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12 JÉRÔME LE ROUSSEAU AND GILLES LEBEAU V 1 V 2 V 3 y r 3r Figure 4: Level sets of the weight function ϕ and regions V 1 , V 2 and V 3 . The red regions, V 1 and V 3 , localise the support of ∇χ. which yields ‖u‖ H 1 (B(y,3r)) ≤ C ( ) δ ‖Au‖ L 2 (Z) + ‖u‖ H 1 (B(y,r)) ‖u‖ 1−δ The H 1 -norm in the ball B(y, 3r) is thus estimated by the H 1 -norm in the ball B(y, r). In particular, we recover the local uniqueness result of Section 4 when Au = 0. This local inequality can be “propagated” and we then obtain a global result. In addition to the Carleman estimate we have proven here, one needs to prove a similar estimate at the boundary (0, S 0 ) × ∂Ω. The “propagation” technique makes use of a finite covering by balls of radius r. The reader is referred to [LR95] for details (pages 353–356). Here, as in [LZ98] (see the proof of theorem 3, pages 312–313), the interpolation inequality can be “initiated” at the boundary s = 0 (again by a Carleman estimate at the boundary). Theorem 5.3 ([LR95, LZ98, JL99]). Let ω be an open set in Ω. There exist C > 0 and δ ∈ (0, 1) such that for u ∈ H 2 (Z) that satisfies u(s, x)| x∈∂Ω = 0, for s ∈ (0, S 0 ) and u(0, x) = 0, for x ∈ Ω, we have ( ) ‖u‖ H 1 (Y) ≤ C‖u‖ 1−δ δ (5.3) ‖Au‖L H 1 (Z) 2 (Z) + ‖∂ s u(0, x)‖ L 2 (ω) . We may now deduce a spectral inequality that, in particular, measures the loss of orthogonality of the H 1 (Z) . eigenfunctions of −∆ in Ω, with homogeneous Dirichlet boundary conditions, when they are restricted to an open subset ω ⊂ Ω such that ω Ω. Let φ j , j ∈ N ∗ , be an orthonormal basis of such eigenfunctions and µ 1 ≤ µ 2 ≤ · · · ≤ µ k ≤ · · · the associated eigenvalues, counted with their multiplicity. Theorem 5.4 ([LZ98],[JL99]). There exists K > 0 such that for all sequences (α j ) j∈N ∗ ⊂ C and all µ > 0 we have ∑ (5.4) |α j | 2 = ∫ ∑ ∣ α j φ j (x) ∣ 2 dx ≤ Ke K √µ ∣ ∣∣∣ ∑ ∫ α j φ j (x) ∣ 2 dx, µ j ≤µ Ω µ j ≤µ ω µ j ≤µ or concisely ‖ ∑ µ j ≤µ α j φ j ‖ 2 L 2 (Ω) ≤ KeK √µ ‖ ∑ µ j ≤µ α j φ j ‖ 2 L 2 (ω) .
CARLEMAN ESTIMATES 13 Proof. We apply Inequality (5.3) to the function u(s, x) = ∑ sinh( √ µ µ j ≤µ α j s) j √ µ j φ j (x) that satisfies Au = 0 as well as the boundary conditions required in Theorem 5.3. We have and also ‖u‖ 2 H 1 (Y) ≥ ‖u‖2 L 2 (Y) = ∑ S 0 −α ∫ |α j | 2 sinh2 ( √ µ j s) ds ≥ ∑ |α j | 2 S 0−α ∫ s 2 ∑ ds = C S µ j ≤µ α µ 0 ,α |α j | 2 , j µ j ≤µ α µ j ≤µ ( ‖u‖ 2 H 1 (Z) = S 0 ∥∥∥∥ ∑ sinh( √ µ j s) ∥ ∥∥∥ 2 ∫ α j √ φ j 0 µ j ≤µ µ j L 2 (Ω) + ∥ ∑ µ j ≤µ + ∥ ∑ µ j ≤µ α j cosh( √ µ j s)φ j ∥ ∥∥∥ 2 α j sinh( √ µ j s) √ µ j ∇ x φ j ∥ ∥∥∥ 2 L 2 (Ω) = ∑ |α j | 2 S ( 0 ∫ (1 + 1 ) sinh 2 ( √ µ j s) + cosh 2 ( √ ) µ j s) µ j ≤µ 0 µ j ) ds L 2 (Ω) ds ≤ Ce C √ µ since ( ∇ xφ i √µi , ∇ xφ j √ µ j ) L 2 (Ω) = δ i j . Finally, we have ‖∂ s u(0, x)‖ 2 L 2 (ω) = ∫ ω ∣ ∣∣ ∑µ j ≤µ α j φ j (x) ∣ ∣ ∣ 2 dx, which yields ∑ µ j ≤µ |α j | 2 , ∑ µ j ≤µ and the conclusion follows. |α j | 2 ≤ Ce C √ ( µ ∑ µ j ≤µ ) 1−δ |α j | (∫ 2 ∑ ∣ α j φ j (x) ∣ 2 δ dx) , ω µ j ≤µ On the one hand, in the case ω = Ω, the result of Theorem 5.4 becomes trivial and the constant Ce C √ µ can be replaced by 1. On the other hand, it is clear that K = K(ω) tends to +∞ as the size of ω goes to zero. An interesting problem would be the precise estimation of K(ω). Some recent results are available with some lower bonds and uperbounds [Mil09, TT09]. When ω Ω, the following proposition shows that the power 1 2 of µ in KeK √µ is optimal (see also [JL99]). Proposition 5.5. Let ω be a non empty open set in Ω with ω Ω. There exist C > 0 and µ 0 > 0 such that for all µ ≥ µ 0 there exists a sequence (α j ) µ j ≤µ, such that ∑ |α j | 2 ≥ Ce C √µ ∣ ∣∣∣ ∑ ∫ α j φ j (x) ∣ 2 dx. ω µ j ≤µ Proof. We denote by P t (x, y) the heat kernel that we can write ∑ j∈N e −tµ j φ j (x)φ j (y) for t > 0; we have e t∆ f (x) = ∫ P t (x, y) f (y) dy. We then write ∣ ∑ e −tµ j φ j (x)φ j (y) ∣ ≤ |P t (x, y)| + ∣ ∑ e −tµ j φ j (x)φ j (y) ∣ . µ j ≤µ For k ∈ N sufficiently large, Sobolev injections give (5.5) ‖φ j ‖ L ∞ ≤ C‖φ j ‖ H 2k ≤ C ′ ‖∆ k φ j ‖ L 2 = C ′ µ k j . For all x, y ∈ Ω we have p t (x, y) ≤ (4πt) −n/2 e − |x−y|2 4t by the maximum principle (see Appendix A.7). Let y 0 be such that d = dist(y 0 , ω) > 0. We then have p t (x, y 0 ) ≤ e −C0/t , with C 0 > 0, uniformly for x in ω. From (5.5) we thus obtain µ j ≤µ µ j ≤µ µ j >µ ∣ ∑ e −tµ j φ j (x)φ j (y 0 ) ∣ ≤ e −C0/t + C ∑ We choose α j = e −tµ j φ j (y 0 ) and we take t = 1/ √ µ. We have ∣ ∑ α j φ j (x) ∣ ≤ Ce −C √ 0 µ + C ∑ µ j ≤µ µ j >µ µ j >µ e −tµ j µ 2k j , x ∈ ω. e −tµ j µ 2k j , x ∈ ω.
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: CARLEMAN ESTIMATES 11 exists R > 0
Page 15 and 16: CARLEMAN ESTIMATES 15 We start with
Page 17 and 18: CARLEMAN ESTIMATES 17 These conditi
Page 19 and 20: CARLEMAN ESTIMATES 19 where the com
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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