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18 JÉRÔME LE ROUSSEAU AND GILLES LEBEAU 7.2. Estimate at the boundary. If we place ourselves in the neighborhood of the boundary we have the following result. Theorem 7.5 (Carleman estimate at the boundary). Let x 0 ∈ ∂Ω and K be a compact set of Ω, x 0 ∈ K, and V an open subset of Ω that is a neighborhood of K in Ω, with K and V chosen sufficiently small. Let ϕ be a weight function that satisfies Assumption 7.1 in V, with (7.1) replaced by (7.3), and ∂ n ϕ| ∂Ω∩V < 0, where n is the outward pointing unit normal to Ω. Then there exist C > 0 and δ 3 > 0 such that ‖h 1 2 e ϕ/h u‖ 2 L 2 (Q) + ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 (Q) ≤ C‖h 2 e ϕ/h Pu‖ 2 L 2 (Q), for 0 < (T + T 2 )ε ≤ δ 3 , h = εt(T − t) and u ∈ C ∞ ([0, T] × Ω), with supp(u(t)) ⊂ K for all t ∈ [0, T], and u| (0,T)×(∂Ω∩V) = 0. The proof of this estimate is more technical than that of Theorem 7.3. We have placed it in Appendix A.9. The idea of the proof is to use the Gårding inequality in the tangential directions, including the time direction. The original proof for this estimate is available in [FI96]. However, following the approach of [FI96] does not put forward the sufficiency of the sub-ellipticity condition (7.3). 7.3. Global estimate. We now focus our attention on global Carleman estimates. We proceed by patching together the local estimates we have presented here, in the interior and at the boundary. The global aspect of the estimate will impose an “observation” term over (0, T) × ω, with ω ⋐ Ω in the r.h.s. of the estimate. To patch these local estimates together we choose a global weight function that can be used to derive each of these local estimates by satisfying the following requirements. Assumption 7.6. Let ω 0 ⋐ ω ⋐ Ω. The weight function ϕ satisfies ϕ| ∂Ω = Cst, ∂ n ϕ| ∂Ω < 0, sup ϕ(x) < 0, |ϕ ′ (x)| 0, x ∈ Ω \ ω 0 , x∈Ω q 2 | ε=0 = 0 ⇒ {q 2 | ε=0 , q 1 | ε=0 } > 0, x ∈ Ω \ ω 0 , Such conditions can be fulfilled by taking ϕ of the form ϕ(x) = e λψ(x) − e λK , with K > ‖ψ‖ ∞ , |ψ ′ (x)| 0, x ∈ Ω \ ω 0 , and ψ| ∂Ω = 0, ∂ n ψ| ∂Ω < 0, ψ(x) > 0, x ∈ Ω, and by taking the positive parameter λ sufficiently large. For the construction of such a function ψ we refer to [FI96, Lemma 1.1]. The construction makes use of Morse functions and the associated approximation theorem [AE84]. Theorem 7.7 (Global Carleman estimate). Let ϕ be a function that satisfies Assumption 7.6. Then there exist δ 4 > 0 and C ≥ 0 such that ( ) ‖h 1 2 e ϕ/h u‖ 2 L 2 (Q) + ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 (Q) ≤ C ‖h 2 e ϕ/h Pu‖ 2 L 2 (Q) + ‖h 1 2 e ϕ/h u‖ 2 L 2 ((0,T)×ω) , for 0 < (T + T 2 )ε ≤ δ 4 , h = εt(T − t) and u ∈ C ∞ ([0, T] × Ω) such that u| (0,T)×∂Ω = 0. Proof. Let ω 1 be such that ω 0 ⋐ ω 1 ⋐ ω. For all x ∈ Ω \ ω 1 , there exist an open subset V x of Ω, with x ∈ V x ⊂ Ω \ ω 0 for which the local Carleman estimate, in the interior or at the boundary, holds with the weight function ϕ, for smooth functions with support in the compact K x = V x . From the covering of Ω \ ω 1 by the open sets V x , x ∈ Ω \ ω 1 we can extract a finite covering (V i ) i∈I , such that for all i ∈ I the Carleman estimate in V i holds for h < h i , C = C i > 0 and supp(u) ⊂ K i = V i . Let (χ i ) i∈I be a partition of unity subordinated to the covering V i , i ∈ I, [Trè67, Hör90], i.e., χ i ∈ C ∞ ∑ (Ω), supp( χ i ) ⊂ K i = V i , 0 ≤ χ i ≤ 1, i ∈ I, and χ i = 1 in Ω \ ω 1 Note that we have supp( χ i ) ∩ ω 0 = ∅. For all i ∈ I, we set u i = χ i u. Then for each u i we have a local Carleman estimate. We now observe that we have Pu i = P( χ i u) = χ i Pu + [P, χ i ]u = χ i Pu − [∆, χ i ]u, i∈I
CARLEMAN ESTIMATES 19 where the commutator is a first-order differential operator in x. For all i ∈ I, we thus obtain (7.4) ‖h 2 e ϕ/h Pu i ‖ 2 L 2 (Q) ≤ C‖h 2 e ϕ/h Pu‖ 2 L 2 (Q) + C‖h 2 e ϕ/h u‖ 2 L 2 (Q) + C‖h 2 e ϕ/h ∇ x u‖ 2 L 2 (Q) We note that we have ≤ C‖h 2 e ϕ/h Pu‖ 2 L 2 (Q) + C(εT 2 ) 3 ‖h 1 2 e ϕ/h u‖ 2 L 2 (Q) + CεT 2 ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 (Q). ‖h 1 2 e ϕ/h u‖ 2 L 2 (Q) + ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 (Q) ≤C ∑ i∈I ( ) ‖h 1 2 e ϕ/h u i ‖ 2 L 2 (Q) + ‖h 3/2 e ϕ/h ∇ x u i ‖ 2 L 2 (Q) + C‖h 1 2 e ϕ/h u‖ 2 L 2 ((0,T)×ω 1 ) + C‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 ((0,T)×ω 1 ), From (7.4) we then obtain ( ‖h 1 2 e ϕ/h u‖ 2 L 2 (Q) + ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 (Q) ≤ C ‖h 2 e ϕ/h Pu‖ 2 L 2 (Q) + (εT 2 ) 3 ‖h 1 2 e ϕ/h u‖ 2 L 2 (Q) + εT 2 ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 (Q) ) + ‖h 1 2 e ϕ/h u‖ 2 L 2 ((0,T)×ω 1 ) + ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 ((0,T)×ω 1 ) . For εT 2 sufficiently small we have ( ‖h 1 2 e ϕ/h u‖ 2 L 2 (Q) + ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 (Q) ≤ C ‖h 2 e ϕ/h Pu‖ 2 L 2 (Q) + ‖h 1 2 e ϕ/h u‖ 2 L 2 ((0,T)×ω 1 ) ) + ‖h 3/2 e ϕ/h ∇ x u‖ 2 L 2 ((0,T)×ω 1 ) . We now aim to remove the last term in the r.h.s. of the previous estimation. Let χ ∈ C ∞ c (ω) be such that χ = 1 in a neighborhood of ω 1 . If Pu = f , after multiplication by e 2ϕ/h h 3 χu, and integration over Q, we obtain (7.5) 1 2 ∫∫ e 2ϕ/h h 3 χ∂ t |u| 2 dt dx − Re ∫∫ e 2ϕ/h h 3 χu∆u dt dx = Re ∫∫ e 2ϕ/h h 3 χu f dt dx Q Q Q For the first term I 1 an integration by parts in t yields |I 1 | = ∣ 1 2 ∫∫ e 2ϕ/h h 3 χ∂ t |u| 2 dt dx ∣ = ∣ 1 Q 2 ∫∫ Q(3εθ ′ h 2 − 2ϕεθ ′ h)e 2ϕ/h χ|u| 2 dt dx ∣ ≤ C‖h 1 2 e ϕ/h u‖ 2 L ((0,T)×ω), 2 since ε|θ ′ | ≤ CεTis bounded. The third term can be estimated as |I 3 | = ∣ ∣Re ∫∫ e 2ϕ/h h 3 χu f dt dx ∣ ≤ C‖h 2 e ϕ/h f ‖ 2 L 2 (Q) + C‖h 1 2 e ϕ/h u‖ 2 L ((0,T)×ω). 2 Q For the second term, with integration by parts in x, we have I 2 = ∫∫ e 2ϕ/h h 3 χ|∇ x u| 2 dt dx + Re ∫∫ h 3 ∇ x (e 2ϕ/h χ)u∇ x u dt dx Q Q ≥ ‖h 3 2 e ϕ/h ∇ x u‖ 2 L 2 ((0,T)×ω 1 ) − 1 2 ∫∫ h 3 ∆(e 2ϕ/h χ)|u| 2 dt dx, Q and ∣ ∣ ∫∫ h 3 ∆(e 2ϕ/h χ)|u| 2 dt dx ∣ 1 ∣ ≤ C‖h 2 e ϕ/h u‖ 2 L ((0,T)×ω). The previous estimates and (7.5) then yield 2 Q ‖h 3 2 e ϕ/h ∇ x u‖ 2 L 2 ((0,T)×ω 1 ) ≤ C‖h 2 e ϕ/h Pu‖ 2 L 2 (Q) + C‖h 1 2 e ϕ/h u‖ 2 L 2 ((0,T)×ω). The proof is complete.
Page 1 and 2: ON CARLEMAN ESTIMATES FOR ELLIPTIC
Page 3 and 4: CARLEMAN ESTIMATES 3 The approach o
Page 5 and 6: CARLEMAN ESTIMATES 5 Lemma 2.2. Let
Page 7 and 8: CARLEMAN ESTIMATES 7 ϕ ′ (x) 0 p
Page 9 and 10: CARLEMAN ESTIMATES 9 Proposition 3.
Page 11 and 12: CARLEMAN ESTIMATES 11 exists R > 0
Page 13 and 14: CARLEMAN ESTIMATES 13 Proof. We app
Page 15 and 16: CARLEMAN ESTIMATES 15 We start with
Page 17: CARLEMAN ESTIMATES 17 These conditi
Page 21 and 22: CARLEMAN ESTIMATES 21 The ψDO symb
Page 23 and 24: CARLEMAN ESTIMATES 23 If we denote
Page 25 and 26: CARLEMAN ESTIMATES 25 We note that
Page 27 and 28: CARLEMAN ESTIMATES 27 Next since (A
Page 29 and 30: CARLEMAN ESTIMATES 29 for all φ

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