DISCRETE CARLEMAN ESTIMATES FOR ELLIPTIC OPERATORS IN ...

DISCRETE CARLEMAN ESTIMATES FOR ELLIPTIC OPERATORS IN ARBITRARY DIMENSION AND APPLICATIONS∗ ´ OME ˆ FRANCK BOYER† § , FLORENCE HUBERT‡ § , AND JER LE ROUSSEAU¶ Abstract. In arbitrary dimension, we consider the semi-discrete elliptic operator −∂t2 + AM , where AM is a finite difference approximation of the operator −∇x (Γ(x)∇x ). For this operator we derive a global Carleman estimate, in which the usual large parameter is connected to the discretization step-size. We address discretizations on some families of smoothly varying meshes. We present consequences of this estimate such as a partial spectral inequality of the form of that proven by G. Lebeau and L. Robbiano for AM and a null controllability result for the parabolic operator ∂t + AM , for the lower part of the spectrum of AM . With the control function that we construct (whose norm is uniformly bounded) we prove that the L2 -norm of the final state converges to zero exponentially, as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced. Key words. Elliptic operator – discrete and semi-discrete Carleman estimates – spectral inequality – control – parabolic equations.

hal-00450854, version 1 - 27 Jan 2010

AMS subject classifications. 35K05 - 65M06 - 93B05 - 93B07 - 93B40

1. Introduction and settings. Let d ≥ 2, L1 , . . . , Ld be positive real numbers, Q [0, Li ]. We set x = (x1 , . . . , xd ) ∈ Ω. With ω b Ω we consider the and Ω = 1≤i≤d

following parabolic problem in (0, T ) × Ω, with T > 0, ∂t y − ∇x · (Γ∇x y) = 1ω v in (0, T ) × Ω,

y|∂Ω = 0,

and y|t=0 = y0 ,

(1.1)

where the diagonal diffusion tensor Γ(x) = Diag(γ1 (x), . . . , γd (x)) with γi (x) > 0 satisfies ³ ´ 1 def reg(Γ) = sup γi (x) + + |∇x γi (x)| < +∞. (1.2) γi (x) x∈Ω i=1,...,d

The null-controllability problem consists in finding v ∈ L2 ((0, T ) × Ω) such that y(T ) = 0. This problem was solved in the 90’s by G. Lebeau and L. Robbiano [LR95] and A. Fursikov and O. Yu. Imanuvilov [FI96]. Let us consider the elliptic operator on Ω given by P ∂xi (γi ∂xi ) A = −∇x · (Γ∇x ) = − 1≤i≤d

with homogeneous Dirichlet boundary conditions on ∂Ω. We shall introduce a finitedifference approximation of the operator A. For a mesh M that we shall describe below, associated with a discretization step h, the discrete operator will be denoted Date: January 27, 2010 three authors were partially supported by l’Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01. † Universit´ e Paul C´ ezanne ([email protected]). ‡ Universit´ e de Provence ([email protected]). § Laboratoire d’Analyse Topologie Probabilit´ es (LATP), CNRS UMR 6632, Universit´ es d’AixMarseille, 39 rue F. Joliot-Curie, 13453 Marseille cedex 13, France. ¶ Universit´ e d’Orl´ eans, Laboratoire Math´ ematiques et Applications, Physique Math´ ematique d’Orl´ eans (MAPMO), CNRS UMR 6628, F´ ed´ eration Denis Poisson, CNRS FR 2964, Bˆ atiment de Math´ ematiques, B.P. 6759, 45067 Orl´ eans cedex 2 ([email protected]). ∗ The

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F. BOYER, F. HUBERT, AND J. LE ROUSSEAU

by AM . It will act on a finite dimensional space CM , of dimension |M|, and will be selfadjoint for a suitable inner product in CM . Our main result is a Carleman-type estimate for the “extended” semi-discrete elliptic operator, −∂t2 + AM . Here, the additional variable t is not directly connected to the time variable in the parabolic problem above. In the discrete setting, such a result was obtained in [BHL09a] in the one-dimensional case. Here, we extend this result to any space dimension. Note that we also prove a Carleman estimate for AM itself. For Carleman estimates in the continuous case we refer to [Hör63, Zui83, Hör85, LR95, FI96, LR97, LL09]. Note that an earlier attempt at deriving discrete Carleman estimates can be found in [KS91]. The result presented in [KS91] cannot be used here as the condition imposed by these authors on the discretization step size, in connection to the large Carleman parameter, is too strong for the applications we have in mind to the problem of uniform controllability properties for semi-discrete parabolic problems. We now describe an important consequence of the Carleman estimate we prove, which was the main motivation of this work. We denote by φM a set of discrete orthonormal eigenfunctions, φj ∈ CM , 1 ≤ j ≤ |M|, of the operator AM , and by µM = {µj , 1 ≤ j ≤ |M|} the set of the associated eigenvalues sorted in a nondecreasing sequence. The following (partial) spectral inequality is then a corollary of the semi-discrete Carleman estimate we prove: ¯2 ¯2 √ R ¯¯ P R ¯¯ P P ¯ ¯ |αk |2 = ¯ αk φk ¯ , αk φk ¯ ≤ CeC µ ¯ ∀(αk )1≤k≤|M| ⊂ C, µk ∈µM µk ≤µ

Ω

µk ∈µM µk ≤µ

ω


(1.3) for µh2 ≤ CS with CS and h sufficiently small (integrals of discrete functions are introduced below). This type of spectral inequality goes back to the work of G. Lebeau and L. Robbiano [LR95] (see also [LZ98a, JL99]). As opposed to the continuous case this inequality is not valid for the whole spectrum. The condition µh2 ≤ CS with CS small, states that it is only valid for a constant lower portion of the spectrum. This condition cannot be relaxed. The optimal value of CS is not known at this point and certainly depends, at least, on the geometry of ω. The spectral inequality (1.3) then implies the null-controllability of system (1.1) for the lower part of the spectrum µ ≤ CS /h2 , i.e., for any initial condition y0 ∈ CM , there exists a control v in L2 ((0, T ) × Ω) (the semi-discrete functional spaces we shall use will be made precise below) with kvkL2 ((0,T )×Ω) ≤ C|y0 |L2 (Ω) such that (y(T ), φk ) = 0 if µk h2 ≤ CS . Moreover, the remainder satisfies |y(T )|L2 (Ω) ≤ 2 e−C/h |y0 |L2 (Ω) . We thus obtain an exponential convergence as h goes to 0. Accurate statements of the results we have just described are given in Section 1.2. The form of the relaxed observability estimate that follows from this controllability result has been the inspiration for the study of Carleman estimates for semi-discrete parabolic operators [BHL10]. The spectral inequality (1.3) is also at the heart of the work carried out by the authors on the numerical analysis of the fully-discretized parabolic control problem in [BHL09b]. In two dimensions, for finite differences, there is a counterexample to the null and approximate controllabilities for uniform grids on a square domain for distributed or boundary controls due to O. Kavian (see [Zua06]). It exploits an explicit eigenfunction of AM in two dimensions that is solely localized on the diagonal of the square domain. This eigenfunction is associated with an eigenvalue in the higher part of the spectrum. Our result may thus seem rather optimal in dimension greater that two.

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DISCRETE CARLEMAN ESTIMATES IN ARBITRARY DIMENSION

In dimension one, there is a null controllability result due to A. Lopez and E. Zuazua [LZ98b] for the entire spectrum in the case of a constant diffusion coefficient and for a constant step size finite-difference discretization. In dimension one, our method based on the proof of discrete Lebeau-Robbiano spectral inequality cannot achieve such a result. In fact, one can notice that (1.3) cannot hold for the full spectrum. In dimension one, the generalization of the result of [LZ98b] to a non constant coefficient and non uniform meshes remains an open problem. We now present the precise settings we shall work with. Q For 1 ≤ i ≤ d, i ∈ N, we set Ωi = [0, Lj ]. For T > 0 we introduce 1≤j≤d j6=i

Q = (0, T ) × Ω,

Qi = (0, T ) × Ωi ,

1 ≤ i ≤ d.

We also set boundaries as (see Figure 1) ∂i− Ω =

Q

[0, Lj ] × {0} ×

1≤j 0, ˜ |∇ψ| ≥ c and ψ > 0 in Q, ∂ni ψ(t, x) < 0 in (0, T ) × V∂i Ω , ∂i2 ψ(t, x) ≥ 0 in (0, T ) × V∂i Ω , ∂t ψ ≥ c on {0} × (Ω \ ω), ψ = Cst and ∂t ψ ≤ −c on {T } × Ω, ˜ in which the outward unit where V∂i Ω is a sufficiently small neighborhood of ∂i Ω in Ω, normal ni to Ω is extended from ∂i Ω. The construction of such a weight function is described in Section A. We then set ϕ = eλψ . To state the Carleman estimate for the semi-discrete operator −∂t2 + AM , we introduce the following discrete gradient operator g = (D1 , . . . , Dd )t . Theorem 1.4. Let ϑi , i ∈ J1, dK satisfy (1.6) and ψ be a weight function satisfying (1.3) for the observation domain ω. For the parameter λ ≥ 1 sufficiently large, there exist C, s0 ≥ 1, h0 > 0, ε0 > 0, depending on ω, T , (ϑi )i∈J1,dK and reg(Γ), such that for any mesh M obtained from (ϑi )i∈J1,dK by (1.7), we have s3 kesϕ uk2L2 (Q) + skesϕ ∂t uk2L2 (Q) + skesϕ guk2L2 (Q) + s|esϕ(0,.) ∂t u(0, .)|2L2 (Ω) + se2sϕ(T ) |∂t u(T , .)|2L2 (Ω) + s3 e2sϕ(T ) |u(T , .)|2L2 (Ω) ³ ≤ C kesϕ (−∂t2 + AM )uk2L2 (Q) + se2sϕ(T ) |gu(T , .)|2L2 (Ω)

´ + s|esϕ(0,.) ∂t u(0, .)|2L2 (ω) , (1.9)

for all s ≥ s0 , 0 < h ≤ h0 and sh ≤ ε0 , and u ∈ C 2 ([0, T ], CM∪∂ M ), satisfying u|{0}×Ω = 0, u|(0,T )×∂Ω = 0. Denoting by φM a set of discrete L2 orthonormal eigenfunctions, φj ∈ CM , 1 ≤ j ≤ |M|, of the operator AM with homogeneous Dirichlet boundary conditions, and by µM the set of the associated eigenvalues sorted in a non-decreasing sequence, µj , 1 ≤ j ≤ |M| we have the following result.


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Theorem 1.5 (Partial discrete Lebeau-Robbiano inequality). Let ϑ satisfying (1.6). There exist C > 0, ε1 > 0 and h0 such that, for any mesh M obtained from ϑ by (1.7) such that h ≤ h0 , for all 0 < µ ≤ ε1 /h2 , we have ¯2 ¯2 √ R ¯¯ P R ¯¯ P P ¯ ¯ |αk |2 = ¯ αk φk ¯ ≤ CeC µ ¯ αk φk ¯ , ∀(αk )1≤k≤|M| ⊂ C. µk ∈µM µk ≤µ

Ω

ω



The proof is given in [BHL09a, Section 6] following the approach introduced in [Le 07]. We introduce the following finite dimensional spaces Ej = Span{φk ; 1 ≤ µk ≤ 22j } ⊂ CM , j ∈ N,

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and denote by ΠEj the L2 -orthogonal projection onto Ej . The controllability result we can deduce from the above results is the following. Theorem 1.6. Let T > 0 and ϑ satisfying (1.6). There exist h0 > 0, CT > 0 and C1 , C2 , C3 > 0 such that for all meshes M defined by (1.7), with 0 < h ≤ h0 , and all initial data y0 ∈ CM , there exists a semi-discrete control function v such that the solution to P Di (γi Di y) = 1ω v, y ∂M = 0, y|t=0 = y0 . (1.10) ∂t y − i∈J1,dK

satisfies ΠEjM y(T ) = 0, for j M = max{j; 22j ≤ C1 /h2 }, with kvkL2 (Q) ≤ CT |y0 |L2 (Ω) 2

and furthermore |y(T )|L2 (Ω) ≤ C2 e−C3 /h |y0 |L2 (Ω) . For a proof see [BHL09a, Section 7]. Finally, in the spirit of the work of [LT06] the controllability result we have obtained yields the following relaxed observability estimate Corollary 1.7. There exist CT > 0 and C > 0 depending on Ω, ω, T , and ϑ, such that the semi-discrete solution q in C ∞ ([0, T ], CM ) to  M  in (0, T ) × Ω, −∂t q + A q = 0 q=0 on (0, T ) × ∂Ω,   M q(T ) = qF ∈ C , in the case h ≤ h0 , satisfies |q(0)|L2 (Ω) ≤ CT

³ RT R 0ω

´ 12 2 |q(t)|2 dt + Ce−C/h |qF |L2 (Ω) .

As mentioned above, these results can also be used for the analysis of the space/time discretized parabolic control problem [BHL09b]. 1.3. Outline. In Section 2 we have gathered preliminary discrete calculus results. Many of the proofs of these results can be found in [BHL09a]. Additional proofs have been placed in Appendix B to ease the reading. Section 3 is devoted to the proof of the semi-discrete elliptic Carleman estimate for uniform meshes. Again, to ease the reading, a large number of proofs of intermediate estimates have been placed in Appendix C. This result is then extended to non-uniform meshes in Section 4. For completeness, in Appendix D we give the counterpart of the Carleman estimate of Theorem 1.4 in the case of a fully-discrete elliptic operator. This result will be used in [BHL10] for the treatment of semi-discrete parabolic operators.

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2. Some preliminary discrete calculus results. Here, to prepare for Section 3, we only consider uniform meshes, i.e., constant-step discretizations in each i , j ∈ J0, Ni K, i ∈ J1, dK. direction, i.e., hi,j+ 21 = hi = NLi +1 This section aims to provide calculus rules for discrete operators such as Di , Di and also to provide estimates for the successive applications of such operators on the weight functions. 2.1. Discrete calculus formulae. We present calculus results for the finitedifference operators that were defined in the introductory section. Proofs are similar to that given in the one-dimension case in [BHL09a]. Lemma 2.1. Let the functions f1 and f2 be continuously defined in a neighborhood of Ω. For i ∈ J1, dK, we have i i Di (f1 f2 ) = Di (f1 ) fˆ2 + fˆ1 Di (f2 ).

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Note that the immediate translation of the proposition to discrete functions f1 , f2 ∈ j i ij CM (resp. CM , j 6= i), and g1 , g2 ∈ CM (resp. CM , j 6= i) is i i Di (f1 f2 ) = Di (f1 ) f˜2 + f˜1 Di (f2 ),

Di (g1 g2 ) = Di (g1 ) g i2 + g i1 Di (g2 ).

Lemma 2.2. Let the functions f1 and f2 be continuously defined in a neighborhood of Ω. For i ∈ J1, dK, we have i h2i î î fd Di (f1 )Di (f2 ). 1 f2 = f 1 f 2 + 4

Note that the immediate translation of the proposition to discrete functions f1 , f2 ∈ j i ij CM (resp. CM , j 6= i), and g1 , g2 ∈ CM (resp. CM , j 6= i) i h2i ˜i ˜i Di (f1 )Di (f2 ), fg 1 f2 = f 1 f 2 + 4

g1 g2 i = g i1 g i2 +

h2i Di (g1 )Di (g2 ). 4

Some of the following properties can be extended in such a manner to discrete functions. We shall not always write it explicitly. Averaging a function twice gives the following formula. Lemma 2.3. Let the function f be continuously defined in a neighborhood of Ω. For i ∈ J1, dK we have h2 bi i A2i f := fˆ = f + i Di Di f. 4 The following proposition covers discrete integrations by parts and related formulae. i Proposition 2.4. Let f ∈ CM∪∂ M and g ∈ CM . For i ∈ J1, dK we have RR R RR f (Di g) = − (Di f )g + (fNi +1 gNi + 12 − f0 g 12 ), Ω

Ω

RR Ω

f gi =

RR Ω

Ωi

hi R f˜ g − (fNi +1 gNi + 12 + f0 g 21 ). 2 Ωi i

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DISCRETE CARLEMAN ESTIMATES IN ARBITRARY DIMENSION j ∪∂ Mj

Lemma 2.5. LetRRi ∈ J1, dK v ∈ CM∪∂ M (resp. CM RR and i that v|∂i Ω = 0. Then Ω v = Ω v˜ .

for j 6= i) be such

Lemma 2.6. Let f be a smooth function defined in a neighborhood of Ω. For i ∈ J1, dK we have τi± f = f ±

hi R1 ∂i f (. ± σhi /2) dσ, 2 0

Dì f = ∂i` f + C`0 h2i

R1 −1

Aì f = f + C` h2i

R1 −1

(1 − |σ|)`+1 ∂i`+2 f (. + l` σhi ) dσ,

(1 − |σ|) ∂i2 f (. + l` σhi ) dσ,

` = 1, 2,

l1 =

1 , l2 = 1, 2

with hi = hi ei . For i, j ∈ J1, dK, i 6= j, we have Di Dj f =

2 ∂ij f

4 1 |h+ ij | R + + +C (1 − |σ|)3 f (4) (. + σh+ ij /2; η , . . . , η ) dσ hi hj −1 00

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+ C 000

4 1 |h+ ij | R − − (1 − |σ|)3 f (4) (x + σh− ij /2; η , . . . , η ) dσ, hi hj −1

± with h± ij = hi ei ± hj ej and η =

Note that

4 |h+ ij | hi hj

1 (h± ij ). |h± ij |

= O(h2 ) by (1.8), for i, j ∈ J1, dK, j 6= i.

Proof. This series of results follow from Taylor formulae, f (x + η) =

n−1 P j=0

R1 (1 − σ)n−1 (n) 1 (j) f (x; η, . . . , η) + f (x + ση; η, . . . , η) dσ, j! (n − 1)! 0

at order n = 1, n = 2, n = 3 or n = 4. 2.2. Calculus results related to the weight functions. We now present some technical lemmata related to discrete operations performed on the Carleman weight function that is of the form esϕ with ϕ = eλψ , ψ ∈ C p , with p sufficiently large. For concision, we set r = esϕ and ρ = r−1 . The positive parameters s and h will be large and small respectively and we are particularly interested in the dependence on s, h and λ in the following basic estimates. We assume s ≥ 1 and λ ≥ 1. We shall use multi-indices of the form α = (αt , αx ) with αt ∈ N and αx ∈ Nd . Lemma 2.7. Let α and β be multi-indices. We have ∂ β (r∂ α ρ) =|α||β| (−sϕ)|α| λ|α+β| (∇ψ)α+β |α| |α+β|−1

+ |α||β|(sϕ)

λ

O(1) + s

(2.1) |α|−1

|α|(|α| − 1)Oλ (1) = Oλ (s

|α|

).

Let σ ∈ [−1, 1] and i ∈ J1, dK. We have ∂ β (r(x)(∂ α ρ)(x + σhi )) = Oλ (s|α| (1 + (sh)|β| )) eOλ (sh) .

(2.2)

Provided sh ≤ K we have ∂ β (r(x)(∂ α ρ)(x+σhi )) = Oλ,K (s|α| ). The same expressions hold with r and ρ interchanged and with s changed into −s. For a proof see [BHL09a, proof of Lemma 3.7].

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With Leibniz formula we have the following estimate. Corollary 2.8. Let α, β and δ be multi-indices. We have ∂ δ (r2 (∂ α ρ)∂ β ρ) =|α + β||δ| (−sϕ)|α+β| λ|α+β+δ| (∇ψ)α+β+δ + |δ||α + β|(sϕ)|α+β| λ|α+β+δ|−1 O(1) + s|α+β|−1 (|α|(|α| − 1) + |β|(|β| − 1))Oλ (1) = Oλ (s|α+β| ). The proofs of the following properties can be found in Appendix B. Proposition 2.9. Let α be a multi-index. Let i, j ∈ J1, dK, provided sh ≤ K, we have rτi± ∂ α ρ = r∂ α ρ + s|α| Oλ,K (sh) = s|α| Oλ,K (1), rAki ∂ α ρ = r∂ α ρ + s|α| Oλ,K ((sh)2 ) = s|α| Oλ,K (1),

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rAki Di ρ k rDki i Dj j ρ

2

= r∂x ρ + sOλ,K ((sh) ) = sOλ,K (1), =

k r∂iki ∂j j ρ

2

2

k = 1, 2,

k = 0, 1,

2

+ s Oλ,K ((sh) ) = s Oλ,K (1),

ki + kj ≤ 2.

The same estimates hold with ρ and r interchanged. Lemma 2.10. Let α and β be multi-indices and k ∈ N. Let i, j ∈ J1, dK, provided sh ≤ K, we have k

k

Dki i Dj j (∂ β (r∂ α ρ)) = ∂iki ∂j j ∂ β (r∂ α ρ) + h2 Oλ,K (s|α| ),

ki + kj ≤ 2,

Aki ∂ β (r∂ α ρ) = ∂ β (r∂ α ρ) + h2 Oλ,K (s|α| ). k

Let σ ∈ [−1, 1], we have Dki i Dj j ∂ β (r(x)∂ α ρ(x + σhi )) = Oλ,K (s|α| ), for ki + kj ≤ 2. The same estimates hold with r and ρ interchanged. Lemma 2.11. Let α, β and δ be multi-indices and k ∈ N. Let i, j ∈ J1, dK, provided sh ≤ K, we have Aki ∂ δ (r2 (∂ α ρ)∂ β ρ) = ∂ δ (r2 (∂ α ρ)∂ β ρ) + h2 Oλ,K (s|α|+|β| ) = Oλ,K (s|α|+|β| ), k

k

Dki i Dj j ∂ δ (r2 (∂ α ρ)∂ β ρ) = ∂iki ∂j j (∂ δ (r2 (∂ α ρ)∂ β ρ)) + h2 Oλ,K (s|α|+|β| ) = Oλ,K (s|α|+|β| ),

ki + kj ≤ 2.

Let σ, σ 0 ∈ [−1, 1]. We have ¡ ¢ Aki ∂ δ r(x)2 (∂ α ρ(x + σhi ))∂ β ρ(x + σ 0 hj ) = Oλ,K (s|α|+|β| ), ¡ ¢ k Dki i Dj j ∂ δ r(x)2 (∂ α ρ(x + σhi ))∂ β ρ(x + σ 0 hj ) = Oλ,K (s|α|+|β| ),

ki + kj ≤ 2.

The same estimates hold with r and ρ interchanged. Proposition 2.12. Let α be a multi-index and k ∈ N. Let i, j ∈ J1, dK, provided sh ≤ K, we have i

k ki kj α 2 d Dki i Dj j Aki ∂ α (rD i ρ ) = ∂i ∂j ∂ (r∂x ρ) + sOλ,K ((sh) ) = sOλ,K (1), k

k

Dki i Dj j (rD2i ρ) = ∂iki ∂j j (r∂i2 ρ) + s2 Oλ,K ((sh)2 ) = s2 Oλ,K (1), k

Dki i Dj j (rA2i ρ) = Oλ,K ((sh)2 ).

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The same estimates hold with r and ρ interchanged. Proposition 2.13. Let α, β be multi-indices, i, j ∈ J1, dK and ki , ki0 , kj , kj0 ∈ N. For ki + kj ≤ 2, provided sh ≤ K we have 0

0

k k k ki kj β 2 α |α|+1 d Oλ,K ((sh)2 ) Ai i Aj j Dki i Dj j ∂ β (r2 (∂ α ρ)D i ρ ) = ∂i ∂j ∂ (r (∂ ρ)∂i ρ) + s i

= s|α|+1 Oλ,K (1), k0

k0

k

k

Ai i Aj j Dki i Dj j ∂ β (r2 (∂ α ρ)A2i ρ) = ∂iki ∂j j ∂ β (r(∂ α ρ)) + s|α| Oλ,K ((sh)2 ) = s|α| Oλ,K (1), k0

k0

k

k

Ai i Aj j Dki i Dj j ∂ β (r2 (∂ α ρ)D2i ρ) = ∂iki ∂j j ∂ β (r2 (∂ α ρ)∂i2 ρ) + s|α|+2 Oλ,K ((sh)2 ) = s|α|+2 Oλ,K (1), and we have k0

k0

k

i

k

k0

k0

k

i

k

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ki j α 2 2 2 3 2 3 d Ai i Aj j Dki i Dj j ∂ α (r2 D i ρ Dj ρ) = ∂i ∂j ∂ (r (∂i ρ)∂j ρ) + s Oλ,K ((sh) ) = s Oλ,K (1), ki j α 2 2 d Ai i Aj j Dki i Dj j ∂ α (r2 D i ρ Aj ρ) = ∂i ∂j ∂ (r∂i ρ) + sOλ,K ((sh) ) = sOλ,K (1).

3. A semi-discrete elliptic Carleman estimate for uniform meshes. Here we consider constant-step discretizations in each direction. The case of regular nonuniform meshes is treated in Section 4. In preparation to this section, we shall prove here the Carleman estimate on uniform meshes, for a slightly more general semi-discrete elliptic operator that we i define now. For all i ∈ J1, dK, let ξ1,i ∈ RM and ξ2,i ∈ RM be two positive discrete functions. We denote by reg(ξ) the following quantity reg(ξ) = max reg(ξ1,i , ξ2,i ),

(3.1)

i∈J1,dK

with µ reg(ξ1,i , ξ2,i ) = max

³ ³ 1 ´ 1 ´ sup ξ1,i + , sup ξ2,i + , ξ1,i ξ2,i M Mi

¶ max sup |Dj ξ1,i |, sup |Di ξ2,i |, max sup |Dj ξ2,i | .

j∈J1,dK

Mj

M

j∈J1,dK Mij i6=j

(3.2) Hence, reg(ξ) measures the boundedness of ξ1,i and ξ2,i and of their discrete derivatives as well as the distance to zero of ξ1,i and ξ2,i , i ∈ J1, dK. By abuse of notation, the letters ξ1,i , ξ2,i will also refer to a Q1 -interpolation of i these values on M and M respectively. Note that the resulting interpolated functions are Lipschitz continuous with kξ1,i kW 1,∞ ≤ Creg(ξ),

kξ2,i kW 1,∞ ≤ Creg(ξ).

We introduce the following notation related to the coefficients ξ1,i and ξ2,i , for

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any function f Di,ξ f =

p

ξ1,i ξ2,i Di f, i ∈ J1, dK ´t ¡ p ¢t gξ f = ξ1,1 ξ2,1 D1 f, . . . , ξ1,d ξ2,d Dd f = D1,ξ f, . . . , Dd,ξ f , ³ ´t µ ∂ f ¶ p p t ∇ξ f = ∂t f, ξ1,1 ξ2,1 ∂x1 f, . . . , ξ1,d ξ2,d ∂xd f = gξ f , P ∆ξ f = ∂t2 f + ξ1,i ξ2,i ∂x2i f. ³p

i∈J1,dK

We let ω b Ω be a nonempty open subset. We set the operator P M to be P P M = −∂t2 − ξ1,i Di (ξ2,i Di ), i∈J1,dK

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continuous in the variable t ∈ (0, T ), with T > 0, and discrete in the variable x ∈ Ω. The Carleman weight function is of the form r = esϕ with ϕ = eλψ , where ψ satisfies Assumption 1.3. ˜ of Ω introduced in Assumption 1.3 allows us to The enlarged neighborhood Ω apply multiple discrete operators such as Di and Ai on the weight functions. In particular, this then yields on ∂i Ω i

i

(rDi ρ )|ki =0 ≤ 0,

(rDi ρ )|ki =Ni +1 ≥ 0,

i ∈ J1, dK.

(3.3)

We are now in position to state and prove the following semi-discrete Carleman estimate. Theorem 3.1. Let reg0 > 0 be given. For the parameter λ ≥ 1 sufficiently large, there exist C, s0 ≥ 1, h0 > 0, ε0 > 0, depending on ω, T , reg0 , such that for any ξ1,i , ξ2,i , i ∈ J1, dK, with reg(ξ) ≤ reg0 we have s3 kesϕ uk2L2 (Q) + skesϕ ∂t uk2L2 (Q) + s

P i∈J1,dK

kesϕ Di uk2L2 (Q) + s|esϕ(0,.) ∂t u(0, .)|2L2 (Ω)

+ se2sϕ(T ) |∂t u(T , .)|2L2 (Ω) + s3 e2sϕ(T ) |u(T , .)|2L2 (Ω) ³ ´ P 2sϕ(T ) ≤ C kesϕ P M uk2L2 (Q) + s e |Di u(T , .)|2L2 (Ω) + s|esϕ(0,.) ∂t u(0, .)|2L2 (ω) , i∈J1,dK

(3.4) for all s ≥ s0 , 0 < h ≤ h0 and sh ≤ ε0 , and u ∈ C 2 ([0, T ], CM∪∂ M ), satisfying u|{0}×Ω = 0, u|(0,T )×∂Ω = 0. Proof. We set f := −P M u. At first, we shall work with the function v = ru, i.e., u = ρv, that satisfies ! Ã ¡ ¢ P 2 = rf. (3.5) r ∂t (ρv) + ξ1,i Di ξ2,i Di (ρv) i∈J1,dK

We have ∂t2 (ρv) = (∂t2 ρ)v + 2(∂t ρ)∂t v + ρ∂t2 v and by Lemma 2.1 i

i

i

i

i

i

Di (ξ2,i Di (ρv)) = (Di (ξ2,i Di ρ)) v˜i + ξ2,i Di ρ Di v + (Di ρ ) ξ2,i Di v + ρ˜i Di (ξ2,i Di v).

19


By Lemma 2.2 we have, for i ∈ J1, dK, hi (Di ξ2,i )(τi+ Di v − τi− Di v), 4 h2 i i i ξ2,i Di ρ = ξ2,i Di ρ + i (Di ξ2,i )(Di Di ρ), 4 i i Di (ξ2,i Di ρ) = (Di ξ2,i )Di ρ + ξ2,i Di Di ρ. i

i

i

ξ2,i Di v = ξ2,i Di v +

Using that ρr = 1 and the above equalities, Equation (3.5) thus reads Av + B1 v = g 0 with Av = A1 v + A2 v where A1 v = ∂t2 v + A2 v =

P

i

ξ1,i rρ˜i Di (ξ2,i Di v),

i∈J1,dK

r(∂t2 ρ) v

+

P

i

ξ1,i ξ2,i r(Di Di ρ) v˜i ,

i∈J1,dK

B1 v = 2r(∂t ρ)∂t v + 2

P

i

i

ξ1,i ξ2,i rDi ρ Di v ,

hal-00450854, version 1 - 27 Jan 2010

i∈J1,dK

g 0 = rf −

P

hi i ξ1,i rDi ρ (Di ξ2,i )(τi+ Di v − τi− Di v) 4 i∈J1,dK

P h2i i i i ξ1,i (Di ξ2,i )r(Di Di ρ)Di v − hi O(1)rDi ρ Di v i∈J1,dK 4 i∈J1,dK ³ ´ i P i − ξ1,i r(Di ξ2,i )Di ρ + hi O(1)r(Di Di ρ) v˜i , −

P

i∈J1,dK i

since kξ2,i − ξ2,i k∞ ≤ Chi . Following [FI96] we now set Bv = B1 −2s(∆ξ ϕ)v , | {z }

g = g 0 − 2s(∆t,x ϕ)v.

=B2 v

An explanation for the introduction of this additional term B2 v is provided in [LL09]. Equation (3.5) now reads Av + Bv = g and we write kAvk2L2 (Q) + kBvk2L2 (Q) + 2 Re (Av, Bv)L2 (Q) = kgk2L2 (Q) .

(3.6)

We shall need the following estimation of kgkL2 (Q) . The proof can be adapted from the one-dimensional case (see Lemma 4.2 and its proof in [BHL09a]). Lemma 3.2 (Estimate of the r.h.s.). For sh ≤ K we have Ã kgk2L2 (Q)

≤ Cλ,K

krf k2L2 (Q)

+s

2

kvk2L2 (Q)

+ (sh)

2

P i∈J1,dK

! kDi vk2L2 (Q)

.

(3.7)

Most of the remaining of the proof will be dedicated to computing the innerproduct Re (Av, Bv)L2 (Q) . Developing this term, we set Iij = Re (Ai v, Bj v)L2 (Q) . Lemma 3.3 (Estimate of I11 ). For sh ≤ K, the term I11 can be estimated from

20


below in the following way ³ 1 ´ 1 I11 ≥ −sλ2 kϕ 2 |∇ξ ψ|∂t vk2L2 (Q) + kϕ 2 |∇ξ ψ|gξ vk2L2 (Q) · ¸T ´ RR RR ³ 2 2 + sλ ϕ(∂t ψ)|gξ v| (T ) − sλ ϕ(∂t ψ)|∂t v| Ω

Ω

0

+ Y11 − X11 − W11 − J11 , with Y11 =

P RR ³¡ 2 2 i¢ (ξ1,i ξ2,i + Oλ,K ((sh)2 )) rDi ρ |k

i =Ni +1

i∈J1,dK Qi

|Di v|2|ki =Ni + 1

2

´ ¡ 2 2 i¢ − (ξ1,i ξ2,i + Oλ,K ((sh)2 )) rDi ρ |ki =0 |Di v|2|ki = 1 dt, 2

and X11 =

RRR

ν11,i |Di v|2 dt +

i∈J1,dK Q

Q

hal-00450854, version 1 - 27 Jan 2010

P RRR

β11 |∂t v|2 dt +

P RRR

i

ν 11,i |Di v |2 dt,

i∈J1,dK Q

with β11 , ν11,i , ν 11,i of the form sλϕO(1) + sOλ,K (sh) and RRR P RRR P RRR γ11,ii |Di Di v|2 dt, W11 = γ11,it |Di ∂t v|2 dt + γ11,ij |Di Dj v|2 dt + Q

i,j∈J1,dK i6=j

i∈J1,dK Q

Q

¡ ¢ with γ11,it , γ11,ij , and γ11,ii of the form h2 sλϕO(1) + sOλ,K (sh) and P RR J11 = δ11,i |Di v|2 (T ) i∈J1,dK Ω

+

´ P RR ³ (2) (2) (δ11,i )|ki =Ni + 12 |Di v|2|ki =Ni + 1 + (δ11,i )|ki = 12 |Di v|2|ki = 1 dt, 2

i∈J1,dK Qi

2

(2)

with δ11,i = sOλ,K (sh), and δ11,i = shi λϕO(1) + shi Oλ,K (sh). found in Appendix C.

The proof can be

The following lemma can be readily adapted from its counterpart in [BHL09a, Lemma 4.4] (use also Lemma 4.8 in [BHL09a]). Lemma 3.4 (Estimate of I12 ). For sh ≤ K, the term I12 is of the following form ³ 1 ´ 1 I12 ≥ 2sλ2 kϕ 2 |∇ξ ψ|∂t vk2L2 (Q) + kϕ 2 |∇ξ ψ|gξ vk2L2 (Q) − X12 − J12 , with X12 =

RRR

β12 |∂t v|2 dt +

Q

J12 =

RR

η12 |v|2 (T ) +

Ω

RR

P RRR

ν12,i |Di v|2 dt +

i∈J1,dK Q

RRR

µ12 |v|2 dt,

Q

O(1)|∂t v|2 (T ),

Ω

where β12 = sλϕO(1),

µ12 = s2 Oλ,K (1),

ν12,i = sλϕO(1) + sOλ,K (sh).

η12 = s2 Oλ,K (1),


21

Lemma 3.5 (Estimate of I21 ). For sh ≤ K, the term I21 can be estimated from below in the following way RR 3 I21 ≥ 3s3 λ4 kϕ 2 |∇ξ ψ|2 vk2L2 (Q) − (sλ)3 (ϕ3 (∂t ψ)|∇ξ ψ|2 )(T ) |v|2 (T ) Ω

+ Y21 − W21 − X21 − J21 , with P RRR

W21 =

i∈J1,dK Q

P RR

Y21 =

i∈J1,dK Qi

X21 =

hal-00450854, version 1 - 27 Jan 2010

RR

Q

γ21,ij |Di Dj v|2 dt,

2

i

Oλ,K ((sh)2 )(rDi ρ )Nx +1 |Di v|2|Nx + 1 dt, 2

2

µ21 |v| dt +

P RRR

2

ν21,i |Di v| dt

i∈J1,dK Q

Q

J21 =

i,j∈J1,dK i6=j

Oλ,K ((sh)2 )(rDi ρ )0 |Di v|2| 1 dt

i∈J1,dK Qi

RRR

RRR

i

P RR

+

P

γ21,it |Di ∂t v|2 dt +

η21 |v|2 (T ) +

P RR

δ21,i |Di v|2 (T ),

i∈J1,dK Ω

Ω

where γ21,it = hO(sh),

γ21,ij = hOλ,K ((sh)2 ),

µ21 = (sλϕ)3 O(1) + s2 Oλ,K (1) + s3 Oλ,K (sh), η21 = s3 Oλ,K ((sh)2 ) + s2 Oλ,K (1),

ν21,i = sOλ,K ((sh)2 ),

and δ21,i = sOλ,K ((sh)2 ).

The proof can be found in Appendix C. The following lemma can be readily adapted from its counterpart in [BHL09a, Lemma 4.6]. Lemma 3.6 (Estimate of I22 ). For sh ≤ K, the term I22 is of the following form 3

I22 = −2s3 λ4 kϕ 2 |∇ξ ψ|2 vk2L2 (Q) − X22 , with X22 =

RRR Q

µ22 |v|2 dt +

P RRR

ν22,i |Di v|2 dt

i∈J1,dK Q

where µ22 = (sλϕ)3 O(1) + s2 Oλ,K (1) + s3 Oλ,K (sh), and ν22,i = sOλ,K (sh). Continuation of the proof of Theorem 3.1. Collecting the terms we have obtained in the previous lemmata, from (3.6) we obtain, for sh ≤ K, ³ 1 ´ 3 1 2s3 λ4 kϕ 2 |∇ξ ψ|2 vk2L2 (Q) + 2sλ2 kϕ 2 |∇ξ ψ|∂t vk2L2 (Q) + kϕ 2 |∇ξ ψ|gξ vk2L2 (Q) ³ P RR iT ´ h RR + 2sλ ξ1,i ξ2,i (ϕ∂t ψ)(T ) |Di v|2 (T ) − ϕ(∂t ψ) |∂t v|2 − 2(sλ)

RR 3 Ω

i∈J1,dK Ω

Ω

0

(ϕ3 (∂t ψ)|∇ξ ψ|2 )(T ) |v|2 (T ) + 2Y ≤ Cλ,K krf k2L2 (Q) + 2X + 2W + 2J, (3.8)

22


³ ´ P where Y = Y11 +Y21 , X = X11 +X12 +X21 +X22 +Cλ,K s2 kvk2L2 (Q) +(sh)2 i∈J1,dK kDi vk2L2 (Q) , W = W11 + W21 , and J = J11 + J12 + J21 . With the following lemma, we may in fact ignore the term Y . Lemma 3.7. Let sh ≤ K. For all λ there exists ε1 (λ) > 0 such that for 0 < sh ≤ ε1 (λ), we have Y ≥ 0. Lemma 3.8. We have ³ 1 ´ 1 ˜ − W, ˜ sλ2 kϕ 2 |∇ξ ψ|∂t vk2L2 (Q) + kϕ 2 |∇ξ ψ|gξ vk2L2 (Q) ≥ CH − X where ³ P RRR ¯ i ¯2 ξ1,i ξ2,i ϕ|∇ξ ψ|2 ¯Di v ¯ dt + sλ2 h2 ϕ|∇ξ ψ|2 |Di ∂t v|2 dt

P RRR

H = sλ2

i∈J1,dK Q

hal-00450854, version 1 - 27 Jan 2010

+

P

RRR

i,j∈J1,dK i6=j

Q

i∈J1,dK Q 2

2

ξ1,i ξ2,i ϕ|∇ξ ψ| |Di Dj v| dt +

P RRR

´ ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di Di v|2 dt ,

i∈J1,dK Q

˜ = sh X

³ RRR

¯ ¯2 P RRR Oλ,K (1)|Di v|2 dt Oλ,K (1)¯∂t v ¯ dt + i∈J1,dK Q

Q

P RRR

+

´ ¯ i ¯2 Oλ,K (1)¯Di v ¯ dt ,

i∈J1,dK Q

and ˜ = sh3 W

³ P RRR

Oλ,K (1)|∂t Di v|2 dt

i∈J1,dK Q

P

RRR

i,j∈J1,dK i6=j

Q

+

Oλ,K (1)|Di Dj v|2 dt +

P RRR

´ Oλ,K (1)|Di Di v|2 dt .

i∈J1,dK Q

End of the proof of Theorem 3.1. Recalling the properties satisfied by ψ listed in Assumption 1.3, if we choose λ1 ≥ 1 sufficiently large, then for λ = λ1 (fixed for the rest of the proof) and sh ≤ ε1 (λ1 ), from (3.8) and Lemmata 3.7 and 3.8, we obtain s3 kvk2L2 (Q) + sk∂t vk2L2 (Q) + s +

P i∈J1,dK

kDi vk2L2 (Q) + H

s|∂t v(0, .)|2L2 (Ω)

≤ Cλ1 ,K

³

+ s|∂t v(T , .)|2L2 (Ω) + s3 |v(T , .)|2L2 (Ω) ´ P krf k2L2 (Q) + s |Di v(T , .)|2L2 (Ω) + s|∂t v(0, .)|2L2 (ω) i∈J1,dK

+ X + W + J, (3.9) where H=s

P i∈J1,dK

i

kDi v k2L2 (Q) + sh2 +

³ P i∈J1,dK

P i∈J1,dK

kDi ∂t vk2L2 (Q) +

´ kDi Di vk2L2 (Q) ,

P i,j∈J1,dK i6=j

kDi Dj vk2L2 (Q)


X=

RRR

µ1 |v|2 dt +

P RRR

ν1,i |Di v|2 dt +

i∈J1,dK Q

Q

P RRR

i

ν 1,i |Di v |2 dt +

RRR

i∈J1,dK Q

23

β1 |∂t v|2 dt,

Q

with µ1 = s2 Oλ1 ,K (1) + s3 Oλ1 ,K (sh) and ν1,i , ν 1,i , β1 , all of the form sOλ1 ,K (sh), and where P RRR P RRR P RRR W = γ1,it |Di ∂t v|2 dt + γ1,ij |Di Dj v|2 dt + γ1,ii |Di Di v|2 dt, i∈J1,dK Q

i,j∈J1,dK i6=j

i∈J1,dK Q

Q

where γ1,it , γ1,ij and γ1,ii are of the form sh2 Oλ1 ,K (sh), and where RR P RR J = η1 |v|2 (T ) + δ1,i |Di v|2 (T ) Ω

+

P RR ³ i∈J1,dK Qi

i∈J1,dK Ω (2) (δ1,i )Ni + 12 |Di v|2Ni + 1 2

(2)

+ (δ1,i ) 21 |Di v|21

´ dt,

2

hal-00450854, version 1 - 27 Jan 2010

(2)

with η1 = s3 Oλ1 ,K (sh) + s2 Oλ1 ,K (1) and δ1,i = sOλ1 ,K (sh), δ1,i = shi Oλ,K (sh). The last term in J was obtained by “absorbing” the following term in J11 ´ P RR ³ hi (ϕ)Ni + 12 O(1)|Di v|2Ni + 1 + (ϕ) 21 O(1)|Di v|21 dt, sλ 2

i∈J1,dK Qi

2

by the volume term sλ2

P RRR

ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di v|2 dt,

i∈J1,dK Q

for λ large. We can now choose ε0 and h0 sufficiently small, with 0 < ε0 ≤ ε1 (λ1 ), and s0 ≥ 1 sufficiently large, such that for s ≥ s0 , 0 < h ≤ h0 , and sh ≤ ε0 , we obtain s3 kvk2L2 (Q) + sk∂t vk2L2 (Q) + s

P i∈J1,dK

kDi vk2L2 (Q) + H

+ s|∂t v(0, .)|2L2 (Ω) + s|∂t v(T , .)|2L2 (Ω) + s3 |v(T , .)|2L2 (Ω) ³ ´ P ≤ Cλ1 ,K,ε0 ,s0 krf k2L2 (Q) + s |Di v(T , .)|2L2 (Ω) + s|∂t v(0, .)|2L2 (ω) . (3.10) i∈J1,dK

To finish the proof, we need to express all the terms in the estimate above in terms of the original function u. We can proceed exactly as in the end of proof of Theorem 4.1 in [BHL09a]. 4. Carleman estimates for non uniform meshes. We consider here the notation introduced in section 1.1.8. i We define, for i ∈ J1, dK, ζi ∈ CM and ζ¯i ∈ CM as follows ζi,k =

hi,ki i , k∈N, ? hi

hi,k ζ¯i,k = ? i , k ∈ N. hi

Even though these two formulae look similar they are in fact different as the indices k are taken in different sets. Lemma 4.1. We have the following properties i

reg(ϑ)−1 ≤ ζi,k ≤ reg(ϑ), i ∈ J1, dK, k ∈ N ,

24


reg(ϑ)−1 ≤ ζ¯i,k ≤ reg(ϑ), i ∈ J1, dK, k ∈ N, |Di ζi |L∞ (Ω) ≤ reg(ϑ)2 , and |Di ζ¯i |L∞ (Ω) ≤ reg(ϑ)2 . M0 ∪∂ M0 0 For u ∈ CM∪∂ M , we define QM to be the discrete function correM u ∈ C sponding to the reference uniform mesh M0 which takes the same values as u for each i i i 0 index k ∈ N. Similarly, for i ∈ J1, dK and u ∈ CM , we denote by QM u ∈ CM0 Mi

i

the discrete function defined on M0 which takes the same values as u for each index i i M0 i 0 k ∈ N . We denote by QM the inverse of the operators QM and QM M M0 and QMi M0 i respectively. Lemma 4.2. i • For any i ∈ J1, dK, any u ∈ CM∪∂ M and any v ∈ CM , we have

hal-00450854, version 1 - 27 Jan 2010

i M0 i 0 0 0 ¯ Di (QM Di (QM v) = QM M u) = QMi (ζi Di u), M (ζi Di v). Mi

• For any u ∈ CM∪∂ M and any i ∈ J1, dK, we have µ µµ ¶ ¶¶ M0 i γi M0 Di (γi Di u) = (ζ¯i )−1 QM D Q D (Q u) . i i M0 M Mi ζi i

Lemma 4.3. For any u ∈ CM , and any v ∈ CM , i ∈ J1, dK, we have 2 2 0 reg(ϑ)−1 |u|2L2 (Ω) ≤ |QM M u|L2 (Ω? ) ≤ reg(ϑ)|u|L2 (Ω) ,

i

0 reg(ϑ)−1 |v|2L2 (Ω) ≤ |QM v|2L2 (Ω? ) ≤ reg(ϑ)|v|2L2 (Ω) . Mi

We can now prove the Carleman estimate of Theorem 1.4 for the semi-discrete elliptic operator P P M = −∂t2 − Di (γi Di ·). i∈J1,dK

We only give a sketch of the proof, since it is very similar to the one which is detailed in [BHL09a] for the one-dimensional case. Proof of Theorem 1.4. The key idea is to perform a change of variables that transforms P M defined on a non-uniform mesh into an semi-discrete elliptic operator defined on a uniform mesh. All the geometric information concerning the initial mesh is then contained in the coefficients of this new operator. 0 More precisely, we consider the discrete function w = QM M u which is defined on the uniform mesh M0 . By using Lemma 4.2 we observe that ¶ µ µµ ¶ ¶¶ µ P M0 i γi M0 M 2 M0 ¯ −1 Di w . QM (P u) = −∂t w − QM (ζi ) Di QMi ζi i∈J1,dK We introduce the operator P M0 = −∂t2 −

P

0 ¯ −1 ξ1,i = QM , M (ζi )

i∈J1,dK ξ1,i

¡ ¢ Di (ξ2,i Di w) with

0 ξ2,i = QM Mi

i

γi , ζi


25

hal-00450854, version 1 - 27 Jan 2010

so that we may now apply the Carleman estimate of Theorem 3.1 to w and P M0 on the uniform mesh M0 and with the weight function x ∈ [0, 1]d 7→ ψ ◦ (ϑ1 (x1 ) . . . ϑd (xd )). We note that reg(ξ) is bounded by some constant depending only on reg(ϑ) and reg(Γ) and independent of the size of the mesh. We can thus find reg0 sufficiently large for which Theorem 3.1 leads to a Carleman inequality for the function w, and the weight function defined above. Using Lemmata 4.2 and 4.3 we then deduce result. Note that the values of h0 , ε0 , may change, depending only on the values of reg(ϑ) and reg(Γ) and not on the mesh size. Appendix A. Construction of a weight function. A weight function that satisfies the conditions listed in Assumption 1.3 can be constructed as follows. We first start with a function φ1 ∈ C ∞ ([0, T ]) such that ∂t φ1 (0) ≥ C > 0, ∂t φ1 (T ) ≤ −C < 0, and φ1 (0) = φ1 (T ) = 0, and φ1 (t) > 0 if t ∈ (0, T ). We choose φ1 with a single critical point. ˜ be such that φ2 ≥ C > 0 and ∂n φ2 ≤ −C 0 < 0 and Let also φ2 ∈ C ∞ (Ω) x 2 00 ∂i φ2 ≥ C > 0 in V∂Ω . ˜ This can be achieved with φ2 (x) = eζ φ2 (x) + C − 1, with φ˜2 = 0 on ∂Ω, φ˜2 > 0, in Ω, φ˜2 = 0 and ∂n φ˜2 ≤ −C˜ < 0, on ∂Ω x

and ζ > 0 sufficiently large and by taking the neighborhood V∂Ω sufficiently small. The function φ2 can be chosen with a finite number of critical points by means of Morse theorem [AE84]. We next set φ(t, x) = φ1 (t)φ2 (x). This function satisfies the desired properties listed in Assumption 1.3 on the boundaries (0, T ) × ∂Ω (and in its neighborhood (0, T ) × V∂Ω ), {0} × (Ω \ ω) and {T } × Ω. It is also characterized by a finite number of critical points. We choose y0 in {0}×ω. We enlarge Q in a small neighborhood of y0 which leaves ∂Q unchanged outside of {0} × ω. We call Q this extension of Q and we extend the function φ to Q in a C k manner. The critical points of φ can be pulled back to the interior of Q \ Q by composing φ with a finite number of diffeomorphisms (see [FI96] for the construction of these diffeomorphisms). The resulting function is the weight function ψ and it satisfies all the properties listed in Assumption 1.3. Appendix B. Proofs of some technical results in Section 2. B.1. Proof of PropositionR 2.9. We recall that rρ = 1. By Lemma 2.6 we have 1 τi+ ∂ α ρ(x) = ∂ α ρ(x) + Chi ρ(x) 0 r(x)∂i ∂ α ρ(x + σhi /2) dσ, which by Lemma 2.7 + α yields rτi ∂ ρ = r∂ α ρ + s|α| Oλ (sh)eOλ (sh) = s|α| Oλ,K (1). The proof is the same i i i [ α ρ , rA2 ∂ α ρ = r ∂ α ρ , and rDki Dkj ρ we prod for rτi− ∂ α ρ. For rDi ρ, rAi ∂ α ρ = r∂d i i j ceed similarly, exploiting the formula in Lemma 2.6 and then applying the result of Lemma 2.7, e.g., Di ρ(x) = ∂i ρ(x) + Ch2i ρ(x)

R1 −1

(1 − |σ|)2 r(x)(∂i3 ρ)(x + σhi /2) dσ

= ∂i ρ(x) + sρ(x)Oλ,K ((sh)2 ) = sr(x)Oλ,K (1). i

−1 d Noting that Ai Di ρ(x) = D (ρ(x + hi ) − ρ(x − hi )) we proceed as we i ρ (x) = (2hi ) did for Di r.

26


B.2. Proof of Lemma 2.10. By Lemma 2.6, we write Di (∂ β (r∂ α ρ))(x) = ∂i ∂ β (r∂ α ρ)(x) + Ch2i

R1 −1

(1 − |σ|)2 ∂i3 ∂ β (r∂ α ρ)(x + σhi /2) dσ.

By Lemma 2.7 we have ∂i3 ∂ β (r∂ α ρ) = Oλ (s|α| ), which yields the first result in the case ki + kj = 1. For the case ki + kj = 2, we proceed similarly, making use of the other formulae listed in Lemma 2.6. For the averaging cases, we make use of the second formula in Lemma 2.6. Following the proof of Lemma 2.7 in [BHL09a] we set ν(x, σhi ) := r(x)ρ(x+σhi ). We have 0 1 R1 (∂i ∂ β ν)(x + σ 0 hi /2, σhi ) dσ 0 = Oλ,K (1), 2 −1

0

Di ∂ β ν(x, σhi ) =

for |β 0 | ≤ |β|,

hal-00450854, version 1 - 27 Jan 2010

(B.1) for sh ≤ K by Lemma 2.7. Next, with µα = r∂ α ρ, we write r(x)∂ α ρ(x + σhi ) = ν(x, σhi )µα (x + σhi ), which gives Di ∂ β (r(x)∂ α ρ(x + σhi )) as a linear combination of terms of the form 0

00

0

00

Ai (∂ β ν(., σhi )) Di (∂ β µα (.+σhi ))+Di (∂ β ν(., σhi )) Ai (∂ β µα (.+σhi )),

β 0 +β 00 = β,

by the continuous and discrete Leibniz rules (Lemma 2.1). By the first part and 00 0 Lemma 2.7 we have Di (∂ β µα (x+σhi )) = Oλ,K (s|α| ). By Lemma 2.7, ∂ β ν(x, σhi ) = 00 Oλ,K (1) and ∂ β µα (x + σhi ) = Oλ,K (s|α| ). The last result hence follows from (B.1). We proceed in a similar way for the case ki + kj = 2. B.3. Proof of Lemma 2.11. For the first two results, we proceed as in Lemma 2.10 and use Corollary 2.8. For the last results we use the continuous and discrete Leibniz rules (Lemma 2.1) and Lemma 2.10. B.4. Proof of Proposition 2.12. Taylor formulae yield R1 i ρ(x + hi ) − ρ(x − hi ) d D = ∂i ρ(x) + Ch2i (1 − |σ|)2 ∂i3 ρ(x + σhi ) dσ, (B.2) i ρ (x) = 2hi −1 which in turn gives i

k ki kj k α d Dki i Dj j Aki ∂ α (rD i ρ ))(x) = Di Dj Ai ∂ (r∂i ρ)(x)

R1 k + Ch2i (1 − |σ|)2 Dki i Dj j Aki ∂ α (r(x)∂i3 ρ(x + σhi )) dσ, −1

and the first result follows by Lemma 2.10 (and Lemma 2.7 for the second equality). Next, from Lemma 2.6, we write k

k

Dki i Dj j (rD2i ρ)(x) = Dki i Dj j (r∂i2 ρ)(x) + Ch2i

R1 −1

k

(1 − |σ|)3 Dki i Dj j (r(x)∂i4 ρ(x + σhi )) dσ, k

and the third result follows as above. For Dki i Dj j (rA2 ρ) we use the formula for A2 ρ given in Lemma 2.6 and proceed as above.


27

B.5. Proof of Proposition 2.13. From (B.2) we write ³ ´ 0 0 ¡ ¢ i k0 k k0 k k k d Ai i Aj j Dki i Dj j ∂ β r2 (∂ α ρ)D = Ai i Aj j Dki i Dj j ∂ β r2 (∂ α ρ)∂i ρ iρ + Ch2i

R1 −1

0 ¡ ¢ k0 k k (1 − |σ|)2 Ai i Aj j Dki i Dj j ∂ β r2 (∂ α ρ)∂i3 ρ(. + σhi ) dσ,

and we conclude with Lemma 2.11. For the next two results we use the formulae listed in Lemma 2.6 and proceed as above. From Lemma 2.6, equation (B.2), and by Lemma 2.11 we have k0

k0

k0

i

k

k0

k

ki j j α 2 2 2 i d Ai i Aj j Dki i Dj j ∂ α (r2 D i ρ Dj ρ) = Ai Aj Di Dj ∂ (r (∂i ρ)∂j ρ)

+ Ch2i + Ch2j

R1 −1

R1

k0

k0

−1

+ Ch2i h2j

hal-00450854, version 1 - 27 Jan 2010

k0

k

(1 − |σ|)2 Ai i Aj j Dki i Dj j ∂ α (r2 ∂i3 ρ(. + σhi )∂j2 ρ) dσ k0

k

(1 − |σ|)3 Ai i Aj j Dki i Dj j ∂ α (r2 (∂i ρ)∂j4 ρ(. + σhj ) dσ RR

(1 − |σ|)2 (1 − |σ 0 |)3

[−1,1]2

k0

k0

k

× Ai i Aj j Dki i Dj j ∂ α (r2 ∂i3 ρ(. + σhi )∂j4 ρ(. + σ 0 hj )) dσ dσ 0 k

= ∂iki ∂j j ∂ α (r2 (∂i ρ)∂j2 ρ) + s3 Oλ,K ((sh)2 ). The last result follows similarly. Appendix C. Proofs of intermediate results in Section 3. P C.1. Proof of Lemma 3.3. From the forms of A1 v and B1 v we have I11 = k,l∈{t,1,...,d} Qkl with RRR Qtt = 2 Re r(∂t ρ) (∂t2 v)∂t v ∗ dt, Q

Qti = 2 Re

RRR

i

Q

Qit = 2 Re

i

ξ1,i ξ2,i rDi ρ (∂t2 v)Di v ∗ dt,

RRR

i ∈ J1, dK,

i

ρi Di (ξ2,i Di v)∂t v ∗ dt, ξ1,i r2 (∂t ρ)˜

i ∈ J1, dK,

Q

Qii = 2 Re

RR

Q

Qij = 2 Re

i

i

i

2 ξ1,i ξ2,i r2 ρ˜i Di ρ Di (ξ2,i Di v)Di v ∗ dt,

RRR

i

j

i ∈ J1, dK, j

ξ1,i ξ1,j ξ2,j r2 ρ˜i Dj ρ Di (ξ2,i Di v)Dj v ∗ dt,

i, j ∈ J1, dK, i 6= j.

Q

We start by computing each term.

¡ ¢ Computation of Qtt . We set qtt = −∂t r(∂t ρ) . An integration by parts w.r.t. t yields · ¸T RRR RR Qtt = qtt |∂t v|2 dt − sλ ϕ(∂t ψ) |∂t v|2 . Q

Ω

Lemma C.1. We have qtt = sλ2 ϕ(∂t ψ)2 + sλϕO(1). The estimation of follows from Lemma 2.7.

0

28

F. BOYER, F. HUBERT, AND J. LE ROUSSEAU i

Computation of Qti . Setting pti = −ξ1,i ξ2,i rDi ρ and qti = ∂t pti we have, by integration by parts w.r.t. t since v|t=0 = 0, Qti = 2 Re

RRR

¡ RR ¡ i¢ i¢ pti (∂t v)Di v ∗ (T ) (∂t v)∂t pti Di v ∗ dt − 2 Re

Q

= 2 Re

RRR

i

qti (∂t v)Di v ∗ dt + 2 Re

Q

RRR

Ω i

pti (∂t v)∂t Di v ∗ dt,

Q

|

{z

}

a Qti

using that pti (T ) = 0 for ψ|t=T = Cst. As v|∂Ω = 0 with Proposition 2.4, Lemma 2.2, and a discrete integration by parts w.r.t. xi , we then write a Qti = 2 Re

=−

RRR

RRR Q

i

∗ p^ ti (∂t v) ∂t Di v dt = 2 Re

Q

RRR Q

i h2i RRR i f ∗ pf (Di pti )|∂t Di v|2 dt ti ∂t v ∂t Di v dt + 2 Q

h2i RRR i 2 (Di pti )|∂t Di v|2 dt. (Di pf ti )|∂t v| dt + 2 Q

hal-00450854, version 1 - 27 Jan 2010

We thus have Qti = −

RRR

2 (Di pf ti )|∂t v| dt + 2 Re i

RRR

i

qti (∂t v)Di v ∗ dt +

Q

Q

h2i RRR (Di pti )|∂t Di v|2 dt. 2 Q (C.1)

Lemma C.2. We have Di pti = sλ2 ξ1,i ξ2,i ϕ(∂i ψ)2 + sλϕO(1) + sOλ,K (sh), i 2 2 Di pf ti = sλ ξ1,i ξ2,i ϕ(∂i ψ) + sλϕO(1) + sOλ,K (sh), 2 qti = sλ ξ1,i ξ2,i ϕ(∂t ψ)(∂i ψ) + sλϕO(1) + sOλ,K ((sh)2 ). ¡ i¢ î Proof. We set α = −ξ1,i ξ2,i . Then Di pti = (Di α)rDi ρ + α ˜ i Di rDi ρ . With Proposition 2.12 we find i

Di pti = (Di α)r∂i ρ + α ˜ i (∂i (r∂i ρ)) + sOλ,K ((sh)2 ).

(C.2)

Then with Lemma 2.7 we obtain the estimate of Di pti as Di α = O(1). Averaging (C.2) we obtain i

i

i

i

i

Di pti = Di α r∂i ρ + α ˜ i ∂i (r∂i ρ) + +

h2 (Di Di α)(Di (r∂i ρ)) 4

h2 (Di α ˜ i )Di (∂i (r∂i ρ)) + sOλ,K ((sh)2 ). 4

By Lemma 2.10 we have i

i

Di α r∂i ρ = sλϕO(1) + h2 Oλ,K (s).

(C.3)

i

i

as Di α = O(1). Note also that α ˜ i = α+hO(1). Then by Lemma 2.10 and Lemma 2.7 we have i

i

α ˜ i ∂i (r∂i ρ) = −αsλ2 ϕ(∂i ψ)2 + sλϕO(1) + Oλ,K (sh).

(C.4)


29

Since hDi Di α = O(1), by Lemma 2.10 we obtain h2 (Di Di α)(Di (r∂i ρ)) = Oλ,K (sh). 4

(C.5)

h2 (Di α ˜ i )Di (∂i (r∂i ρ)) = hOλ,K (sh), 4

(C.6)

Similarly we have

i

i

as α ˜ i = Di α = O(1). Collecting estimates (C.3)–(C.6), we obtain the second result. i Finally we write qti = α∂t (rDi ρ ); Proposition 2.12 and Lemma 2.7 yield the estimates for qti . i

Computation of Qit . We set pit = −ξ1,i r2 (∂t ρ)˜ ρi and qit = ξ2,i Di pit . Since v|∂Ω = 0, with a discrete integration by parts w.r.t. xi (Proposition 2.4) we then write Qit = 2 Re

RRR

Di (pit ∂t v ∗ ) ξ2,i Di v dt = 2 Re

hal-00450854, version 1 - 27 Jan 2010

Q

= 2 Re

RRR

i

∗

qit Di v ∂t v dt −

RRR

Q

RRR ³

´ i i ∗ ∗ +ξ p qit ∂g v f (∂ D v ) Di v dt t 2,i it t i

Q 2 ξ2,i (∂t pf it )|Di v| dt + i

RR

¢ ¡ i 2 ξ2,i pf it |Di v| (T ),

Ω

Q

after an integration by parts w.r.t. t, to yield RRR

RRR h2i Di (qit )(Di Di v)∂t v ∗ dt Re 2 Q Q ¡ i ¢ RRR RR i 2 2 − ξ2,i (∂t pf ξ2,i pf it )|Di v| dt + it |Di v| (T ).

Qit = 2 Re

i

qit i Di v ∂t v ∗ dt +

Q

Ω

Lemma C.3. We have i

ξ2,i pf it = sλξ1,i ξ2,i ϕ(∂t ψ) + sOλ,K (sh), i 2 2 ξ2,i ∂t pf it = sλ ξ1,i ξ2,i ϕ(∂t ψ) + sλϕO(1) + sOλ,K (sh), qit i = sλ2 ξ1,i ξ2,i ϕ(∂i ψ)(∂t ψ) + sλξ1,i ξ2,i ϕO(1) + sOλ,K (sh), hDi (qit ) = sλϕO(1) + Oλ,K (sh). Proof. The first three estimates follow from Proposition 2.13 and Corollary 2.8 following the method of the proof of Lemma C.2 (see also the proof of similar technical lemmata in [BHL09a, Appendix B]). For the fourth estimate we first write i

i

hi Di qit = hi Di (ξ2,i Di pit ) = hi (Di ξ2,i )Di pit + hi ξ2,i Di Di pit = Oλ,K (sh) + hi O(1)Di Di pit , following the method of the proof of Lemma C.2. We then write i i

i

i

2 ρi − ξf ρi ). Di pit = −(Di ξ1,i )r2 (∂t ρ)˜ 1,i Di (r (∂t ρ)˜

30


and obtain i

hi Di Di pit =

−hi (Di Di ξ1,i )r2 (∂t ρ)˜ ρi i

i

i

i

i

i

i

− 2hi Di ξ1,i Di (r2 (∂t ρ)˜ ρi ) i

2 − hi ξf ρi ) 1,i Di Di (r (∂t ρ)˜ = sλϕO(1) + Oλ,K (sh),

arguing as in the proof of Lemma C.2, as Di ξ1,i = O(1). The result follows. i

i

i

2 ξ2,i r2 ρ˜i Di ρ and qii = Di (ξ2,i pii ). Computation of Qii . We set pii = −ξ1,i By Lemmata 2.1 and 2.4, we have ´ ¡ i ¢ RRR RR ³¡ i ¢ Qii = qii |Di v|2 dt − ξ2,i pii Ni +1 |Di v|2Ni + 1 − ξ2,i pii 0 |Di v|21 dt Q

RRR

−2

2

Qi

2

¯ i ¯2 pii Di (ξ2,i )¯Di v ¯ dt.

Q

hal-00450854, version 1 - 27 Jan 2010

For the first term we write RRR

qii |Di v|2 dt = −

RRR

qii |Di v|2 dt + 2

Q

Q

|

RRR

i

qii |Di v|2 dt

Q

{z

}

a =Qii

+ hi

¢ RR ¡ (qii ) 12 |Di v|21 + (qii )Ni + 21 |Di v|2Ni + 1 2

Qi

2

by Proposition 2.4 and with Lemma 2.2 we further have a Qii =2

RRR

i

qii i |Di v|2 dt +

Q

h2i RRR (Di qii ) Di |Di v|2 dt. 2 Q

A further use of Lemma 2.2 and a discrete integration by parts w.r.t. xi (Proposition 2.4) yield, ¯ h2 RRR h2 RRR i i ¯2 qii i ¯Di v ¯ dt + i qii |Di Di v|2 dt − i (Di Di qii )|Di v|2 dt 2 Q 2 Q Q ´ h2 RR ³ + i (Di qii )Ni +1 |Di v|2Ni + 1 − (Di qii )0 |Di v|21 dt. 2 2 2 Qi

a =2 Qii

RRR

We thus have Qii = −

RRR

qii |Di v|2 dt + 2

Q

−

RR ³¡ Qi

+ hi +

ξ2,i pii

RR ³ Qi

i

RRR

¯ ¯ RRR i ¯2 i ¯2 qii i ¯Di v ¯ dt − 2 pii Di (ξ2,i )¯Di v ¯ dt

Q

Q

´ 2 2 dt |D v| |D v| 1 1 − ξ2,i pii i i Nx + Ni +1 0

¢

¡

i

¢

2

2

´

(qii i )Ni +1 |Di v|2Nx + 1 + (qii i )0 |Di v|21 dt 2

2

h2i RRR i h2 RRR qii |Di Di v|2 dt − i (Di Di qii )|Di v|2 dt. 2 Q 2 Q

31


Lemma C.4. We have ¡ 2 2 ¢ i i ξ2,i pii = − ξ1,i ξ2,i + Oλ,K (sh) rDi ρ , pii Di (ξ2,i ) = sλϕO(1) + sOλ,K (sh), 2 2 qii = sλ2 ξ1,i ξ2,i ϕ(∂i ψ)2 + sλϕO(1) + sOλ,K (sh), 2 2 qii i = sλ2 ξ1,i ξ2,i ϕ(∂i ψ)2 + sλϕO(1) + sOλ,K (sh),

h2i Di Di qii = sλϕO(1) + sOλ,K (sh). Moreover for hi sufficiently small we have (qii i )Ni +1 ≥ sλ(ϕ)Ni + 21 O(1) + sOλ,K (sh),

(C.7)

(qii )0 ≥ sλ(ϕ) 12 O(1) + sOλ,K (sh).

hal-00450854, version 1 - 27 Jan 2010

i

Proof. The first estimate follows from Proposition 2.9. The next three estimates all follow from Proposition 2.13 and Corollary 2.8, following the method of the proof of Lemma C.2. i i i 2 ξ2,i ξ2,i and β = r2 ρ˜i Di ρ we first To estimate h2i Di Di qii , introducing α = −ξ1,i write i

i

i i i i fi fi ]i ]i ^ ^ Di Di qii = (Di Di Di α)γ˜ i + 3D ˜ i (Di Di Di γ). i Di α Di γ + 3Di α Di Di γ + α

We note that we have i

i

i

fi α ˜ i = O(1),

]i Di α = O(1),

^ hD i Di α = O(1),

h2 Di Di Di α = O(1), and, with Proposition 2.13, i

fi γ˜ i = sλϕO(1) + sOλ,K ((sh)2 ), hDi Di Di γ = sOλ,K (1).

i

]i Di γ = sOλ,K (1),

i

^ D i Di γ = sOλ,K (1),

The estimate for h2i Di Di qii then follows. For the second part of the proof we only address the first inequality in (C.7). The second inequality follows similarly. We have qii = i

¡

2 ξ ξ i −Di ξ1,i 2,i 2,i

|

{z

¢

i i

i

r2 ρ˜ Di ρ i

i

i

i i 2 ξ2,i ξ2,i +ξ1,i

} |

=sλϕO(1)+sOλ,K (sh)

{z

}

bii

≥0

where the estimation of the first term follows as in the proof of Lemma C.2 and with i i bii = Di (−r2 ρ˜i Di ρ ). It remains thus to prove that (bii )ki ≥ 0, ki = Ni + 12 , Ni + 32 , for hi sufficiently small. Observing that ∂i2 ϕ(x) = λ2 (∂i ψ)2 ϕ + λ(∂i2 ψ)ϕ, with the assumption made on ψ in the neighborhood of the boundary ∂i Ω, we see that the function xi 7→ ϕ(t, x1 , . . . , xd ) is convex in a neighborhood of {xi = Li }. It thus follows that ϕki +1 + ϕki −1 − 2ϕki ≥ 0,

32


for ki hi close to Li = (Ni + 1)hi . As ρ = e−sϕ it follows that ρki +1 ρk i ≤ , ρki ρki −1

for ki hi close to (Ni + 1)hi .

(C.8)

We now write i

i

(−r2 ρ˜i Di ρ )ki =

1 ³¡ ρk −1 ¢2 ¡ ρk +1 ¢2 ´ 1+ i − 1+ i , 8hi ρki ρki

which gives hi (bii )ki + 12 =

ρki ¢2 ¡ ρk −1 ¢2 ¡ ρk +1 ¢2 ¡ 1 ³¡ ρk +2 ¢2 ´ 1+ − 1+ i + 1+ i − 1+ i , 8hi ρki +1 ρk i ρki ρki +1 | {z } | {z } ≥0

≥0

by (C.8) if ki hi close to Li = (Ni + 1)hi . Inequality (C.7) thus follows for hi small, noting that (ϕ)ki +1 = (ϕ)ki + 12 + h2 Oλ (1).

hal-00450854, version 1 - 27 Jan 2010

i

j

Computation of Qij , i 6= j. We set pij = −ξ1,i ξ1,j ξ2,j r2 ρ˜i Dj ρ and qij = ξ2,i Di pij . As v|∂Ω = 0, a discrete integration by parts w.r.t. xi (see Lemma 2.4) yields ¡ RRR j¢ Qij =2 Re ξ2,i Di pij Dj v ∗ Di v dt, Q a b which can be written as Qij = Qij + Qij with a Qij = 2 Re

RRR

i

^j qij Dj v ∗ Di v dt,

Q

b Qij = 2 Re

RRR

j

i

∗ ξ2,i pf ij Dj Di v Di v dt.

Q

By Proposition 2.4 we write RRR i j a Qij = 2 Re qij Di v Dj v ∗ dt Q

= 2 Re

RRR

i

j

qij i Di v Dj v ∗ dt +

Q

RRR h2i j Re (Di qij ) (Di Di v) Dj v ∗ dt. 2 Q

We also have b Qij = 2 Re

RRR

j

∗ ξ2,i pf ij Di v Dj Di v dt i

Q

RRR î j h2j RRR j i 2 ∗ g Dj (ξ2,i pf = 2 Re ξ2,i pf D v D D v dt + ij ) |Dj Di v| dt ij i j i 2 Q Q =−

RRR

j

2 Dj (ξ2,i pf ij ) |Di v| dt + i

Q

h2j RRR i 2 Dj (ξ2,i pf ij ) |Dj Di v| dt. 2 Q

We thus have Qij = −

RRR Q

+

j

2 Dj (ξ2,i pf ij ) |Di v| dt + 2 Re i

RRR

i

j

qij i Di v Dj v ∗ dt

(C.9)

Q

RRR h2j RRR h2i j i 2 Re (Di qij ) (Di Di v) Dj v ∗ dt + Dj (ξ2,i pf ij ) |Dj Di v| dt 2 2 Q Q

33


Lemma C.5. We have j

2 2 Dj (ξ2,i pf ij ) = sλ ξ1,i ξ2,i ξ1,j ξ2,j ϕ(∂j ψ) + sλϕO(1) + sOλ,K (sh), qij i = sλ2 ξ1,i ξ2,i ξ1,j ξ2,j ϕ(∂i ψ)(∂j ψ) + sλϕO(1) + sOλ,K (sh), i

2 2 Dj (ξ2,i pf ij ) = sλ ξ1,i ξ2,i ξ1,j ξ2,j ϕ(∂j ψ) + sλϕO(1) + sOλ,K (sh), hDi qij = sλϕO(1) + Oλ,K (sh). i

The estimates all follow from Proposition 2.13 and Corollary 2.8, arguing as in the proof of Lemma C.2. Estimate of I11 . We now collect the different terms that we have just computed and use Lemmata C.1 to C.5 to write 0 00 000 0 00 I11 = I11 + Y11 + I11 + I11 − (J11 + Z11 + Z11 + Z11 ),

where

hal-00450854, version 1 - 27 Jan 2010

0 = −sλ2 I11

RRR

ϕξ1,i ξ2,i |∇ξ ψ|2 |Di v|2 dt

i∈J1,dK Q

Q

P RR ³

+ sλ

P RRR

ϕ|∇ξ ψ|2 |∂t v|2 dt − sλ2

· ¸T ´ RR ϕξ1,i ξ2,i (∂t ψ)|Di v|2 (T ) − sλ ϕ(∂t ψ)|∂t v|2

i∈J1,dK Ω

Ω

0

and Y11 =

P RR ³¡ 2 2 i¢ (ξ1,i ξ2,i + Oλ,K (sh)) rDi ρ Ni +1 |Di v|2Ni + 1 2

i∈J1,dK Qi

´ ¡ 2 2 i¢ ξ2,i + Oλ,K (sh)) rDi ρ 0 |Di v|21 dt, − (ξ1,i 2

and 00 = 2sλ2 I11

RRR Q

³ ´ ¯ P 2 2 i ¯2 ϕ (∂t ψ)2 |∂t v|2 + ξ1,i ξ2,i (∂i ψ)2 ¯Di v ¯ dt

+ 2sλ2 Re

RRR Q

P

+

³

i∈J1,dK

P

ϕ 2(∂t ψ)∂t v

ξ1,i ξ2,i (∂i ψ)Di v ∗

i∈J1,dK i

ξ1,i ξ2,i ξ1,j ξ2,j (∂i ψ)(∂j ψ)Di v Dj v ∗

j

i

´ dt,

i,j∈J1,dK i6=j

= 2sλ2

RRR

¯ ¯2 P i¯ ¯ ϕ¯(∂t ψ)∂t v + ξ1,i ξ2,i (∂i ψ)Di v ¯ dt

Q

i∈J1,dK

≥ 0, and 000 I11 =

sλ2 h2i RRR ϕξ1,i ξ2,i (∂i ψ)2 |Di ∂t v|2 dt 2 i∈J1,dK Q P

+

P i,j∈J1,dK i6=j

+ ≥ 0,

sλ2 h2j RRR ϕξ1,i ξ2,i ξ1,j ξ2,j (∂j ψ)2 |Di Dj v|2 dt 2 Q

sλ2 h2i RRR 2 2 ϕξ1,i ξ2,i (∂i ψ)2 |Di Di v|2 dt 2 i∈J1,dK Q P

34


and

P RR

J11 =

δ11,i |Di v|2 (T )

i∈J1,dK Ω

P RR ³

+

(2)

(2)

(δ11,i )Nx + 12 |Di v|2Nx + 1 + (δ11,i ) 12 |Di v|21 2

i∈J1,dK Qi

2

´ dt

(2)

with δ11,i = sOλ,K (sh), and δ11,i = h(sλϕO(1) + sOλ,K (sh)), and RRR 0 P RRR 0 P RRR 0 ¯¯ i ¯2 Z11 = β11 |∂t v|2 dt + ν11,i |Di v|2 dt + ν 11,i Di v ¯ dt i∈J1,dK Q

Q

and 0 = Re Z11

P

RRR

i,j∈J1,dK i6=j

Q

i∈J1,dK Q

i

P RRR

j

α11,ij Di v Dj v ∗ dt + Re

i

α11,ti Di v ∂t v ∗ dt

i∈J1,dK Q

0 0 where β11 , ν11,i , ν 011,i , α11,ij , and α11,ti are of the form sλϕO(1) + sOλ,K (sh), and P RRR 00 P RRR 00 P RRR 00 00 = γ11,ti |Di ∂t v|2 dt + Z11 γ11,ij |Di Dj v|2 dt + γ11,ii |Di Di v|2 dt

hal-00450854, version 1 - 27 Jan 2010

i∈J1,dK Q

P RRR

+ Re

i∈J1,dK Q

i,j∈J1,dK i6=j

i∈J1,dK Q

Q

00 γ11,iit (Di Di v)∂t v ∗ dt + Re

P

RRR

i,j∈J1,dK i6=j

Q

j

00 γ11,iij (Di Di v)Dj v ∗ dt,

¡ 2

¢ 00 00 00 00 , are of the form h sλϕO(1) + sOλ,K (sh) , and γ11,iit , and γ11,ii , γ11,ij where γ11,ti ¡ ¢ 00 γ11,iij are of the form h sλϕO(1) + Oλ,K (sh) . We conclude with Cauchy-Schwarz inequalities that yields ¯ ¯ ¯2 RRR P RRR i ¯2 0 |Z11 |≤ α11,i ¯Di v ¯ dt + α11,t ¯∂t v ¯ dt, i∈J1,dK Q

Q

with α11,i and α11,t of the form sλϕO(1) + sOλ,K (sh), and P RRR 000 P RRR 000 P RRR 000 00 γ11,ij |Di Dj v|2 dt + γ11,ii |Di Di v|2 dt |Z11 |≤ γ11,ti |Di ∂t v|2 dt + i∈J1,dK Q

+

RRR Q

i,j∈J1,dK i6=j

000 γ11,t |∂t v|2 dt +

P RRR i∈J1,dK Q

i∈J1,dK Q

Q

i

000 γ11,i |Di v |2 dt,

¡ ¢ 000 000 000 000 and are of the form h2 sλϕO(1) + sOλ,K (sh) and γ11,t , and γ11,ii , γ11,ij with γ11,ti 000 γ11,i are of the form sλϕO(1) + Oλ,K (sh). C.2. Proof of Lemma 3.5. As compared to the computation of the counterpart of I21 in the proof of the semi-discrete Carleman estimate in [BHL09a] (also denoted I21 there) we need to compute the following additional terms, RRR j i Qij = −2 Re pij v˜j Di v ∗ dt, Q i

2

for i 6= j, where pij = −ξ1,i ξ2,i ξ1,j ξ2,j r (Dj Dj ρ) Di ρ . With Proposition 2.4, we have Qij = −2 Re

RRR

j i

pij Di v ∗ v˜j dt

Q

= −2 Re |

RRR Q

RRR h i j^ ∗i v ˜j dt − Re (Dj pij )(Dj Di v ∗ )˜ pf v j dt. ij Di v 2 Q {z } j

a Qij

2


35

We now write a Qij = −2 Re

RRR

i

j

j j ∗ ] pf ˜ D ij v i v dt

Q

RRR gj i ej i j h2 RRR j ∗ ] = −2 Re pf ˜ D (Di pf ˜j |2 dt, ij ) |Di v ij v i v dt − 2 Q Q | {z } b Qij

and with a discrete integration by parts in xi (Proposition 2.4) and Lemma 2.2 we have, as v˜j = 0 on ∂i Q, b =− Qij

Q

Q

³ 2 i´ j j j g2 dt − h RRR Di pg Di pg f |v| f |Dj v|2 dt ij ij 4 Q Q j i RRR h2 RRR ³ gj i ´ j = Di pg f |v|2 dt − Di pf |Dj v|2 dt ij ij 4 Q Q =

hal-00450854, version 1 - 27 Jan 2010

RRR gj i RRR ³ gj i ´ j 2 pf v j |2 dt = Di pf |˜ v | dt ij Di |˜ ij i´

RRR ³

We thus have j i h2 RRR j j (Di pf ˜j |2 dt Di pg f |v|2 dt − ij ) |Di v ij 2 Q Q RRR h2 RRR ³ gj i ´ h2 i − Di pf Re (Dj pij )(Dj Di v ∗ ) v˜j dt, |Dj v|2 dt − ij 4 Q 2 Q

Qij =

RRR

Lemma C.6. We have j Di pg f ij

i

j

= 3s3 λ4 ξ1,i ξ2,i ξ1,j ξ2,j ϕ3 (∂i ψ)2 (∂j ψ)2 + (sλϕ)3 O(1) + s2 Oλ,K (1) + s3 Oλ,K ((sh)2 ),

3 Di pf ij = s Oλ,K (1), j

i

Di pg f = s3 Oλ,K (1), ij j

Dj pij = s3 Oλ,K (1).

The estimations follow from Proposition 2.13 and Corollary 2.8 arguing as in the proof of Lemmata C.2. By Young’s inequality we now note that ¯ RRR h2 ¯¯ j ¯ (Di pij )(Di Dj v ∗ ) v˜i dt¯ ¯ Re 2 Q RRR RRR j ≤ s3 (sh) Oλ,K (1)|˜ v i |2 dt + sh2 (sh) Oλ,K (1)|Di Dj v |2 dt Q

≤ s3 (sh)

RRR

Q

i

g2 dt + sh2 (sh) Oλ,K (1)|v|

Q

3

= s (sh)

RRR Q

i

RRR

j

Oλ,K (1)|Di Dj v|2 dt

Q

2

2

Oλ,K (1)|v| dt + sh (sh)

RRR

Oλ,K (1)|Di Dj v|2 dt,

Q

g2 and using Proposition 2.4. Proceeding similarly for the term in since |˜ v i |2 ≤ |v|

36


¯ j ¯2 g ¯ we then obtain |Di v˜j |2 = ¯D iv Qij ≥ 3s3 λ4

RRR

ξ1,i ξ2,i ξ1,j ξ2,j ϕ3 (∂i ψ)2 (∂j ψ)2 |v|2 dt +

Q

+

RRR

RRR

µ |v|2 dt +

Q 2

νj |Dj v| dt +

Q

RRR

RRR

νi |Di v|2 dt

Q

(C.10) 2

γ|Di Dj v| dt,

Q

with µ = (sλϕ)3 O(1) + s2 Oλ,K (1) + s3 Oλ,K (sh), νi = sOλ,K ((sh)2 ),

νj = sOλ,K ((sh)2 ),

γ = sh2 Oλ,K (sh).

hal-00450854, version 1 - 27 Jan 2010

With the computation performed in [BHL09a] (See Lemma 4.5 and its proof in Section B.4 in [BHL09a]) we then obtain the sought estimate from below for I21 . C.3. Proof of Lemma 3.7. We see that ¢ P RR ¡ Y = (qi )Nx +1 |Di v|2Ni + 1 − (qi )0 |Di v|21 dt 2

i∈J1,dK Qi

2

i

with qi = (1 + Oλ,K ((sh)2 )) rDi ρ . By (3.3) we have Y ≥ 0 for sh sufficiently small. C.4. Proof of Lemma 3.8. We choose i ∈ J1, dK. With Lemmata 2.5 and 2.2 and Proposition 2.4, we have RRR

ϕ|∇ξ ψ|2 |∂t v|2 dt =

Q

RRR

i

ϕ|∇ξ ψ|2 |∂t v|2 dt

Q

RRR

h2i RRR 2 ^ 2 ^ ϕ|∇ Di (ϕ|∇ξ ψ|2 )Di |∂t v|2 dt ξ ψ| |∂t v| dt + 4 Q Q i ¯ RRR i¯ 2 ¯∂fv ¯2 dt ^ = ϕ|∇ ξ ψ| t =

i

i

Q

+

´ RRR h2i ³ RRR ^ 2 i Di Di (ϕ|∇ξ ψ|2 )|∂t v|2 dt . ϕ|∇ξ ψ| |Di ∂t v|2 dt − 4 Q Q

We thus have RRR

ϕ|∇ξ ψ|2 |∂t v|2 dt ≥

Q

h2i RRR ^ 2 i ϕ|∇ξ ψ| |Di ∂t v|2 dt 4 Q ´ hi RRR ³ + − (τi − τi− )Di (ϕ|∇ξ ψ|2 ) |∂t v|2 dt. 4 Q

(C.11)

Similarly, for i, j ∈ J1, dK with i 6= j, we obtain RRR Q

ϕξ1,i ξ2,i |∇ξ ψ|2 |Di v|2 dt ≥

j h2j RRR ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Dj Di v|2 dt (C.12) 4 Q ´ hj RRR ³ + − (τj − τj− )Dj (ξ1,i ξ2,i ϕ|∇ξ ψ|2 ) |Di v|2 dt. 4 Q


37

For i ∈ J1, dK, we also write RRR

ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di v|2 dt =

Q

´ hi RR ³ (ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di v|2 ) 21 + (ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di v|2 )Ni + 21 dt 2 Qi RRR i + ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di v|2 dt, Q

|

{z

}

=Qi

by Proposition 2.4, and Lemma 2.2 yields Qi =

RRR

i

i

ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di v|2 dt +

Q

hal-00450854, version 1 - 27 Jan 2010

=

h2i RRR Di (ξ1,i ξ2,i ϕ|∇ξ ψ|2 ) Di |Di v|2 dt 4 Q

¯2 i ¯ i ¯ h2 RRR i ¯2 ξ1,i ξ2,i ϕ|∇ξ ψ|2 ¯Di v ¯ dt + i ξ1,i ξ2,i ϕ|∇ξ ψ|2 ¯Di Di v ¯ dt 4 Q Q ³ ´ RRR hi − (τi+ − τi− )Di (ξ1,i ξ2,i ϕ|∇ξ ψ|2 ) |Di v|2 dt 4 Q ´ h2 RR ³ + i Di (ξ1,i ξ2,i ϕ|∇ξ ψ|2 )Ni +1 |Di v|2Ni + 1 − Di (ξ1,i ξ2,i ϕ|∇ξ ψ|2 )0 |Di v|21 dt. 2 2 4 Qi RRR

Observing that |∇ξ ψ| ≥ C > 0 we find that ´ hi RR ³ (ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di v|2 ) 12 + (ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di v|2 )Ni + 12 dt 2 Qi ´ h2 RR ³ + i Di (ξ1,i ξ2,i ϕ|∇ξ ψ|2 )Ni +1 |Di v|2Ni + 1 − Di (ξ1,i ξ2,i ϕ|∇ξ ψ|2 )0 |Di v|21 dt ≥ 0, 2 2 4 Qi for h sufficiently small, as Di (ξ1,i ξ2,i ϕ|∇ξ ψ|2 ) = Oλ (1). It follows that RRR

ξ1,i ξ2,i ϕ|∇ξ ψ|2 |Di v|2 dt ≥

Q

¯2 i ¯ i ¯ h2 RRR i ¯2 ξ1,i ξ2,i ϕ|∇ξ ψ|2 ¯Di v ¯ dt + i ξ1,i ξ2,i ϕ|∇ξ ψ|2 ¯Di Di v ¯ dt 4 Q Q (C.13) ´ hi RRR ³ + − (τi − τi− )Di (ξ1,i ξ2,i ϕ|∇ξ ψ|2 ) |Di v|2 dt, 4 Q RRR

We have i

2 2 ^ ϕ|∇ ξ ψ| = ϕ|∇ξ ψ| + hOλ,K (1),

i

ξ1,i ξ2,i ϕ|∇ξ ψ|2 = ξ1,i ξ2,i ϕ|∇ξ ψ|2 + hOλ,K (1),

i

ξ1,i ξ2,i ϕ|∇ξ ψ|2 = ξ1,i ξ2,i ϕ|∇ξ ψ|2 + hOλ,K (1), (τj+ − τj− )Di (ξ1,i ξ2,i ϕ|∇ξ ψ|2 ) = Oλ,K (1),

i, j ∈ J1, dK.

The result follows. Appendix D. A fully-discrete elliptic Carleman estimate for uniform meshes. In Section 3 we have derived a Carleman estimate for a semi-discrete elliptic operator having in mind applications to the controllability of semi-discrete and discrete parabolic equations. For completeness, in the present section we treat the case of

38


d fully discrete elliptic operator. Here P we thus only consider variables in Ω ⊂ R . The M operator we consider is A = − i∈J1,dK ξ1,i Di ξ2,i Di . The case of a non uniform mesh can be treated as in Section 4. We choose here to treat the case of an inner-observation in ω b Ω. The weight function we choose is different from that introduced in Section 3. It is of the form r = esϕ with ϕ = eλψ , with ψ fulfilling the following assumption. Construction of such a weight function is classical (see e.g. [FI96]).

˜ be a smooth open and conAssumption D.1. Let ω0 b ω be an open set. Let Ω d ˜ R), p sufficiently nected neighborhood of Ω in R . The function ψ = ψ(x) is in C p (Ω, large, and satisfies, for some c > 0, ˜ ψ > 0 in Ω,

hal-00450854, version 1 - 27 Jan 2010

∂i2 ψ(x)

˜ \ ω0 , |∇ψ| ≥ c in Ω

∂ni ψ(t, x) ≤ −c < 0 in (0, T ) × V∂i Ω ,

≥ 0 in V∂i Ω .

˜ in which the outward unit where V∂i Ω is a sufficiently small neighborhood of ∂i Ω in Ω, normal ni to Ω is extended from ∂i Ω. We also set ρ = r−1 . The following notation is adapted to the fully-discrete setting of the present section ³p ´t p P ∇ξ f = ξ1,1 ξ2,1 ∂x1 f, . . . , ξ1,d ξ2,d ∂xd f , ∆ξ f = ξ1,i ξ2,i ∂x2i f. i∈J1,dK

As in Section 3 we use reg(ξ) to measure the boundedness of ξ1,i and ξ2,i and of their discrete derivatives as well as the distance to zero of ξ1,i and ξ2,i , i ∈ J1, dK (see (3.1)-(3.2)). Here, by abuse of notation, the letters ξ1,i , ξ2,i will also refer to i a Q1 -interpolation on M and M respectively. Note that the resulting interpolated functions are Lipschitz continuous with kξ1,i kW 1,∞ ≤ Creg(ξ),

kξ2,i kW 1,∞ ≤ Creg(ξ).

˜ of Ω introduced in Assumption 1.3 allows us to The enlarged neighborhood Ω apply multiple discrete operators such as Di and Ai on the weight functions. In particular, this then yields on ∂i Ω i

i

(rDi ρ )|ki =0 ≤ 0,

(rDi ρ )|ki =Ni +1 ≥ 0,

i ∈ J1, dK.

Theorem D.2. Let reg0 > 0 be given. For the parameter λ ≥ 1 sufficiently large, there exist C, s0 ≥ 1, h0 > 0, ε0 > 0, depending on ω and reg0 , such that for any ξ1,i , ξ2,i , i ∈ J1, dK, with reg(ξ) ≤ reg0 we have s3 kesϕ uk2L2 (Ω) + s

P i∈J1,dK

kesϕ Di uk2L2 (Ω) + s

P i∈J1,dK

|esϕ Di u|2L2 (∂i Ω)

³ ´ ≤ Cλ1 ,K,ε0 ,s0 kesϕ AM uk2L2 (Ω) + s3 kesϕ uk2L2 (ω) .

for all s ≥ s0 , 0 < h ≤ h0 and sh ≤ ε0 , and u ∈ CM∪∂ M , satisfying u|∂Ω = 0. Proof. We set f := −AM u and v = ru that satisfies P r ξ1,i Di ξ2,i Di (ρv) = rf. i∈J1,dK

39


Arguing as in the proof of Theorem 3.1 we then write Av + Bv = g with A = A1 + A2 and B = B1 + B2 and P P i i A1 v = ξ1,i rρ˜i Di (ξ2,i Di v), A2 v = ξ1,i ξ2,i r(Di Di ρ) v˜i , i∈J1,dK

i∈J1,dK

P

B1 v = 2

i

i

ξ1,i ξ2,i rDi ρ Di v ,

B2 v = −2s(∆ξ ϕ)v

i∈J1,dK

g = rf −

P

hi i ξ1,i rDi ρ (Di ξ2,i )(τi+ Di v − τi− Di v) i∈J1,dK 4

P h2i i i i ξ1,i (Di ξ2,i )r(Di Di ρ)Di v − hi O(1)rDi ρ Di v 4 i∈J1,dK i∈J1,dK ³ ´ i P i − ξ1,i r(Di ξ2,i )Di ρ + hi O(1)r(Di Di ρ) v˜i − 2s(∆t,x ϕ)v. P

−

i∈J1,dK

hal-00450854, version 1 - 27 Jan 2010

The proof of Lemma 3.2 can be directly adapted and we have Ã ! 2 2 2 2 2 P 2 kgkL2 (Ω) ≤ Cλ,K krf kL2 (Ω) + s kvkL2 (Ω) + (sh) kDi vkL2 (Ω) .

(D.1)

i∈J1,dK

Developing the inner-product Re (Av, Bv)L2 (Ω) , we set Iij = Re (Ai v, Bj v)L2 (Ω) . Lemma D.3 (Estimate of I11 ). For sh ≤ K, the term I11 can be estimated from below in the following way 1

I11 ≥ −sλ2 kϕ 2 |∇ξ ψ|gξ vk2L2 (Ω) + Y11 − X11 − W11 − J11 , with Y11 =

P

R ³¡ 2 2 i¢ (ξ1,i ξ2,i + Oλ,K ((sh)2 )) rDi ρ Ni +1 |Di v|2Ni + 1 2

i∈J1,dK Ωi

¡

2 2 − (ξ1,i ξ2,i + Oλ,K ((sh)2 )) rDi ρ

i

¢

´ 2 |D v| , 1 i 0 2

and X11 =

P RR

ν11,i |Di v|2 +

i∈J1,dK Ω

P RR

i

ν 11,i |Di v |2 ,

i∈J1,dK Ω

with ν11,i and ν 11,i of the form sλϕO(1) + sOλ,K (sh) and P RR P RR W11 = γ11,ij |Di Dj v|2 + γ11,ii |Di Di v|2 , i,j∈J1,dK i6=j

i∈J1,dK Ω

Ω

¡ ¢ with γ11,ij and γ11,ii of the form h2 sλϕO(1) + sOλ,K (sh) and ´ P R ³ (2) (2) J11 = (δ11,i )Ni + 21 |Di v|2Ni + 1 + (δ11,i ) 12 |Di v|21 , 2

i∈J1,dK Ωi

2

(2)

with δ11,i = shi λϕO(1) + shi Oλ,K (sh). For a proof, see the proof of Lemma 3.3 in Appendix C and only consider the terms Qii and Qij . Lemma D.4 (Estimate of I12 ). For sh ≤ K, the term I12 is of the following form 1

I12 ≥ 2sλ2 kϕ 2 |∇ξ ψ|gξ vk2L2 (Ω) − X12 ,

40


with P RR

X12 =

ν12,i |Di v|2 +

i∈J1,dK Ω

RR

µ12 |v|2 ,

Ω

where µ12 = s2 Oλ,K (1), and ν12,i = sλϕO(1) + sOλ,K (sh). Lemma D.5 (Estimate of I21 ). For sh ≤ K, the term I21 can be estimated from below in the following way 3

I21 ≥ 3s3 λ4 kϕ 2 |∇ξ ψ|2 vk2L2 (Ω) + Y21 − W21 − X21 , with R

P

Y21 =

i

i∈J1,dK Ωi

+

P

Oλ,K ((sh)2 )(rDi ρ )0 |Di v|21 2

R

i

i∈J1,dK Ωi

hal-00450854, version 1 - 27 Jan 2010

W21 =

P

RR

i,j∈J1,dK i6=j

Ω

2 Oλ,K ((sh)2 )(rDi ρ )Nx +1 |Di v|N 1, x+ 2

γ21,ij |Di Dj v|2 ,

X21 =

RR

P RR

µ21 |v|2 +

Ω

ν21,i |Di v|2 ,

i∈J1,dK Ω

where γ21,ij = hOλ,K ((sh)2 ),

µ21 = (sλϕ)3 O(1) + s2 Oλ,K (1) + s3 Oλ,K (sh),

ν21,i = sOλ,K ((sh)2 ). For a proof, adapt the proof of Lemma 3.5 in Appendix C as was done for Lemma D.3. Lemma D.6 (Estimate of I22 ). For sh ≤ K, the term I22 is of the following form 3

I22 = −2s3 λ4 kϕ 2 |∇ξ ψ|2 vk2L2 (Ω) − X22 , with X22 =

RRR Ω

µ22 |v|2 +

P RRR

ν22,i |Di v|2

i∈J1,dK Ω

where µ22 = (sλϕ)3 O(1) + s2 Oλ,K (1) + s3 Oλ,K (sh), and ν22,i = sOλ,K (sh). With the previous lemmata, arguing as in the proof of Theorem 3.1, using that i

i

(rDi ρ )Ni +1 ≥ c > 0 and − (rDi ρ )0 ≥ c > 0 onΩi ,

1≤i≤d

i

by Assumption D.1 since rDi ρ = −sλ(∂i ψ)ϕ + sOλ,K (sh), and recalling that |∇ψ| ≥ c > 0 in Ω \ ω0 we obtain that for some λ1 ≥ 1 sufficiently large, s1 (λ1 ) > 1 and eps1 (λ1 ) > 0 then for λ = λ1 (fixed for the rest of the proof), s ≥ s1 (λ1 ) and sh ≤ ε1 (λ1 ) we have s3 kvk2L2 (Ω) + s

P i∈J1,dK

kDi vk2L2 (Ω) + s|Di v|2L2 (∂i Ω)

³ ´ P ≤ Cλ1 ,K,ε0 ,s0 krf k2L2 (Ω) + s3 kvk2L2 (ω0 ) + s kDi vk2L2 (ω0 ) . i∈J1,dK

41


we proceeding as in the end of proof of Theorem 4.1 in [BHL09a] we obtain s3 kesϕ uk2L2 (Ω) + s

P i∈J1,dK

kesϕ Di uk2L2 (Ω) + s

P i∈J1,dK

|esϕ Di u|2L2 (∂i Ω)

´ ³ P ≤ Cλ0 1 ,K,ε0 ,s0 kesϕ f k2L2 (Ω) + s3 kesϕ uk2L2 (ω0 ) + s kesϕ Di uk2L2 (ω0 ) . i∈J1,dK

It thus remains to eliminate the last term in the r.h.s.. To that purpose we adapt the procedure followed in the continuous case (see e.g. [FI96, FCG06, LL09]). We multiply the equation satisfied by u, i.e. AM u = f , by sr2 χu∗ , where χ ∈ Cc∞ (ω) is such that χ ≥ 0 and χ = 1 in a neighborhood of ω0 . We then integrate over Ω: RR P RR − Re s ξ1,i r2 χu∗ Di (ξ2,i Di u) = Re s r2 χu∗ f. (D.2) i∈J1,dK Ω

Ω

We first note that the r.h.s. can be estimated by ¯ ¯ RR ¯ ¯ ¯ Re s r2 χu∗ f ¯ ≤ Ckrf k2L2 (Ω) + s2 Ckruk2L2 (ω) .

(D.3)

hal-00450854, version 1 - 27 Jan 2010

Ω

In the l.h.s. of (D.2) we perform a discrete integration by parts to yield P RR P RR Di (ξ1,i r2 χu∗ )ξ2,i Di u − Re s ξ1,i r2 χu∗ Di (ξ2,i Di u) = Re s i∈J1,dK Ω

i∈J1,dK Ω

P RR

=s

i

2 ξ2,i ξ^ 1,i r χ |Di u| + Re s 2

i∈J1,dK Ω

P RR

i

f∗ Di u ξ2,i Di (ξ1,i r2 χ)u

(D.4)

i∈J1,dK Ω

In ω0 , for h sufficiently small, we have 2 i i i e2 i hi (Di ξ1,i )(Di r2 ). 2 ^2 = ξf ξ^ 1,i r χ ≥ ξ1,i r 1,i r + 4

The results of the lemmata of Section 2.2 remain valid for r2 in place of r, i.e. for s i changed into 2s. As ξf 1,i = ξ1,i + hO(1) and Di ξ1,i = O(1) we thus find i

2 2 2 ξ^ 1,i r χ ≥ r (ξ1,i + hO(1) + Oλ,K ((sh) ).

For the first term in the r.h.s. of (D.4) it follows that, for h and sh sufficiently small, s

P RR i∈J1,dK Ω

P

i

2 2 ξ2,i ξ^ 1,i r χ |Di u| ≥ Cs

i∈J1,dK

krDi uk2L2 (ω0 ) .

(D.5)

For second term in the r.h.s. of (D.4) we write P RR i f∗ Di u Re s ξ2,i Di (ξ1,i r2 χ) u i∈J1,dK Ω

P RR

= Re s

i∈J1,dK Ω

i i P RR i 2 f∗ Di u + 1 s ξ2,i re2 Di (ξ1,i χ) u ξ2,i Di (r2 )ξ] 1,i χ Di |u| . 2 i∈J1,dK Ω (D.6)

Arguing as above, the first term in the r.h.s. of (D.6) can be estimated by ¯ ¯ RR i i ¯ f∗ Di u¯¯ ≤ CkrDi uk2L2 (Ω) + Cs2 kruk2L2 (ω) , ¯ Re s ξ2,i re2 Di (ξ1,i χ) u Ω

(D.7)

42


for h and sh sufficiently small, as supp(χ) b ω. For the second term in the r.h.s. of (D.6) a discrete integration by parts yields i i 1 P RR 1 P RR 2 2 s ξ2,i Di (r2 )ξ] Di (ξ2,i Di (r2 )ξ] 1,i χ Di |u| = − s 1,i χ )|u| 2 i∈J1,dK Ω 2 i∈J1,dK Ω

With the results of Section 2.2, using that Di ξ1,i = O(1) and Di ξ2,i = O(1) we find ¯ 1 P RR ¯ i ¯ 2¯ ξ2,i Di (r2 )ξ] χ D |u| (D.8) ¯ ≤ Cs3 kruk2L2 (ω) , ¯ s 1,i i 2 i∈J1,dK Ω for h and sh sufficiently small. With (D.2)–(D.8) we conclude that ³ ´ P P s krDi uk2L2 (ω0 ) ≤ C krf k2L2 (Ω) + s3 kruk2L2 (ω) + krDi uk2L2 (Ω) . i∈J1,dK

i∈J1,dK

hal-00450854, version 1 - 27 Jan 2010

For s sufficiently large we thus obtain the desired Carleman estimate. REFERENCES [AE84] [BHL09a]

[BHL09b] [BHL10] [FCG06]

[FI96] [H¨ or63] [H¨ or85] [JL99]

[KS91] [Le 07]

[LL09]

[LR95] [LR97] [LT06] [LZ98a]

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DISCRETE CARLEMAN ESTIMATES IN ARBITRARY DIMENSION [LZ98b]

[Zua06]

hal-00450854, version 1 - 27 Jan 2010

[Zui83]

43

A. Lopez and E. Zuazua, Some new results to the null controllability of the 1-d heat equation, Sminaire sur les quations aux Drives Partielles, 1997–1998, Exp. No. VIII, 22 pp., cole Polytech., Palaiseau (1998). E. Zuazua, Control and numerical approximation of the wave and heat equations, International Congress of Mathematicians, Madrid, Spain III (2006), 1389–1417. C. Zuily, Uniqueness and Non Uniqueness in the Cauchy Problem, Birkhauser, Progress in mathematics, 1983.