x,0) +esϕ1(x,0))2|f(x)|2|R(x, 0)|2dx ≥CZΩ (e2sϕ0(x,0) +e2sϕ1(x,0))|f(x)|2dx ≥Ce2sexp(λc)||f||L2.(195) Let us fix λ=λ. From (168), (195) and (169), (192), (193) and (194) we get e2sexp(λc)||f||L2≤C(a, b, Ω, T, x0, x1, M, M1, α0)e2sexp(λ(c−ε))||f||L2 +C(a, b, Ω, x0, x1, T)s ZZγ×(0,T )¯¯¯¯ ∂ty ∂ν ¯¯¯¯ 2 dσdt (196) for all s≥s. Thus, for slarge enough the inequality (196) implies (25). Acknowledgements. The authors want to acknowledge Jean-Pierre Puel for fruitful discussions. [1] H. Ammari, G. Uhlmann, Reconstruction of the potential from partial Cauchy data for the Schr¨odinger equation, Indiana Univ. Math. J., 53(1), (2004), 169–183. [2] C. Bardos, G. Lebeau, and J. 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