Logarithmic decay of the energy for an hyperbolic-parabolic coupled system
aa r X i v : . [ m a t h . A P ] J un Logarithmic decay of the energy for anhyperbolic-parabolic coupled system
Ines Kamoun Fathallah ∗ Abstract
This paper is devoted to the study of a coupled system consisting in a waveand heat equations coupled through transmission condition along a steadyinterface. This system is a linearized model for fluid-structure interactionintroduced by Rauch, Zhang and Zuazua for a simple transmission conditionand by Zhang and Zuazua for a natural transmission condition.Using an abstract Theorem of Burq and a new Carleman estimate shownnear the interface, we complete the results obtained by Zhang and Zuazuaand by Duyckaerts. We show, without any geometric restriction, a logarith-mic decay result.
Keywords : Fluid-structure interaction; Wave-heat model; Stability; Log-arithmic decay. : 37L15; 35B37; 74F10; 93D20
In this work, we are interested with a linearized model for fluid-structure inter-action introduced by Zhang and Zuazua in [14] and Duyckaerts in [6]. This modelconsists of a wave and heat equations coupled through an interface with suitabletransmission conditions. Our purpose is to analyze the stability of this system andso to determine the decay rate of energy of solution as t → ∞ .Let Ω ⊂ R n be a bounded domain with a smooth boundary Γ = ∂ Ω. Let Ω and Ω be two bounded open sets with smooth boundary such that Ω ⊂ Ω andΩ = Ω \ Ω . We denote by γ = ∂ Ω ∩ ∂ Ω the interface, γ ⊂⊂ Ω, Γ j = ∂ Ω j \ γ , j = 1 , ∂ n and ∂ n ′ the unit outward normal vectors of Ω and Ω respectively ∗ Laboratoire LMV, Universit´e de Versailles Saint-Quentin-En-Yvelines, 45 Avenue des Etats-Unis Batiment Fermat 78035 Versailles (France). E-mail address: [email protected];Tel:+33139253629; Fax:+33139254645. ∂ n ′ = − ∂ n on γ ). ∂ t u − △ u = 0 in (0 , ∞ ) × Ω ,∂ t v − △ v = 0 in (0 , ∞ ) × Ω ,u = 0 on (0 , ∞ ) × Γ ,v = 0 on (0 , ∞ ) × Γ ,u = ∂ t v, ∂ n u = − ∂ n ′ v on (0 , ∞ ) × γ,u | t =0 = u ∈ L (Ω ) in Ω ,v | t =0 = v ∈ H (Ω ) , ∂ t v | t =0 = v ∈ L (Ω ) in Ω . (1)In this system, u may be viewed as the velocity of fluid; while v and ∂ t v repre-sent respectively the displacement and velocity of the structure. That’s why thetransmission condition u = ∂ t v is considered as the natural condition. For themodelisation subject, we refer to [11] and [14].System (1) is introduced by Zhang and Zuazua [14]. The same system wasconsidered by Rauch, Zhang and Zuazua in [11] but for simplified transmissioncondition u = v on the interface instead of u = ∂ t v . They prove, under a suitableGeometric Control Condition (GCC) (see [1]), a polynomial decay result. Zhangand Zuazua in [14] prove, without GCC, a logarithmic decay result. Duyckaerts in[6] improves these results.For system (1), Zhang and Zuazua in [14], show the lack of uniform decay andthey prove, under GCC, a polynomial decay result. Without geometric conditions,they analyze the difficulty to prove the logarithmic decay result. This difficulty ismainly due to the lack of gain regularity of wave component v near the interface γ (see [14], Remark 19) which means that the embedding of the domain D ( A ) ofdissipative operator in the energy space is not compact (see [14], Theorem 1). In[6], Duyckaerts improves the polynomial decay result under GCC and confirms thesame obstacle to show the logarithmic decay for solution of (1) without GCC. Inthis paper we are interested with this problem.There is an extensive literature on the stabilization of PDEs and on the Loga-rithmic decay of the energy ([2], [3] [4], [8], [10], [12] and the references cited therein)and this paper use a part of the idea developed in [3].Here we recall the mathematical frame work for this problem (see [14]).Define the energy space H and the operator A on H , of domain D ( A ) by H = (cid:8) U = ( u , v , v ) ∈ L (Ω ) × H (Ω ) × L (Ω ) (cid:9) when H (Ω ) is defined as the space H (Ω ) = (cid:8) v ∈ H (Ω ) , v | Γ = 0 (cid:9) , A U = ( △ u , v , △ v ) D ( A ) = { U ∈ H, u ∈ H (Ω ) , △ u ∈ L (Ω ) ,v ∈ H (Ω ) , △ v ∈ L (Ω ) , u | γ = v | γ , ∂ n u | γ = − ∂ n v | γ } . ∂ t U = A U, U ( t ) = ( u ( t ) , v ( t ) , ∂ t v ( t )) . For any solution ( u, v, ∂ t v ) of system (1), we have a natural energy E ( t ) = E ( u, v, ∂ t v )( t ) = 12 (cid:18)Z Ω | u ( t ) | dx + Z Ω | ∂ t v ( t ) | dx + Z Ω |∇ v ( t ) | dx (cid:19) . By means of the classical energy method, we have ddt E ( t ) = − Z Ω |∇ u | dx. Therefore the energy of (1) is decreasing with respect to t , the dissipation comingfrom the heat component u . Our main goal is to prove a logarithmic decay withoutany geometric restrictions.As Duyckaerts [6] did for the simplified model, the idea is, first, to use a knownresult of Burq (see [5]) which links, for dissipative operators, logarithmic decay toresolvent estimates with exponential loss; secondly to prove, following the work ofBellassoued in [3], a new Carleman inequality near the interface γ .The main results are given by Theorem 1.1 for resolvent and Theorem 1.2 fordecay. Theorem 1.1
There exists
C > , such that for every µ ∈ R with | µ | large,we have (cid:13)(cid:13) ( A − iµ ) − (cid:13)(cid:13) L ( H ) ≤ Ce C | µ | . (2) Theorem 1.2
There exists
C > , such that the energy of a smooth solution of (1)decays at logarithmic speed p E ( t ) ≤ C log( t + 2) k U k D ( A ) . (3)Burq in ([5], Theorem 3) and Duyckaerts in ([6], Section 7) show that to proveTheorem 1.2 it suffices to show Theorem 1.1.The strategy of the proof of Theorem 1.1 is the following. A new Carlemanestimate shown near the interface γ implies an interpolation inequality given byTheorem 2.2. Theorem 2.2 implies Theorem 2.1 which gives an estimate of the wavecomponent by the heat one and which is the key point of the proof of Theorem 1.1.The rest of this paper is organized as follows. In section 2, we show, from The-orem 2.1, Theorem 1.1 and we explain how Theorem 2.2 implies Theorem 2.1. Insection 3, we begin by stating the new Carleman estimate and explain how thisestimate implies Theorem 2.2. We give then the proof of this Carleman estimate.Section 4 is devoted to the proof of important estimates stated in Theorem 3.2 inthe proof of Carleman estimate. Appendices A and B are devoted to prove sometechnical results that will be used along the paper.3 cknowledgment Sincere thanks to professor Luc Robbiano for inspiring question, his greatly con-tribution to this work and for his careful reading of the manuscript. I want to thankalso professor Mourad Bellassoued for his proposition to work in this domain.
This section is devoted to the proof of Theorem 1.1. We start by stating Theorem2.1. Then we will explain how this Theorem implies Theorem 1.1. Finally, we givethe proof of Theorem 2.1.Let µ be a real number such that | µ | is large, and assume F = ( A − iµ ) U, U = ( u , v , v ) ∈ D ( A ) , F = ( f , g , g ) ∈ H (4)The equation (4) yields ( △ − iµ ) u = f in Ω , ( △ + µ ) v = g + iµg in Ω ,v = g + iµv in Ω , (5)with the following boundary conditions u | Γ = 0 , v | Γ = 0 op ( b ) u = u − iµv = g | γ ,op ( b ) u = ∂ n u − ∂ n v = 0 | γ . (6)To proof Theorem 1.1, we begin by stating this result Theorem 2.1
Let U = ( u , v , v ) ∈ D ( A ) satisfying equation (5) and (6). Thenthere exists constants C > , c > and µ > such that for any µ ≥ µ we havethe following estimate k v k H (Ω ) ≤ Ce c µ (cid:16) k f k L (Ω ) + k g + iµg k L (Ω ) + k g k H (Ω ) + k u k H (Ω ) (cid:17) . (7)Moreover, from the first equation of system (5), we have Z Ω ( −△ + iµ ) u u dx = k∇ u k L (Ω ) + iµ k u k L (Ω ) − Z γ ∂ n u u dσ. Since u | γ = g + iµv and ∂ n u = − ∂ n ′ v , then Z Ω ( −△ + iµ ) u u dx = k∇ u k L (Ω ) + iµ k u k L (Ω ) − iµ Z γ ∂ n ′ v v dσ + Z γ ∂ n ′ v g dσ. (8)From the second equation of system (5) and multiplying by ( − iµ ), we obtain iµ Z Ω ( △ + µ ) v v dx = − iµ k∇ v k L (Ω ) + iµ k v k L (Ω ) + iµ Z γ ∂ n ′ v v dσ. (9)4dding (8) and (9), we obtain Z Ω ( −△ + iµ ) u u dx + iµ Z Ω ( △ + µ ) v v dx = iµ k u k L (Ω ) + k∇ u k L (Ω ) − iµ k∇ v k L (Ω ) + iµ k v k L (Ω ) + Z γ ∂ n ′ v g dσ. Taking the real part of this expression, we get k∇ u k L (Ω ) ≤ k ( △ − iµ ) u k L (Ω ) k u k L (Ω ) + (cid:13)(cid:13) ( △ + µ ) v (cid:13)(cid:13) L (Ω ) k v k L (Ω ) + (cid:12)(cid:12)(cid:12)(cid:12)Z γ ∂ n ′ v g dσ (cid:12)(cid:12)(cid:12)(cid:12) . (10)Recalling that △ v = g + iµg − µ v and using the trace lemma (Lemma 3.4 in[6]), we obtain k ∂ n v k H − ( γ ) ≤ C (cid:16) µ k v k H (Ω ) + k g + iµg k L (Ω ) (cid:17) . Combining with (10), we obtain k∇ u k L (Ω ) ≤ k f k L (Ω ) k u k L (Ω ) + k g + iµg k L (Ω ) k v k L (Ω ) + (cid:16) µ k v k H (Ω ) + k g + iµg k L (Ω ) (cid:17) k g k H ( γ ) . Then k∇ u k L (Ω ) ≤ Cǫ k f k L (Ω ) + ǫ k u k L (Ω ) + Cǫ k g + iµg k L (Ω ) + ǫ k v k L (Ω ) + (cid:16) µ k v k H (Ω ) + k g + iµg k L (Ω ) (cid:17) k g k H ( γ ) . Now we need to use this result shown in Appendix A.
Lemma 2.1
Let O be a bounded open set of R n . Then there exists C > such thatfor u and f satisfying ( △ − iµ ) u = f in O , µ ≥ , we have the following estimate k u k H ( O ) ≤ C (cid:16) k∇ u k L ( O ) + k f k L ( O ) (cid:17) . (11)Using this Lemma, we obtain, for ǫ small enough k u k H (Ω ) ≤ C k f k L (Ω ) + C ǫ k g + iµg k L (Ω ) + ǫ k v k L (Ω ) + (cid:16) µ k v k H (Ω ) + k g + iµg k L (Ω ) (cid:17) k g k H ( γ ) . Then there exists c >> c such that k u k H (Ω ) ≤ C (cid:16) k f k L (Ω ) + ǫe − c µ k v k H (Ω ) + C ǫ e − c µ k g + iµg k L (Ω ) + e c µ k g k H (Ω ) (cid:17) . (12)Inserting in (7), we obtain, for ǫ small enough k v k H (Ω ) ≤ Ce cµ (cid:16) k f k L (Ω ) + k g k H (Ω ) + k g + iµg k L (Ω ) (cid:17) . (13)5ombining (12) and (13), we obtain k u k H (Ω ) ≤ Ce cµ (cid:16) k f k L (Ω ) + k g k H (Ω ) + k g + iµg k L (Ω ) (cid:17) . (14)Recalling that v = g + iµv and using (13), we obtain k v k H (Ω ) ≤ Ce cµ (cid:16) k f k L (Ω ) + k g k H (Ω ) + k g + iµg k L (Ω ) (cid:17) . (15)Combining (13), (14) and (15), we obtain Theorem 1.1. (cid:3) Proof of Theorem 2.1
Estimate (7) is consequence of two important results. The first is a known re-sult shown by Lebeau and Robbiano in [9] and the second one is given by Theorem2.2 and proved in section 3.Let 0 < ǫ < ǫ and V ǫ j , j = 1 ,
2, such that V ǫ j = { x ∈ Ω , d ( x, γ ) < ǫ j } .Recalling that ( △ + µ ) v = g + iµg , then for all D >
0, there exists
C > ν ∈ ]0 ,
1[ such that we have the following estimate (see [9]) k v k H (Ω \ V ǫ ) ≤ Ce Dµ k v k − νH (Ω ) (cid:16) k g + iµg k L (Ω ) + k v k H ( V ǫ ) (cid:17) ν (16)Moreover we have the following result shown in section 3. Theorem 2.2
There exists
C > , c > , c > , ǫ > and µ > such that forany µ ≥ µ , we have the following estimate k v k H ( V ǫ ) ≤ Ce c µ h k f k L (Ω ) + k g + iµg k L (Ω ) + k g k H (Ω ) + k u k H (Ω ) i + Ce − c µ k v k H (Ω ) . (17)Combining (16) and (17) we obtain k v k H (Ω \ V ǫ ) ≤ Cǫe Dµ k v k H (Ω ) + Cǫ k g + iµg k L (Ω ) + Cǫ e − c µ k v k H (Ω ) + Cǫ e c µ h k f k L (Ω ) + k g + iµg k L (Ω ) + k g k H (Ω ) + k u k H (Ω ) i . (18)Adding (17) and (18), we obtain k v k H (Ω ) ≤ Cǫe Dµ k v k H (Ω ) + C ǫ k g + iµg k L (Ω ) + C ǫ e − c µ k v k H (Ω ) + C ǫ e c µ h k f k L (Ω ) + k g + iµg k L (Ω ) + k g k H (Ω ) + k u k H (Ω ) i . We fixe ǫ small enough and D < c , then there exists µ > µ ≥ µ , we obtain (7). (cid:3) Carleman estimate and Consequence
In this part, we show the new Carleman estimate and we prove Theorem 2.2which is consequence of this estimate.
In this subsection we state the Carleman estimate which is the starting point ofthe proof of the main result. Let u = ( u , v ) satisfies the equation − ( △ + µ ) u = f in Ω , − ( △ + µ ) v = f in Ω ,op ( B ) u = u − iµv = e on γ,op ( B ) u = ∂ n u − ∂ n v = e on γ, (19)We will proceed like Bellassoued in [3], we will reduce the problem of transmissionas a particular case of a diagonal system define only on one side of the interface withboundary conditions.We define the Sobolev spaces with a parameter µ , H sµ by u ( x, µ ) ∈ H sµ ⇐⇒ h ξ, µ i s b u ( ξ, µ ) ∈ L , h ξ, µ i = | ξ | + µ , b u denoted the partial Fourier transform with respect to x .For a differential operator P ( x, D, µ ) = X | α | + k ≤ m a α,k ( x ) µ k D α , we note the associated symbol by p ( x, ξ, µ ) = X | α | + k ≤ m a α,k ( x ) µ k ξ α . The class of symbols of order m is defined by S mµ = n p ( x, ξ, µ ) ∈ C ∞ , (cid:12)(cid:12)(cid:12) D αx D βξ p ( x, ξ, µ ) (cid:12)(cid:12)(cid:12) ≤ C α,β h ξ, µ i m −| β | o and the class of tangential symbols of order m by T S mµ = n p ( x, ξ ′ , µ ) ∈ C ∞ , (cid:12)(cid:12)(cid:12) D αx D βξ ′ p ( x, ξ ′ , µ ) (cid:12)(cid:12)(cid:12) ≤ C α,β h ξ ′ , µ i m −| β | o . We denote by O m (resp. T O m ) the set of differentials operators P = op ( p ), p ∈ S mµ (resp. T S mµ ).We shall frequently use the symbol Λ = h ξ ′ , µ i = ( | ξ ′ | + µ ) .We shall need to use the following G˚arding estimate: if p ∈ T S µ satisfies for C > p ( x, ξ ′ , µ ) + p ( x, ξ ′ , µ ) ≥ C Λ , then ∃ C > , ∃ µ > , ∀ µ > µ , ∀ u ∈ C ∞ ( K ) , Re ( P ( x, D ′ , µ ) u, u ) ≥ C k op (Λ) u k L . (20)7et x = ( x ′ , x n ) ∈ R n − × R . In the normal geodesic system given locally byΩ = { x ∈ R n , x n > } , x n = dist ( x, ∂ Ω ) = dist ( x, x ′ ) , the Laplacian is written in the form △ = − A ( x, D ) = − (cid:0) D x n + R (+ x n , x ′ , D x ′ ) (cid:1) . The Laplacian on Ω can be identified locally to an operator in Ω gives by △ = − A ( x, D ) = − (cid:0) D x n + R ( − x n , x ′ , D x ′ ) (cid:1) . We denote the operator, with C ∞ coefficients defined in Ω = { x n > } , by A ( x, D ) = diag (cid:16) A ( x, D x ) , A ( x, D x ) (cid:17) and the tangential operator by R ( x, D x ′ ) = diag (cid:16) R ( − x n , x ′ , D x ′ ) , R (+ x n , x ′ , D x ′ ) (cid:17) = diag (cid:16) R ( x, D x ′ ) , R ( x, D x ′ ) (cid:17) . The principal symbol of the differential operator A ( x, D ) satisfies a ( x, ξ ) = ξ n + r ( x, ξ ′ ), where r ( x, ξ ′ ) = diag (cid:16) r ( x, ξ ′ ) , r ( x, ξ ′ ) (cid:17) is the principalsymbol of R ( x, D x ′ ) and the quadratic form r j ( x, ξ ′ ), j = 1 ,
2, satisfies ∃ C > , ∀ ( x, ξ ′ ) , r j ( x, ξ ′ ) ≥ C | ξ ′ | , j = 1 , . We denote P ( x, D ) the matrix operator with C ∞ coefficients defined inΩ = { x n > } , by P ( x, D ) = diag( P ( x, D ) , P ( x, D )) = (cid:18) A ( x, D ) − µ A ( x, D ) − µ (cid:19) . Let ϕ ( x ) = diag( ϕ ( x ) , ϕ ( x )), with ϕ j , j = 1 ,
2, are C ∞ functions in Ω j . For µ large enough, we define the operator A ( x, D, µ ) = e µϕ A ( x, D ) e − µϕ := op ( a )where a ∈ S µ is the principal symbol given by a ( x, ξ, µ ) = (cid:16) ξ n + iµ ∂ϕ∂x n (cid:17) + r (cid:16) x, ξ ′ + iµ ∂ϕ∂x ′ (cid:17) . Let op (˜ q ,j ) = 12 ( A j + A ∗ j ) , op (˜ q ,j ) = 12 i ( A j − A ∗ j ) , j = 1 , A j = op (˜ q ,j ) + iop (˜ q ,j ) , ˜ q ,j = ξ n + q ,j ( x, ξ ′ , µ ) , ˜ q ,j = 2 µ ∂ϕ j ∂x n ξ n + 2 µq ,j ( x, ξ ′ , µ ) , j = 1 , , (21)8here q ,j ∈ T S µ and q ,j ∈ T S µ are two tangential symbols given by q ,j ( x, ξ ′ , µ ) = r j ( x, ξ ′ ) − ( µ ∂ϕ j ∂x n ) − µ r j ( x, ∂ϕ j ∂x ′ ) ,q ,j ( x, ξ ′ , µ ) = ˜ r j ( x, ξ ′ , ∂ϕ j ∂x ′ ) , j = 1 , , (22)with ˜ r ( x, ξ ′ , η ′ ) is the bilinear form associated to the quadratic form r ( x, ξ ′ ).In the next, P ( x, D, µ ) is the matrix operator with C ∞ coefficients defined inΩ = { x n > } by P ( x, D, µ ) = diag( P ( x, D, µ ) , P ( x, D, µ )) = (cid:18) A ( x, D, µ ) − µ A ( x, D, µ ) − µ (cid:19) (23)and u = ( u , v ) satisfies the equation P u = f in { x n > } ,op ( b ) u = u | x n =0 − iµv | x n =0 = e on { x n = 0 } ,op ( b ) u = (cid:16) D x n + iµ ∂ϕ ∂x n (cid:17) u | x n =0 + (cid:16) D x n + iµ ∂ϕ ∂x n (cid:17) v | x n =0 = e on { x n = 0 } , (24)where f = ( f , f ), e = ( e , e ) and B = ( op ( b ) , op ( b )). We note p j ( x, ξ, µ ), j = 1 ,
2, the associated symbol of P j ( x, D, µ ).We suppose that ϕ satisfies ϕ ( x ) = ϕ ( x ) on { x n = 0 } ∂ϕ ∂x n > { x n = 0 } (cid:18) ∂ϕ ∂x n (cid:19) − (cid:18) ∂ϕ ∂x n (cid:19) > { x n = 0 } (25)and the following condition of hypoellipticity of H¨ormander: ∃ C > , ∀ x ∈ K ∀ ξ ∈ R n \{ } , (cid:18) Re p j = 0 et 12 µ Im p j = 0 (cid:19) ⇒ (cid:26) Re p j , µ Im p j (cid:27) ≥ C h ξ, µ i , (26)where { f, g } ( x, ξ ) = P (cid:16) ∂f∂ξ j ∂g∂x j − ∂f∂x j ∂g∂ξ j (cid:17) is the Poisson bracket of two functions f ( x, ξ ) and g ( x, ξ ) and K is a compact in Ω .We denote by k u k L (Ω ) = k u k , k u k k,µ = k X j =0 µ k − j ) k u k H j (Ω ) , k u k k = (cid:13)(cid:13) op (Λ k ) u (cid:13)(cid:13) , | u | k,µ = k u | x n =0 k k,µ , | u | k = | u | x n =0 | k , k ∈ R and | u | , ,µ = | u | + | D x n u | . We are now ready to state our result. 9 heorem 3.1
Let ϕ satisfies (25) and (26). Let w ∈ C ∞ (Ω ) and χ ∈ C ∞ ( R n +1 ) such that χ = 1 in the support of w . Then there exists constants C > and µ > such that for any µ ≥ µ we have the following estimate µ k w k ,µ + µ | w | + µ | D x n w | − ≤ C (cid:16) k P ( x, D, µ ) w k + | op ( b ) w | + µ | op ( b ) w | (cid:17) . (27) Corollary 3.1
Let ϕ satisfies (25) and (26). Then there exists constants C > and µ > such that for any µ ≥ µ we have the following estimate µ k e µϕ h k H ≤ C (cid:16) k e µϕ P ( x, D ) h k + | e µϕ op ( B ) h | H + µ | e µϕ op ( B ) h | (cid:17) , (28) for any h ∈ C ∞ (Ω ) . Proof.
Let w = e µϕ h . Recalling that P ( x, D, µ ) w = e µϕ P ( x, D ) e − µϕ w and using (27), weobtain (28). We denote x = ( x ′ , x n ) a point in Ω. Let x = (0 , − δ ), δ >
0. We set ψ ( x ) = | x − x | − δ and ϕ ( x ) = e − βψ ( x ′ , − x n ) , ϕ ( x ) = e − β ( ψ ( x ) − αx n ) , β > , and δ < α < δ. The weight function ϕ = diag ( ϕ , ϕ ) has to satisfy (25) and (26). With thesechoices, we have ϕ | x n =0 = ϕ | x n =0 and ∂ϕ ∂x n | x n =0 >
0. It remains to verify (cid:18) ∂ϕ ∂x n (cid:19) − (cid:18) ∂ϕ ∂x n (cid:19) > { x n = 0 } (29)and the condition (26). We begin by condition (26) and we compute for ϕ and p (the computation for ϕ and p is made in the same way). Recalling that (cid:26) Re p , µ Im p (cid:27) ( x, ξ ) = Im2 µ [ ∂ ξ p ( x, ξ − iµϕ ′ ( x )) ∂ x p ( x, ξ + iµϕ ′ ( x ))]+ t [ ∂ ξ p ( x, ξ − iµϕ ′ ( x ))] ϕ ′′ ( x ) [ ∂ ξ p ( x, ξ − iµϕ ′ ( x ))] . We replace ϕ ( x ) by ϕ ( x ) = e − βψ ( x ′ , − x n ) , β >
0, we obtain, by noting ξ = − βϕ ( x ) η (cid:26) Re p , µ Im p (cid:27) ( x, ξ )= ( − βϕ ) (cid:20)(cid:26) Re p ( x, η − iµψ ′ ) , µ Im p ( x, η + iµψ ′ ) (cid:27) ( x, η ) − β | ψ ′ ( x ) ∂ η p ( x, η + iµψ ′ ) | (cid:21) | ψ ′ ( x ) ∂ η p ( x, η + iµψ ′ ) | = 4 h µ | p ( x, ψ ′ ) | + | ˜ p ( x, η, ψ ′ ) | i where ˜ p ( x, η, ψ ′ ) is the bilinear form associated to the quadratic form p ( x, η ). Wehave (cid:18) Re p = 0 et 12 µ Im p = 0 (cid:19) ⇐⇒ p ( x, η + iµψ ′ ) = 0 , • If µ = 0, we have p ( x, ξ ) = 0 which is impossible. Indeed, we have p ( x, ξ ) ≥ C | ξ | , ∀ ( x, ξ ) ∈ K × R n , K compact in Ω . • If µ = 0, we have ˜ p ( x, η, ψ ′ ) = 0.Then | ψ ′ ( x ) ∂ η p ( x, η + iµψ ′ ) | = 4 µ | p ( x, ψ ′ ) | >
0. On the other hand, wehave (cid:26) Re p ( x, η − iµψ ′ ) , µ Im p ( x, η + iµψ ′ ) (cid:27) ( x, η ) ≤ C ( | η | + µ | ψ ′ | )where C is a positive constant independent of ψ ′ . Then for β ≥ C , we satisfythe condition (26).Now let us verify (29). We have, on { x n = 0 } , (cid:18) ∂ϕ ∂x n (cid:19) − (cid:18) ∂ϕ ∂x n (cid:19) = β α (4 δ − α ) e − βψ . Then to satisfy (29), it suffices to choose β = Mδ where M > Mδ ≥ C .We now choose r < r ′ < r < ψ (0) < r ′ < r < r ′ . We denote w j = { x ∈ Ω , r j < ψ ( x ) < r ′ j } and T x = w ∩ Ω . We set R j = e − βr j , R ′ j = e − βr ′ j , j = 1 , , R ′ < R < R ′ < R < R ′ < R . We need also to introduce a cut-off function˜ χ ∈ C ∞ ( R n +1 ) such that ˜ χ ( ρ ) = ρ ≤ r , ρ ≥ r ′ ρ ∈ [ r ′ , r ] . Let ˜ u = (˜ u , ˜ v ) = ˜ χu = ( ˜ χu , ˜ χv ). Then we get the following system ( △ − iµ )˜ u = ˜ χf + [ △ − iµ, ˜ χ ] u ( △ + µ )˜ v = ˜ χ ( g + iµg ) + [ △ + µ , ˜ χ ] v , ˜ v = g + iµ ˜ v , with the following boundary conditions ˜ u | Γ = ˜ v | Γ = 0 ,op ( b )˜ u = ˜ u − iµ ˜ v = ( ˜ χg ) | γ ,op ( b )˜ u = ([ ∂ n , ˜ χ ] u − [ ∂ n , ˜ χ ] v ) | γ . µ k e µϕ ˜ u k H ≤ C (cid:16) k e µϕ ( △ − iµ )˜ u k + (cid:13)(cid:13) e µϕ ( △ + µ )˜ v (cid:13)(cid:13) + | e µϕ op ( b )˜ u | H + µ | e µϕ op ( b )˜ u | (cid:17) . (30)Using the fact [ △ − iµ, ˜ χ ] is the first order operator supported in ( w ∪ w ) ∩ Ω , wehave k e µϕ ( △ − iµ )˜ u k ≤ C (cid:16) e µR k f k L (Ω ) + e µR k u k H (Ω ) (cid:17) . (31)Recalling that [ △ + µ , ˜ χ ] is the first order operator supported in ( w ∪ w ) ∩ Ω , weshow (cid:13)(cid:13) e µϕ ( △ + µ )˜ v (cid:13)(cid:13) ≤ C (cid:16) e µ k g + iµg k L (Ω ) + e µR k v k H (Ω ) (cid:17) . (32)From the trace formula and recalling that op ( b )˜ u is an operator of order zero andsupported in { x n = 0 } ∩ w , we show µ | e µϕ op ( b )˜ u | ≤ Ce µR k u k H (Ω) ≤ C (cid:16) e µR k u k H (Ω ) + e µR k v k H (Ω ) (cid:17) . (33)Now we need to use this result shown in Appendix B Lemma 3.1
There exists
C > such that for all s ∈ R and u ∈ C ∞ (Ω) , we have k op (Λ s ) e µϕ u k ≤ Ce µC k op (Λ s ) u k . (34)Following this Lemma, we obtain | e µϕ op ( b )˜ u | H ≤ Ce µc | g | H ≤ Ce µc k g k H (Ω ) . (35)Combining (30), (31), (32), (33) and (35), we obtain Cµe µR ′ k u k H ( w ∩ Ω ) + Cµe µR ′ k v k H ( T x ) ≤ C ( e µR k f k L (Ω ) + e µR k u k H (Ω ) + e µ k g + iµg k L (Ω ) + e µR k v k H (Ω ) + e µR k u k H (Ω ) + e µc k g k H (Ω ) ) . Since R ′ < R . Then we have k v k H ( T x ) ≤ Ce c µ h k f k L (Ω ) + k g + iµg k L (Ω ) + k g k H (Ω ) + k u k H (Ω ) i + Ce − c µ k v k H (Ω ) . (36)Since γ is compact, then there exists a finite number of T x . Let V ǫ ⊂ ∪ T x . Thenwe obtain (17) 12 .3 Proof of Carleman estimate (Theorem 3.1) In the first step, we state the following estimates
Theorem 3.2
Let ϕ satisfies (25) and (26). Then there exists constants C > and µ such that for any µ ≥ µ we have the following estimates µ k u k ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ | u | , ,µ (cid:17) (37) and µ k u k ,µ + µ | u | , ,µ ≤ C (cid:0) k P ( x, D, µ ) u k + µ − | op ( b ) u | + µ | op ( b ) u | (cid:1) , (38) for any u ∈ C ∞ (Ω ) . In the second step, we need to prove this Lemma
Lemma 3.2
There exists constants
C > and µ > such that for any µ ≥ µ wehave the following estimate (cid:13)(cid:13)(cid:13) D x n op (Λ − ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) D x n op (Λ ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) + µ | u | , ,µ ≤ C (cid:0) k P ( x, D, µ ) u k + µ − | op ( b ) u | + µ | op ( b ) u | (cid:1) , (39) for any u ∈ C ∞ (Ω ) . Proof.
We have P ( x, D, µ ) = D x n + R + µC + µ C , where R ∈ T O , C = c ( x ) D x n + T , with T ∈ T O and C ∈ T O . Then we have (cid:13)(cid:13)(cid:13) ( D x n + R ) op (Λ − ) u (cid:13)(cid:13)(cid:13) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) P op (Λ − ) u (cid:13)(cid:13)(cid:13) + µ (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) + µ (cid:13)(cid:13)(cid:13) D x n op (Λ − ) u (cid:13)(cid:13)(cid:13) + µ (cid:13)(cid:13)(cid:13) op (Λ − ) u (cid:13)(cid:13)(cid:13) (cid:19) . Since µ (cid:13)(cid:13)(cid:13) op (Λ − ) u (cid:13)(cid:13)(cid:13) ≤ Cµ k u k ,µ (cid:13)(cid:13)(cid:13) D x n op (Λ − ) u (cid:13)(cid:13)(cid:13) ≤ Cµ k D x n u k and µ (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) = µ ( √ µ op (Λ) u, √ µu ) ≤ C (cid:0) µ k op (Λ) u k + µ k u k (cid:1) . Using the fact that k u k ,µ ≃ k op (Λ) u k + k D x n u k , we obtain (cid:13)(cid:13)(cid:13) ( D x n + R ) op (Λ − ) u (cid:13)(cid:13)(cid:13) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) P op (Λ − ) u (cid:13)(cid:13)(cid:13) + µ k u k ,µ (cid:19) . (cid:13)(cid:13)(cid:13) ( D x n + R ) op (Λ − ) u (cid:13)(cid:13)(cid:13) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) P op (Λ − ) u (cid:13)(cid:13)(cid:13) + k P u k + µ | u | , ,µ (cid:19) . (40)We can write P op (Λ − ) u = op (Λ − ) P u + [
P, op (Λ − )] u = op (Λ − ) P u + [
R, op (Λ − )] u + µ [ C , op (Λ − )] u + µ [ C , op (Λ − )] u = op (Λ − ) P u + t + t + t . (41)Let us estimate t , t and t . We have [ R, op (Λ − )] ∈ T O , then following (37), wehave k t k ≤ C (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) ≤ C (cid:0) k op (Λ) u k + k u k (cid:1) ≤ C (cid:16) k P u k + µ | u | , ,µ (cid:17) . (42)We have t = µ [ C , op (Λ − )] u = µ [ c ( x ) D x n , op (Λ − )] u + µ [ T , op (Λ − )] u . Thenfollowing (37), we obtain k t k ≤ C (cid:0) µ − k D x n u k + µ k u k (cid:1) ≤ C (cid:16) k P u k + µ | u | , ,µ (cid:17) . (43)We have [ C , op (Λ − )] ∈ T O − , then following (37), we obtain (cid:13)(cid:13)(cid:13) µ [ C , op (Λ − )] u (cid:13)(cid:13)(cid:13) ≤ Cµ k u k ≤ C (cid:16) k P u k + µ | u | , ,µ (cid:17) (44)From (41), (42), (43) and (44), we have (cid:13)(cid:13)(cid:13) P op (Λ − ) u (cid:13)(cid:13)(cid:13) ≤ C (cid:16) k P u k + µ | u | , ,µ (cid:17) . Inserting this inequality in (40), we obtain (cid:13)(cid:13)(cid:13) ( D x n + R ) op (Λ − ) u (cid:13)(cid:13)(cid:13) ≤ C (cid:16) k P u k + µ | u | , ,µ (cid:17) . (45)Moreover, we have (cid:13)(cid:13)(cid:13) ( D x n + R ) op (Λ − ) u (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) D x n op (Λ − ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) Rop (Λ − ) u (cid:13)(cid:13)(cid:13) +2 R e ( D x n op (Λ − ) u, Rop (Λ − ) u ) , where ( ., . ) denoted the scalar product in L . By integration by parts, we find (cid:13)(cid:13)(cid:13) ( D x n + R ) op (Λ − ) u (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) D x n op (Λ − ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) Rop (Λ − ) u (cid:13)(cid:13)(cid:13) +2 R e (cid:16) i ( D x n u, Rop (Λ − ) u ) + i ( D x n u, [ op (Λ − ) , R ] op (Λ − ) u ) (cid:17) +2 R e (cid:16) ( RD x n op (Λ − ) u, D x n op (Λ − ) u ) + ( D x n op (Λ − ) u, [ D x n , R ] op (Λ − ) u ) (cid:17) . (46)14ince, we have (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) = ( op (Λ ) op (Λ ) u, op (Λ ) u ) = X j ≤ n − ( D j op (Λ ) u, op (Λ ) u )+ µ ( op (Λ ) u, op (Λ ) u ) . By integration by parts, we find (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) = X j ≤ n − ( D j op (Λ ) u, D j op (Λ ) u )+ µ (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) = k + µ (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) . (47)Let χ ∈ C ∞ ( R n +1 ) such that χ = 1 in the support of u . We have k = X j ≤ n − ( χ D j op (Λ ) u, D j op (Λ ) u ) + X j ≤ n − ((1 − χ ) D j op (Λ ) u, D j op (Λ ) u ) . Recalling that χ u = u , we obtain k = X j ≤ n − ( χ D j op (Λ ) u, D j op (Λ ) u )+ X j ≤ n − ([(1 − χ ) , D j op (Λ )] u, D j op (Λ ) u ) = k ′ + k ” . (48)Using the fact that [(1 − χ ) , D j op (Λ )] ∈ T O and D j op (Λ ) ∈ T O , we show k ” ≤ C k op (Λ) u k . (49)Using the fact that P j,k ≤ n − χ a j,k D j vD k v ≥ δχ P j ≤ n − | D j v | , δ >
0, we obtain k ′ ≤ C X j,k ≤ n − ( χ a jk D j op (Λ ) u, D k op (Λ ) u ) ≤ C X j,k ≤ n − ([ χ , a jk D j op (Λ )] u, D k op (Λ ) u ) + X j,k ≤ n − ( a jk D j op (Λ ) u, D k op (Λ ) u ) . Using the fact that [ χ , a jk D j op (Λ )] ∈ T O and D k op (Λ ) u ∈ T O , we obtain k ′ ≤ C X j,k ≤ n − ( a jk D j op (Λ ) u, D k op (Λ ) u ) + k op (Λ) u k ! . (50)By integratin by parts and recalling that R = P j,k ≤ n − a j,k D j D k , we have X j,k ≤ n − ( a jk D j op (Λ ) u, D k op (Λ ) u ) = ( Rop (Λ ) u, op (Λ ) u (51)+ X j,k ≤ n − ([ D k , a jk ] D j op (Λ ) u, op (Λ ) u ) . Since [ D k , a jk ] D j op (Λ ) ∈ T O , then X j,k ≤ n − ([ D k , a jk ] D j op (Λ ) u, op (Λ ) u ) ≤ C k op (Λ) u k . X j,k ≤ n − ( a jk D j op (Λ ) u, D k op (Λ ) u ) ≤ C (cid:16) ( Rop (Λ ) u, op (Λ ) u ) + k op (Λ) u k (cid:17) . (52)Since( Rop (Λ ) u, op (Λ ) u ) = ( Rop (Λ − ) u, op (Λ ) u ) + ([ op (Λ − ) , R ] op (Λ ) u, op (Λ ) u )) . Using the fact that [ op (Λ − ) , R ] op (Λ ) ∈ T O and the Cauchy Schwartz inequality,we obtain( Rop (Λ ) u, op (Λ ) u ) ≤ ǫC (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) + Cǫ (cid:13)(cid:13)(cid:13) Rop (Λ − ) u (cid:13)(cid:13)(cid:13) + C k op (Λ) u k (53)Combining (47), (48), (49), (50), (52) and (53), we obtain (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) ≤ ǫC (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) + Cǫ (cid:13)(cid:13)(cid:13) Rop (Λ − ) u (cid:13)(cid:13)(cid:13) + C k op (Λ) u k . For ǫ small enough, we obtain (cid:13)(cid:13)(cid:13) Rop (Λ − ) u (cid:13)(cid:13)(cid:13) ≥ C (cid:18)(cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) − µ (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) (cid:19) . (54)Using the same computations, we show( RD x n op (Λ − ) u, D x n op (Λ − ) u ) ≥ C (cid:18)(cid:13)(cid:13)(cid:13) D x n op (Λ ) u (cid:13)(cid:13)(cid:13) − µ k D x n u k (cid:19) . (55)Combining (46), (54) and (55), we obtain (cid:13)(cid:13)(cid:13) ( D x n + R ) op (Λ − ) u (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12) ( D x n u, Rop (Λ − ) u ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( D x n u, [ op (Λ − ) , R ] op (Λ − ) u ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( D x n op (Λ − ) u, [ D x n , R ] op (Λ − ) u ) (cid:12)(cid:12)(cid:12) + µ k u k ,µ (56) ≥ C (cid:18)(cid:13)(cid:13)(cid:13) D x n op (Λ − ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) D x n op (Λ ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) (cid:19) . Since (cid:12)(cid:12) ( D x n u, Rop (Λ − ) u ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( D x n u, [ op (Λ − ) , R ] op (Λ − ) u ) (cid:12)(cid:12)(cid:12) ≤ C (cid:0) | D x n u | + | u | (cid:1) = C | u | , ,µ (57)and (cid:12)(cid:12)(cid:12) ( D x n op (Λ − ) u, [ D x n , R ] op (Λ − ) u ) (cid:12)(cid:12)(cid:12) ≤ Cµ k u k ,µ . (58)From (45), (56), (57), (58) and (37), we obtain (cid:13)(cid:13)(cid:13) D x n op (Λ − ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) D x n op (Λ ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) ≤ C (cid:16) k P ( x, D, µ ) u k + µ | u | , ,µ (cid:17) . (cid:3) We are now ready to prove Theorem 3.1.Let χ ∈ C ∞ ( R n +1 ) such that χ = 1 in the support of w and u = χop (Λ − ) w .Then P u = op (Λ − ) P w + [
P, op (Λ − )] w + P [ χ, op (Λ − )] w = op (Λ − ) P w + [
P, op (Λ − )] w + D x n [ χ, op (Λ − )] w + R [ χ, op (Λ − )] w + µc ( x ) D x n [ χ, op (Λ − )] w + µT [ χ, op (Λ − )] w + µ C [ χ, op (Λ − )] w = op (Λ − ) P w + [
P, op (Λ − )] w + a + a + a + a + a . (59)Let us estimate a , a , a , a and a . Recalling that [ χ, op (Λ − )] ∈ T O − and χw = w . Using the fact that [ D x n , T k ] ∈ T O k for all T k ∈ T O k , we show k a k ≤ C (cid:18)(cid:13)(cid:13)(cid:13) D x n op (Λ − ) w (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) D x n op (Λ − ) w (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) op (Λ − ) w (cid:13)(cid:13)(cid:13) (cid:19) (60)and k a k ≤ C (cid:18) µ (cid:13)(cid:13)(cid:13) D x n op (Λ − ) w (cid:13)(cid:13)(cid:13) + µ (cid:13)(cid:13)(cid:13) op (Λ − ) w (cid:13)(cid:13)(cid:13) (cid:19) . (61)We have R [ χ, op (Λ − )] ∈ T O , T [ χ, op (Λ − )] ∈ T O − and C [ χ, op (Λ − )] ∈ T O − . Then we obtain k a k + k a k + k a k ≤ C (cid:13)(cid:13)(cid:13) op (Λ ) w (cid:13)(cid:13)(cid:13) . (62)Using the same computations made in the proof of Lemma 3.2 (cf t , t and t of(41)), we show (cid:13)(cid:13)(cid:13) [ P, op (Λ − )] w (cid:13)(cid:13)(cid:13) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) op (Λ ) w (cid:13)(cid:13)(cid:13) + µ − k D x n w k (cid:19) . (63)Following (59), (60), (61), (62) and (63), we obtain k P u k ≤ C (cid:18) µ − k P w k + (cid:13)(cid:13)(cid:13) op (Λ ) w (cid:13)(cid:13)(cid:13) + µ − k D x n w k + µ − (cid:13)(cid:13) D x n op (Λ − ) w (cid:13)(cid:13) (cid:19) . (64)We have op ( b ) u = op ( b ) χop (Λ − ) w = op (Λ − ) op ( b ) w + op ( b )[ χ, op (Λ − )] w. Recalling that op ( b ) ∈ T O , we obtain µ − | op ( b ) u | = µ − | op (Λ) op ( b ) u | ≤ C (cid:18) µ − (cid:12)(cid:12)(cid:12) op (Λ ) op ( b ) w (cid:12)(cid:12)(cid:12) + µ − (cid:12)(cid:12)(cid:12) op (Λ ) w (cid:12)(cid:12)(cid:12) (cid:19) . (65)17e have op ( b ) u = op ( b ) χop (Λ − ) w = op (Λ − ) op ( b ) w + op ( b )[ χ, op (Λ − )] w +[ op ( b ) , op (Λ − )] w. Recalling that op ( b ) ∈ D x n + T O , we obtain µ | op ( b ) u | ≤ C (cid:18) µ (cid:12)(cid:12)(cid:12) op (Λ − ) op ( b ) w (cid:12)(cid:12)(cid:12) + µ (cid:12)(cid:12)(cid:12) op (Λ − ) w (cid:12)(cid:12)(cid:12) + µ (cid:12)(cid:12)(cid:12) D x n op (Λ − ) w (cid:12)(cid:12)(cid:12) (cid:19) . (66)Moreover, we have µ | u | , ,µ = µ | u | + µ | D x n u | = µ | op (Λ) u | + µ | D x n u | . We can write op (Λ) u = op (Λ) χop (Λ − ) w = op (Λ ) w + op (Λ)[ χ, op (Λ − )] w. Then µ | op (Λ) u | ≥ µ (cid:12)(cid:12)(cid:12) op (Λ ) w (cid:12)(cid:12)(cid:12) − Cµ (cid:12)(cid:12)(cid:12) op (Λ − ) w (cid:12)(cid:12)(cid:12) ≥ µ (cid:12)(cid:12)(cid:12) op (Λ ) w (cid:12)(cid:12)(cid:12) − Cµ − (cid:12)(cid:12)(cid:12) op (Λ ) w (cid:12)(cid:12)(cid:12) . For µ large enough, we obtain µ | op (Λ) u | ≥ Cµ (cid:12)(cid:12)(cid:12) op (Λ ) w (cid:12)(cid:12)(cid:12) . (67)By the same way, we prove, for µ large enough µ | D x n u | ≥ Cµ (cid:12)(cid:12)(cid:12) D x n op (Λ − ) w (cid:12)(cid:12)(cid:12) . (68)Combining (67) and (68), we obtain µ | u | , ,µ ≥ C (cid:18) µ (cid:12)(cid:12)(cid:12) op (Λ ) w (cid:12)(cid:12)(cid:12) + µ (cid:12)(cid:12)(cid:12) D x n op (Λ − ) w (cid:12)(cid:12)(cid:12) (cid:19) . (69)By the same way, we prove (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) ≥ k op (Λ) w k − C k w k , (70) (cid:13)(cid:13)(cid:13) D x n op (Λ ) u (cid:13)(cid:13)(cid:13) ≥ k D x n w k − C (cid:13)(cid:13) op (Λ − ) D x n w (cid:13)(cid:13) − C (cid:13)(cid:13) op (Λ − ) w (cid:13)(cid:13) (71)and (cid:13)(cid:13)(cid:13) D x n op (Λ − ) u (cid:13)(cid:13)(cid:13) ≥ (72) (cid:13)(cid:13) D x n op (Λ − ) w (cid:13)(cid:13) − C (cid:13)(cid:13) D x n op (Λ − ) w (cid:13)(cid:13) − C (cid:13)(cid:13) D x n op (Λ − ) w (cid:13)(cid:13) − C (cid:13)(cid:13) op (Λ − ) w (cid:13)(cid:13) . Combining (70), (71) and (72), we obtain for µ large enough (cid:13)(cid:13)(cid:13) D x n op (Λ − ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) D x n op (Λ ) u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) op (Λ ) u (cid:13)(cid:13)(cid:13) ≥ C (cid:16)(cid:13)(cid:13) D x n op (Λ − ) w (cid:13)(cid:13) + k D x n w k + k op (Λ) w k (cid:17) . (73)Combining (39), (64), (65), (66), (69) and (73), we obtain (27), for µ large enough. (cid:3) Proof of Theorem 3.2
This section is devoted to the proof of Theorem 3.2.
The proof is based on a cutting argument related to the nature of the roots ofthe polynomial p j ( x, ξ ′ , ξ n , µ ), j = 1 ,
2, in ξ n . On x n = 0, we note q ( x ′ , ξ ′ , µ ) = q , (0 , x ′ , ξ ′ , µ ) = q , (0 , x ′ , ξ ′ , µ ) . Let us introduce the following micro-local regions E +1 / = ( ( x, ξ ′ , µ ) ∈ K × R n , q , / + q ( ∂ϕ / ∂x n ) > ) , Z / = ( ( x, ξ ′ , µ ) ∈ K × R n , q , / + q ( ∂ϕ / ∂x n ) = 0 ) , E − / = ( ( x, ξ ′ , µ ) ∈ K × R n , q , / + q ( ∂ϕ / ∂x n ) < ) . (Here and in the following the index 1 / using for telling 1 or 2).We decompose p / ( x, ξ, µ ) as a polynomial in ξ n . Then we have the following lemmadescribing the various types of the roots of p / . Lemma 4.1
We have the following1. For ( x, ξ ′ , µ ) ∈ E +1 / , the roots of p / denoted z ± / satisfy ± Im z ± / > .2. For ( x, ξ ′ , µ ) ∈ Z / , one of the roots of p / is real.3. For ( x, ξ ′ , µ ) ∈ E − / , the roots of p / are in the half- plane Im ξ n > if ∂ϕ / ∂x n < (resp. in the half-plane Im ξ n < if ∂ϕ / ∂x n > ). Proof.
Using (21) and (22), we can write p ( x ′ , ξ, µ ) = (cid:18) ξ n + iµ ∂ϕ ∂x n − iα (cid:19) (cid:18) ξ n + iµ ∂ϕ ∂x n + iα (cid:19) ,p ( x ′ , ξ, µ ) = (cid:18) ξ n + iµ ∂ϕ ∂x n − iα (cid:19) (cid:18) ξ n + iµ ∂ϕ ∂x n + iα (cid:19) , (74)where α j ∈ C , j = 1 ,
2, defined by α ( x ′ , ξ ′ , µ ) = (cid:18) µ ∂ϕ ∂x n (cid:19) + q , + 2 iµq ,α ( x ′ , ξ ′ , µ ) = (cid:18) µ ∂ϕ ∂x n (cid:19) − µ + q , + 2 iµq . (75)19e set z ± / = − iµ ∂ϕ / ∂x n ± iα / , (76)the roots of p / . The imaginary parts of the roots of p / are − µ ∂ϕ / ∂x n − Re α / , − µ ∂ϕ / ∂x n + Re α / . The signs of the imaginary parts are opposite if (cid:12)(cid:12) ∂ϕ / /∂x n (cid:12)(cid:12) < (cid:12)(cid:12) Re α / (cid:12)(cid:12) , equal tothe sign of − ∂ϕ / /∂x n if (cid:12)(cid:12) ∂ϕ / /∂x n (cid:12)(cid:12) > (cid:12)(cid:12) Re α / (cid:12)(cid:12) and one of the imaginary partsis null if (cid:12)(cid:12) ∂ϕ / /∂x n (cid:12)(cid:12) = (cid:12)(cid:12) Re α / (cid:12)(cid:12) . However the lines Re z = ± µ ∂ϕ / /∂x n changeby the application z z ′ = z into the parabolic curve Re z ′ = (cid:12)(cid:12) µ ∂ϕ / /∂x n (cid:12)(cid:12) −| Im z ′ | / µ ∂ϕ / /∂x n ) . Thus we obtain the lemma by replacing z ′ by α / . (cid:3) Lemma 4.2
If we assume that the function ϕ satisfies the following condition (cid:18) ∂ϕ ∂x n (cid:19) − (cid:18) ∂ϕ ∂x n (cid:19) > , (77) then the following estimate holds q , − µ + q ( ∂ϕ /∂x n ) > q , + q ( ∂ϕ /∂x n ) . (78) Proof.
Following (22), on { x n = 0 } , we have q , ( x, ξ ′ , µ ) − q , ( x, ξ ′ , µ ) = (cid:18) µ ∂ϕ ∂x n (cid:19) − (cid:18) µ ∂ϕ ∂x n (cid:19) . (79)Using (77), we have (78). (cid:3) Remark 4.1
The result of this lemma imply that E +1 ⊂ E +2 . E +1 In this part we study the problem in the elliptic region E + . In this region wecan inverse the operator and use the Calderon projectors. Let χ + ( x, ξ ′ , µ ) ∈ T S µ such that in the support of χ + we have q , + q ( ∂ϕ /∂x n ) ≥ δ >
0. Then we have thefollowing partial estimate.
Proposition 4.1
There exists a constant
C > and µ > such that for any µ ≥ µ , we have µ (cid:13)(cid:13) op ( χ + ) u (cid:13)(cid:13) ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + k u k ,µ + µ | u | , ,µ (cid:17) , (80) for any u ∈ C ∞ (Ω ) . f we suppose moreover that ϕ satisfies (77) then the following estimate holds µ (cid:12)(cid:12) op ( χ + ) u (cid:12)(cid:12) , ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ − | op ( b ) u | + µ | op ( b ) u | + k u k ,µ + µ − | u | , ,µ (cid:17) , (81) for any u ∈ C ∞ (Ω ) and b j , j = 1 , , defined in (24). Proof
Let ˜ u = op ( χ + ) u . Then we get P ˜ u = ˜ f in { x n > } ,op ( b )˜ u = ˜ u | x n =0 − iµ ˜ v | x n =0 = ˜ e on { x n = 0 } ,op ( b )˜ u = (cid:16) D x n + iµ ∂ϕ ∂x n (cid:17) ˜ u | x n =0 + (cid:16) D x n + iµ ∂ϕ ∂x n (cid:17) ˜ v | x n =0 = ˜ e on { x n = 0 } , (82)with ˜ f = op ( χ + ) f + [ P, op ( χ + )] u . Since [ P, op ( χ + )] ∈ ( T O ) D x n + T O , we have k ˜ f k L ≤ C (cid:16) k P ( x, D, µ ) u k L + k u k ,µ (cid:17) (83)and ˜ e = op ( χ + ) e satisfying | ˜ e | ≤ C | e | (84)and˜ e = h ( D x n + iµ ∂ϕ ∂x n ) , op ( χ + ) i u | x n =0 + h ( D x n + iµ ∂ϕ ∂x n ) , op ( χ + ) i v | x n =0 + op ( χ + ) e . Since [ D x n , op ( χ + )] ∈ T O , we have | ˜ e | ≤ C (cid:0) | u | + | e | (cid:1) . (85)Let ˜ u the extension of ˜ u by 0 in x n <
0. According to (21), (22) and (23), we obtain,by noting ∂ϕ/∂x n = diag ( ∂ϕ /∂x n , ∂ϕ /∂x n ), γ j (˜ u ) = t (cid:0) D jx n (˜ u ) | x n =0 + , D jx n (˜ v ) | x n =0 + (cid:1) , j = 0 , δ ( j ) = ( d/dx n ) j ( δ x n =0 ), P ˜ u = ˜ f − γ (˜ u ) ⊗ δ ′ + 1 i (cid:18) γ (˜ u ) + 2 iµ ∂ϕ∂x n (cid:19) ⊗ δ (86)Let χ ( x, ξ, µ ) ∈ S µ equal to 1 for sufficiently large | ξ | + µ and in a neighborhood ofsupp( χ + ) and satisfies that in the support of χ we have p is elliptic. These conditionsare compatible from the choice made for supp( χ + ) and Remark 4.1. Let m largeenough chosen later, by the ellipticity of p on supp( χ ) there exists E = op ( e ) aparametrix of P . We recall that e ∈ S − µ , of the form e ( x, ξ, µ ) = P mj =0 e j ( x, ξ, µ ),where e = χp − and e j = diag( e j, , e j, ) ∈ S − − jµ such that e j, and e j, are rationalfractions in ξ n . Then we have EP = op ( χ ) + R m , R m ∈ O − m − . (87)21ollowing (86) and (87), we obtain ˜ u = E ˜ f + E (cid:20) − h ⊗ δ ′ + 1 i h ⊗ δ (cid:21) + w ,h = γ (˜ u ) + 2 iµ ∂ϕ∂x n γ (˜ u ) , h = γ (˜ u ) ,w = (Id − op ( χ )) ˜ u − R m ˜ u. (88)Using the fact that supp(1 − χ ) ∩ supp( χ + ) = ∅ and symbolic calculus (See Lemma2.10 in [7]), we have (Id − op ( χ )) op ( χ + ) ∈ O − m , then we obtain k w k ,µ ≤ Cµ − k u k L . (89)Now, let us look at this term E (cid:20) − h ⊗ δ ′ + 1 i h ⊗ δ (cid:21) . For x n >
0, we get E (cid:20) − h ⊗ δ ′ + 1 i h ⊗ δ (cid:21) = ˆ T h + ˆ T h , ˆ T j ( h ) = (cid:18) π (cid:19) n − Z e i ( x ′ − y ′ ) ξ ′ ˆ t j ( x, ξ ′ , µ ) h ( y ′ ) dy ′ dξ ′ = op (ˆ t j ) h ˆ t j = 12 πi Z γ e ix n ξ n e ( x, ξ, µ ) ξ jn dξ n where γ is the union of the segment { ξ n ∈ R , | ξ n | ≤ c p | ξ ′ | + µ } and the halfcircle { ξ n ∈ C , | ξ n | = c p | ξ ′ | + µ , Imξ n > } , where the constant c is chosensufficiently large so as to have the roots z +1 and z +2 inside the domain with boundary γ (If c is large enough, the change of contour R −→ γ is possible because the symbol e ( x, ξ, µ ) is holomorphic for large | ξ n | ; ξ n ∈ C ). In particular we have in x n ≥ (cid:12)(cid:12)(cid:12) ∂ kx n ∂ αx ′ ∂ βξ ′ ˆ t j (cid:12)(cid:12)(cid:12) ≤ C α,β,k h ξ ′ , µ i j − −| β | + k , j = 0 , . (90)We now choose χ ( x, ξ ′ , µ ) ∈ T S µ , satisfying the same requirement as χ + , equal to1 in a neighborhood of supp( χ + ) and such that the symbol χ be equal to 1 in aneighborhood of supp( χ ). We set t j = χ ˆ t j , j = 0 ,
1. Then we obtain˜ u = E ˜ f + op ( t ) h + op ( t ) h + w + w (91)where w = op ((1 − χ )ˆ t ) h + op ((1 − χ )ˆ t ) h . By using the composition formulaof tangential operator, estimate (90), the fact that supp(1 − χ ) ∩ supp( χ + ) = ∅ and the following trace formula | γ ( u ) | j ≤ Cµ − k u k j +1 ,µ , j ∈ N , (92)22e obtain k w k ,µ ≤ Cµ − (cid:0) k u k ,µ + | u | , ,µ (cid:1) . (93)Since χ = 1 in the support of χ , we have e ( x, ξ, µ ) is meromorphic w.r.t ξ n in thesupport of χ . z +1 / are in Im ξ n ≥ c p | ξ ′ | + µ ( c > c is small enough wecan choose γ / in Im ξ n ≥ c p | ξ ′ | + µ and we can write t j = diag( t j, , t j, ) , t j, / ( x, ξ ′ , µ ) = χ ( x, ξ ′ , µ ) 12 πi Z γ / e ix n ξ n e / ( x, ξ, µ ) ξ jn dξ n , j = 0 , . (94)Then there exists c > x n ≥
0, we obtain (cid:12)(cid:12)(cid:12) ∂ kx n ∂ αx ′ ∂ βξ ′ t j (cid:12)(cid:12)(cid:12) ≤ C α,β,k e − c x n h ξ ′ ,µ i h ξ ′ , µ i j − −| β | + k . (95)In particular, we have e c x n µ ( ∂ kx n ) t j is bounded in T S j − kµ uniformly w.r.t x n ≥ k ∂ x ′ op ( t j ) h j k L + k op ( t j ) h j k L ≤ C Z x n > e − c x n µ | op ( e c x n µ t j ) h j | ( x n ) dx n ≤ Cµ − | h j | j and k ∂ x n op ( t j ) h j k L ≤ C Z x n > e − c x n µ | op ( e c x n µ ∂ x n t j ) h j | L ( x n ) dx n ≤ Cµ − | h j | j . Using the fact that h = γ (˜ u ) + 2 iµ ∂ϕ∂x n γ (˜ u ) and h = γ (˜ u ), we obtain k op ( t j ) h j k ,µ ≤ Cµ − | u | , ,µ . (96)From (91) and estimates (83), (89), (93) and (96), we obtain (80).It remains to proof (81). We recall that, in supp( χ ), we have e = diag ( e , , e , ) = diag (cid:18) p , p (cid:19) = diag (cid:18) ξ n − z +1 )( ξ n − z − ) , ξ n − z +2 )( ξ n − z − ) (cid:19) . Using the residue formula, we obtain e − ix n z +1 / t j, / = χ ( z +1 / ) j z +1 / − z − / + λ / , j = 0 , , λ / ∈ T S − jµ . (97)Taking the traces of (91), we obtain γ (˜ u ) = op ( c ) γ (˜ u ) + op ( d ) γ (˜ u ) + w , (98)where w = γ ( E ˜ f + w + w ) satisfies, according to the trace formula (92), theestimates (83), (89) and (93), the following estimate µ | w | ≤ C (cid:16) k P ( x, D, µ ) u k + k u k ,µ + µ − | u | , ,µ (cid:17) (99)23nd following (96), c and d are two tangential symbols of order respectively 0 and − c = diag( c , , c , ) with c , / = − χ z − / z +1 / − z − / ! ,d − = diag( d − , , d − , ) with d − , / = χ z +1 / − z − / ! . Following (82), the transmission conditions give γ (˜ u ) − iµγ (˜ v ) = ˜ e γ (˜ u ) + γ (˜ v ) + iµ ∂ϕ ∂x n γ (˜ u ) + iµ ∂ϕ ∂x n γ (˜ v ) = ˜ e . (100)We recall that ˜ u = (˜ u , ˜ v ), combining (98) and (100) we show that op ( k ) t (cid:0) γ (˜ u ) , γ (˜ v ) , Λ − γ (˜ u ) , Λ − γ (˜ v ) (cid:1) = w + 1 µ op ˜ e + op Λ − ˜ e , (101)where k is a 4 × k + 1 µ r = − c , − Λ d − ,
00 1 − c , − Λ d − , − i iµ Λ − ∂ϕ ∂x n iµ Λ − ∂ϕ ∂x n + 1 µ r , where r is a tangential symbol of order 0.We now choose χ ( x, ξ ′ , µ ) ∈ T S µ , satisfying the same requirement as χ + , equal to1 in a neighborhood of supp( χ + ) and such that the symbol χ be equal to 1 in aneighborhood of supp( χ ). In supp( χ ), we obtain k | supp ( χ ) = z +1 z +1 − z − − Λ z +1 − z − z +2 z +2 − z − − Λ z +2 − z − − i iµ Λ − ∂ϕ ∂x n iµ Λ − ∂ϕ ∂x n . k ) | supp ( χ ) = − (cid:0) z +1 − z − (cid:1) − (cid:0) z +2 − z − (cid:1) − Λ α . To prove that there exists c > (cid:12)(cid:12) det( k ) | supp ( χ ) (cid:12)(cid:12) ≥ c , by homogeneity itsuffices to prove that det( k ) | supp ( χ ) = 0 if | ξ ′ | + µ = 1.If we suppose that det( k ) | supp ( χ ) = 0, we obtain α = 0 and then α = 0.Following (75),we obtain q = 0 and (cid:18) µ ∂ϕ ∂x n (cid:19) + q , = 0 . Combining with the fact that q , + q ( ∂ϕ /∂x n ) >
0, we obtain − (cid:18) µ ∂ϕ ∂x n (cid:19) > . Therefore det( k ) | supp ( χ ) = 0. It follows that, for large µ , k = k + µ r is ellipticin supp( χ ). Then there exists l ∈ T S µ , such that op ( l ) op ( k ) = op ( χ ) + ˜ R m , with ˜ R m ∈ T O − m − , for m large. This yields t ( γ (˜ u ) , γ (˜ v ) , Λ − γ (˜ u ) , Λ − γ (˜ v )) = op ( l ) w + µ op ( l ) op ˜ e + op ( l ) op Λ − ˜ e + ( op (1 − χ ) − ˜ R m ) t ( γ (˜ u ) , γ (˜ v ) , Λ − γ (˜ u ) , Λ − γ (˜ v )) . Since supp(1 − χ ) ∩ supp( χ + ) = ∅ and by using (99), we obtain µ | ˜ u | , ,µ ≤ C (cid:16) µ − | ˜ e | + µ | ˜ e | + k P ( x, D, µ ) u k L + k u k ,µ + µ − | u | , ,µ (cid:17) . From estimates (84) and (85) and the trace formula (92), we obtain (81). (cid:3) Z The aim of this part is to prove the estimate in the region Z . In this region, if ϕ satisfies (77), the symbol p ( x, ξ, µ ) admits a real roots and p ( x, ξ, µ ) admits tworoots z ± satisfy ± Im( z ± ) >
0. Let χ ( x, ξ ′ , µ ) ∈ T S µ equal to 1 in Z and suchthat in the support of χ we have q , − µ + q ( ∂ϕ /∂x n ) ≥ δ >
0. Then we have thefollowing partial estimate. 25 roposition 4.2
There exists constants
C > and µ > such that for any µ ≥ µ we have the following estimate µ (cid:13)(cid:13) op ( χ ) u (cid:13)(cid:13) ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ | u | , ,µ + k u k ,µ (cid:17) , (102) for any u ∈ C ∞ (Ω ) .If we assume moreover that ϕ satisfies (77) then we have µ (cid:12)(cid:12) op ( χ ) u (cid:12)(cid:12) , ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ − | op ( b ) u | + µ | op ( b ) u | + k u k ,µ + µ − | u | , ,µ (cid:17) , (103) for any u ∈ C ∞ (Ω ) and b j , j = 1 , , defined in (24). Let u ∈ C ∞ ( K ), ˜ u = op ( χ ) u and P the differential operator with principalsymbol given by p ( x, ξ, µ ) = ξ n + q ( x, ξ ′ , µ ) ξ n + q ( x, ξ ′ , µ )where q j = diag( q j, , q j, ), j = 1 ,
2. Then we have the following system P ˜ u = ˜ f in { x n > } ,B ˜ u = ˜ e = (˜ e , ˜ e ) on { x n = 0 } , (104)where ˜ f = op ( χ ) f + [ P, op ( χ )] u . Since [ P, op ( χ )] ∈ ( T O ) D x n + T O , we have k ˜ f k L ≤ C (cid:16) k P ( x, D, µ ) u k L + k u k ,µ (cid:17) , (105) B defined in (24) and ˜ e = op ( χ ) e satisfying | ˜ e | ≤ C | e | (106)and˜ e = h ( D x n + iµ ∂ϕ ∂x n ) , op ( χ ) i u | x n =0 + h ( D x n + iµ ∂ϕ ∂x n ) , op ( χ ) i v | x n =0 + op ( χ ) e . Since [ D x n , op ( χ + )] ∈ T O , we have | ˜ e | ≤ C (cid:0) | u | + | e | (cid:1) . (107)Let us reduce the problem (104) to a first order system. Put v = t ( h D ′ , µ i ˜ u, D x n ˜ u ).Then we obtain the following system D x n v − op ( P ) v = F in { x n > } ,op ( B ) v = ( µ Λ˜ e , ˜ e ) on { x n = 0 } , (108)26here P is a 4 × P = (cid:18) Λ − q − q (cid:19) , (cid:18) Λ = h ξ ′ , µ i = (cid:16) | ξ ′ | + µ (cid:17) (cid:19) , B is a tangential symbol of order 0, with principal symbol given by B + 1 µ r = (cid:18) − i iµ Λ − ∂ϕ ∂x n iµ Λ − ∂ϕ ∂x n (cid:19) + 1 µ r ( r a tangential symbol of order 0) and F = t (0 , ˜ f ).For a fixed ( x , ξ ′ , µ ) in supp χ , the generalized eigenvalues of the matrix P arethe zeroes in ξ n of p and p i.e z ± = − iµ ∂ϕ ∂x n ± iα and z ± = − iµ ∂ϕ ∂x n ± iα with ± Im( z ± ) > z +1 ∈ R .We note s ( x, ξ ′ , µ ) = ( s − , s − , s +1 , s +2 ) a basis of the generalized eigenspace of P ( x , ξ ′ , µ ) corresponding to eigenvalues with positive or negative imaginary parts. s ± j ( x, ξ ′ , µ ), j = 1 , C ∞ function on a conic neighborhood of ( x , ξ ′ , µ ) of a de-gree zero in ( ξ ′ , µ ). We denote op ( s )( x, D x ′ , µ ) the pseudo-differential operator asso-ciated to the principal symbol s ( x, ξ ′ , µ ) = (cid:0) s − ( x, ξ ′ , µ ) , s − ( x, ξ ′ , µ ) , s +1 ( x, ξ ′ , µ ) , s +2 ( x, ξ ′ , µ ) (cid:1) .Let ˆ χ ( x, ξ ′ , µ ) ∈ T S µ equal to 1 in a conic neighborhood of ( x , ξ ′ , µ ) and in a neigh-borhood of supp( χ ) and satisfies that in the support of ˆ χ , s is elliptic. Then thereexists n ∈ T S µ , such that op ( s ) op ( n ) = op ( ˆ χ ) + ˆ R m , with ˆ R m ∈ T O − m − , for m large.Let V = op ( n ) v . Then we have the following system D x n V = GV + AV + F in { x n > } ,op ( B ) V = ( µ Λ˜ e , ˜ e ) + v on { x n = 0 } , (109)where G = op ( n ) op ( P ) op ( s ), A = [ D x n , op ( n )] op ( s ), F = op ( n ) F + op ( n ) op ( P )( op (1 − ˆ χ ) − ˆ R m ) v + [ D x n , op ( n )] ( op (1 − ˆ χ ) − ˆ R m ) v , op ( B ) = op ( B ) op ( s ) and v = op ( B )( op ( ˆ χ −
1) + ˆ R m ) v .Using the fact that supp(1 − ˆ χ ) ∩ supp( χ ) = ∅ , ˆ R m ∈ T O − m − , for m large andestimate (105), we show k F k ≤ C (cid:16) k P ( x, D, µ ) u k L + k u k ,µ (cid:17) . (110)Using the fact that supp(1 − ˆ χ ) ∩ supp( χ ) = ∅ , ˆ R m ∈ T O − m − , for m large andthe trace formula (92), we show µ | v | ≤ C (cid:16) µ − | u | , ,µ + k u k ,µ (cid:17) . (111)Here we need to recall an argument shown in Taylor [13] given by this lemma27 emma 4.3 Let v solves the system ∂∂y v = Gv + Av where G = (cid:18) E F (cid:19) and A are pseudo-differential operators of order and ,respectively. We suppose that the symbols of E and F are two square matrices andhave disjoint sets of eigenvalues. Then there exists a pseudo-differential operator K of order − such that w = ( I + K ) v satisfies ∂∂y w = Gw + (cid:18) α α (cid:19) w + R w + R v where α j and R j , j = 1 , are pseudo-differential operators of order and −∞ ,respectively. By this argument, there exists a pseudo-differential operator K ( x, D x ′ , µ ) of order − D x n w − op ( H ) w = ˜ F in { x n > } ,op ( ˜ B ) w = ( µ Λ˜ e , ˜ e ) + v + v on { x n = 0 } , (112)where w = ( I + K ) V , ˜ F = ( I + K ) F , op ( H ) is a tangential of order 1 with principalsymbol H = diag( H − , H + ) and − Im( H − ) ≥ C Λ, op ( ˜ B ) = op ( B )( I + K ′ ) with K ′ is such that ( I + K ′ )( I + K ) = Id + R ′ m ( R ′ m ∈ O − m − , for m large) and v = op ( B ) R ′ m V .According to (110), we have k ˜ F k ≤ C (cid:16) k P ( x, D, µ ) u k L + k u k ,µ (cid:17) . (113)Using the fact that R ′ m ∈ O − m − , for m large, the trace formula (92) and estimates(106), (107) and (111), we show µ (cid:12)(cid:12)(cid:12) op ( ˜ B ) w (cid:12)(cid:12)(cid:12) ≤ C (cid:18) µ | e | + µ | e | + µ − | u | , ,µ + k u k ,µ (cid:19) . (114) Lemma 4.4
Let R = diag ( − ρ Id , , ρ > . Then there exists C > such that1. Im ( RH ) = diag ( e ( x, ξ ′ , µ ) , , with e ( x, ξ ′ , µ ) = − ρIm ( H − ) ,2. e ( x, ξ ′ , µ ) ≥ C Λ in supp ( χ ) ,3. −R + ˜ B ⋆ ˜ B ≥ C. Id on { x n = 0 } ∩ supp ( χ ) . roof Denote the principal symbol ˜ B of the boundary operator op ( ˜ B ) by (cid:16) ˜ B − , ˜ B + (cid:17) where˜ B + is the restriction of ˜ B to subspace generated by (cid:0) s +1 , s +2 (cid:1) . We begin by provingthat ˜ B + is an isomorphism. Denote w = t (1 ,
0) and w = t (0 , . Then s +1 = (cid:0) w , z +1 Λ − w (cid:1) s +2 = (cid:0) w , z +2 Λ − w (cid:1) are eigenvectors of z +1 and z +2 . We have ˜ B + = ( B + µ r )( s +1 s +2 ) = B +0 + µ r +0 .To proof that ˜ B + is an isomorphism it suffices, for large µ , to proof that B +0 is anisomorphism. Following (76), we obtain B +0 = (cid:18) − i Λ − iα Λ − iα (cid:19) . Then det( B +0 ) = − Λ − α . If we suppose that det( B +0 ) = 0, we obtain α = 0 and then α = 0. Following (75),we obtain q = 0 and (cid:18) µ ∂ϕ ∂x n (cid:19) + q , = 0 . Combining with the fact that q , + q ( ∂ϕ /∂x n ) = 0, we obtain (cid:16) µ ∂ϕ ∂x n (cid:17) = 0, that isimpossible because following (77), we have (cid:16) ∂ϕ ∂x n (cid:17) = 0 and following ( 22), we have µ = 0. We deduce that ˜ B + is an isomorphism.Let us show the Lemma 4.4. We haveIm( RH ) = diag (cid:0) − ρ Im( H − ) , (cid:1) = diag ( e ( x, ξ ′ , µ ) , , (115)where e ( x, ξ ′ , µ ) = − ρ Im( H − ) ≥ C Λ, C >
0. It remains to proof 3.Let w = ( w − , w + ) ∈ C = C ⊕ C . Then we have ˜ B w = ˜ B − w − + ˜ B + w + . Since ˜ B + is an isomorphism, then there exists a constant C > (cid:12)(cid:12)(cid:12) ˜ B + w + (cid:12)(cid:12)(cid:12) ≥ C (cid:12)(cid:12) w + (cid:12)(cid:12) . Therefore, we have (cid:12)(cid:12) w + (cid:12)(cid:12) ≤ C (cid:18)(cid:12)(cid:12)(cid:12) ˜ B w (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) w − (cid:12)(cid:12) (cid:19) . We deduce − ( R w, w ) = ρ (cid:12)(cid:12) w − (cid:12)(cid:12) ≥ C (cid:12)(cid:12) w + (cid:12)(cid:12) + ( ρ − (cid:12)(cid:12) w − (cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ˜ B w (cid:12)(cid:12)(cid:12) . Then, we obtain the result, if ρ is large enough. (cid:3) .3.2 Proof of proposition 4.2 We start by showing (102). We have k P ( x, D, µ ) u k = k (Re P ) u k + k (Im P ) u k + i (cid:20)(cid:16) (Im P ) u , (Re P ) u (cid:17) − (cid:16) (Re P ) u , (Im P ) u (cid:17)(cid:21) . By integration by parts we find k P ( x, D, µ ) u k = k (Re P ) u k + k (Im P ) u k + i (cid:16) [Re P , Im P ] u , u (cid:17) + µQ ( u ) , where Q ( u ) = ( − ∂ϕ ∂x n D x n u , D x n u ) + ( op ( r ) u , D x n u ) + ( op ( r ′ ) D x n u , u ) + ( op ( r ) u , u ) + µ ( ∂ϕ ∂x n u , u ) ,r = r ′ = 2 q , , r = − ∂ϕ ∂x n q , . Then we have | Q ( u ) | ≤ C | u | , ,µ . We show the same thing for P ( x, D, µ ) v . In addition we know that the principalsymbol of the operator [Re P j , Im P j ], j = 1 ,
2, is given by { Re P j , Im P j } . Proceedinglike Lebeau and Robbiano in paragraph 3 in [9], we obtain (102).It remains to prove (103). Following Lemma 4.4, let G ( x n ) = d/dx n ( op ( R ) w, w ) L ( R n − ) .Using D x n w − op ( H ) = ˜ F , we obtain G ( x n ) = − op ( R ) ˜ F , w ) − op ( R ) op ( H ) w, w ) . The integration in the normal direction gives( op ( R ) w, w ) = Z ∞ Im( op ( R ) op ( H ) w, w ) dx n + 2 Z ∞ Im( op ( R ) ˜ F , w ) dx n . (116)From Lemma 4.4 and the G˚arding inequality, we obtain, for µ large,Im( op ( R ) op ( H ) w, w ) ≥ C (cid:12)(cid:12) w − (cid:12)(cid:12) , (117)moreover we have for all ǫ > Z ∞ (cid:12)(cid:12)(cid:12) ( op ( R ) ˜ F , w ) (cid:12)(cid:12)(cid:12) dx n ≤ ǫCµ (cid:13)(cid:13) w − (cid:13)(cid:13) + C ǫ µ k ˜ F k . (118)Applying Lemma 4.4 and the G˚arding inequality, we obtain, for µ large, − ( op ( R ) w, w ) + | op ( ˜ B ) w | ≥ C | w | . (119)30ombining (119), (118), (117) and (116), we get C (cid:12)(cid:12) w − (cid:12)(cid:12) + C | w | ≤ Cµ k ˜ F k + | op ( ˜ B ) w | . (120)Then µ | w | ≤ C k ˜ F k + µ | op ( ˜ B ) w | . Recalling that w = ( I + K ) V , V = op ( n ) v , v = t ( h D ′ , µ i ˜ u, D x n ˜ u ) and ˜ u = op ( χ ) u and using estimates (113) and (114), we prove (103). (cid:3) E − This part is devoted to estimate in region E − .Let χ − ( x, ξ ′ , µ ) ∈ T S µ equal to 1 in E − and such that in the support of χ − we have q , + q ( ∂ϕ /∂x n ) ≤ − δ <
0. Then we have the following partial estimate.
Proposition 4.3
There exists constants
C > and µ > such that for any µ ≥ µ we have the following estimate µ (cid:13)(cid:13) op ( χ − ) u (cid:13)(cid:13) ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ | u | , ,µ + k u k ,µ (cid:17) , (121) for any u ∈ C ∞ (Ω ) .If we assume moreover that ∂ϕ ∂x n > then we have µ (cid:12)(cid:12) op ( χ − ) u (cid:12)(cid:12) , ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ − | u | , ,µ + k u k ,µ (cid:17) (122) for any u = ( u , v ) ∈ C ∞ (Ω ) . Proof.
Let ˜ u = op ( χ − ) u = ( op ( χ − ) u , op ( χ − ) v ) = (˜ u , ˜ v ).In this region we have not a priori information for the roots of p ( x, ξ, µ ). Using thesame technique of the proof of (102), we obtain µ (cid:13)(cid:13) op ( χ − ) v (cid:13)(cid:13) ,µ ≤ C (cid:16) k P ( x, D, µ ) v k + µ | v | , ,µ + k v k ,µ (cid:17) (123)In supp( χ − ) the two roots z ± of p ( x, ξ, µ ) are in the half-plane Imξ n <
0. Then wecan use the Calderon projectors. By the same way that the proof of (80) and usingthe fact that the operators t , and t , vanish in x n > Imξ n <
0, see (94)), the counterpart of (91) is then˜ u = E ˜ f + w , + w , , for x n > . (124)We then obtain (see proof of (80)) µ (cid:13)(cid:13) op ( χ − ) u (cid:13)(cid:13) ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ | u | , ,µ + k u k ,µ (cid:17) . (125)31ombining (123) and (125), we obtain (121).It remains to proof (122). We take the trace at x n = 0 + of (124), γ (˜ u ) = w , = γ ( E ˜ f + w , + w , ) , which, by the counterpart of (99), gives µ | γ (˜ u ) | ≤ C (cid:16) k P ( x, D, µ ) u k + k u k ,µ + µ − | u | , ,µ (cid:17) . (126)From (124) we also have D x n ˜ u = D x n E ˜ f + D x n w , + D x n w , , for x n > . We take the trace at x n = 0 + and obtain γ (˜ u ) = γ ( D x n ( E ˜ f + w , + w , )) . Using the trace formula (92), we obtain | γ (˜ u ) | ≤ Cµ − (cid:13)(cid:13)(cid:13) D x n ( E ˜ f + w , + w , ) (cid:13)(cid:13)(cid:13) ,µ ≤ Cµ − (cid:13)(cid:13)(cid:13) E ˜ f + w , + w , (cid:13)(cid:13)(cid:13) ,µ and, by the counterpart of (83), (89) and (93), this yields µ | γ (˜ u ) | ≤ C (cid:16) k P ( x, D, µ ) u k + k u k ,µ + µ − | u | , ,µ (cid:17) . (127)Combining (126) and (127), we obtain µ (cid:12)(cid:12) op ( χ − ) u (cid:12)(cid:12) , ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + k u k ,µ + µ − | u | , ,µ (cid:17) . Then we have (122). (cid:3)
We choose a partition of unity χ + + χ + χ − = 1 such that χ + , χ and χ − satisfythe properties listed in proposition 4.1, 4.2 and 4.3 respectively. We have k u k ,µ ≤ (cid:13)(cid:13) op ( χ + ) u (cid:13)(cid:13) ,µ + (cid:13)(cid:13) op ( χ ) u (cid:13)(cid:13) ,µ + (cid:13)(cid:13) op ( χ − ) u (cid:13)(cid:13) ,µ . Combining this inequality and (80), (102) and (121), we obtain, for large µ , the firstestimate (37) of Theorem 3.2. i.e. µ k u k ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ | u | , ,µ (cid:17) . It remains to estimate µ | u | , ,µ . We begin by giving an estimate of µ | u | , ,µ .We have | u | , ,µ ≤ (cid:12)(cid:12) op ( χ + ) u (cid:12)(cid:12) , ,µ + (cid:12)(cid:12) op ( χ ) u (cid:12)(cid:12) , ,µ + (cid:12)(cid:12) op ( χ − ) u (cid:12)(cid:12) , ,µ , (cid:12) op ( χ + ) u (cid:12)(cid:12) , ,µ ≤ (cid:12)(cid:12) op ( χ + ) u (cid:12)(cid:12) , ,µ and (cid:12)(cid:12) op ( χ ) u (cid:12)(cid:12) , ,µ ≤ (cid:12)(cid:12) op ( χ ) u (cid:12)(cid:12) , ,µ . Combining these inequalities, (81), (103), (122) and the fact that µ − | u | , ,µ = µ − | u | , ,µ + µ − | v | , ,µ , we obtain, for large µµ | u | , ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ − | op ( b ) u | + µ | op ( b ) u | + µ − | v | , ,µ + k u k ,µ (cid:17) . (128)For estimate µ | v | , ,µ , we need to use the transmission conditions given by (24).We have op ( b ) u = u | x n =0 − iµv | x n =0 on { x n = 0 } . Then µ | v | ≤ C (cid:0) µ − | u | + µ − | op ( b ) u | (cid:1) . Since we have µ − | u | ≤ µ | u | , ,µ . Then using (128), we obtain µ | v | ≤ C (cid:16) k P ( x, D, µ ) u k + µ − | op ( b ) u | + µ | op ( b ) u | + µ − | v | , ,µ + k u k ,µ (cid:17) . (129)We have also op ( b ) u = (cid:18) D x n + iµ ∂ϕ ∂x n (cid:19) u | x n =0 + (cid:18) D x n + iµ ∂ϕ ∂x n (cid:19) v | x n =0 on { x n = 0 } . Then µ | D x n v | ≤ C (cid:0) µ | op ( b ) u | + µ | D x n u | + µ | u | + µ | v | (cid:1) . Using the fact that | u | k − ≤ µ − | u | k , we obtain µ | D x n v | ≤ C (cid:0) µ | op ( b ) u | + µ | D x n u | + µ | u | + µ | v | (cid:1) . Since we have µ | u | , ,µ = µ | D x n u | + µ | u | . Then using (128) and (129), weobtain µ | D x n v | ≤ C (cid:16) k P ( x, D, µ ) u k + µ − | op ( b ) u | + µ | op ( b ) u | + µ − | v | , ,µ + k u k ,µ (cid:17) . (130)Combining (129) and (130), we have µ | v | , ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ − | op ( b ) u | + µ | op ( b ) u | + k u k ,µ (cid:17) . (131)Combining (128) and (131), we obtain µ | u | , ,µ ≤ C (cid:16) k P ( x, D, µ ) u k + µ − | op ( b ) u | + µ | op ( b ) u | + k u k ,µ (cid:17) . (132)Inserting (132) in (37) and for large µ , we obtain (38). (cid:3) ppendix A This appendix is devoted to prove Lemma 2.1. For this, we need to distinguish twocases.1.
Inside O To simplify the writing, we note k u k L ( O ) = k u k .Let χ ∈ C ∞ ( O ). We have by integration by part(( △ − iµ ) u, χ u ) = ( −∇ u, χ ∇ u ) − ( ∇ u, ∇ ( χ ) u ) − iµ k χu k . Then µ k χu k ≤ C (cid:0) k f k (cid:13)(cid:13) χ u (cid:13)(cid:13) + k∇ u k + k∇ u k k χu k (cid:1) . Then µ k χu k ≤ C (cid:18) ǫ k f k + ǫ (cid:13)(cid:13) χ u (cid:13)(cid:13) + k∇ u k + 1 ǫ k∇ u k + ǫ k χu k (cid:19) . Recalling that µ ≥
1, we have for ǫ small enough k χu k ≤ C (cid:0) k∇ u k + k f k (cid:1) . (133)Hence the result inside O .2. In the neighborhood of the boundary
Let x = ( x ′ , x n ) ∈ R n − × R . Then ∂ O = { x ∈ R n , x n = 0 } . Let ǫ > < x n < ǫ . Then we have u ( x ′ , ǫ ) − u ( x ′ , x n ) = Z ǫx n ∂ x n u ( x ′ , σ ) dσ. Then | u ( x ′ , x n ) | ≤ | u ( x ′ , ǫ ) | + 2 (cid:18)Z ǫx n | ∂ x n u ( x ′ , σ ) | dσ (cid:19) . Using the Cauchy Schwartz inequality, we obtain | u ( x ′ , x n ) | ≤ | u ( x ′ , ǫ ) | + 2 ǫ Z ǫ | ∂ x n u ( x ′ , x n ) | dx n . Integrating with regard to x ′ , we obtain Z | x ′ | <ǫ | u ( x ′ , x n ) | dx ′ ≤ Z | x ′ | <ǫ | u ( x ′ , ǫ ) | dx ′ +2 ǫ Z | x ′ | <ǫ, | x n | <ǫ (cid:16) | ∂ x n u ( x ′ , x n ) | dx n (cid:17) dx ′ . (134)Using the trace Theorem, we have Z | x ′ | <ǫ | u ( x ′ , ǫ ) | dx ′ ≤ C Z | x ′ | < ǫ, | x n − ǫ | < ǫ ( | u ( x ) | + |∇ u ( x ) | ) dx. (135)34ow we need to introduce the following cut-off functions χ ( x ) = < x n < ǫ , x n> ǫ and χ ( x ) = ǫ < x n < ǫ , x n < ǫ , x n> ǫ. Combining (134) and (135), we obtain for ǫ small enough k χ u k ≤ C (cid:0) k χ u k + k∇ u k (cid:1) . (136)Since following (133), we have k χ u k ≤ C (cid:0) k f k + k∇ u k (cid:1) . Inserting in (136), we obtain k χ u k ≤ C (cid:0) k f k + k∇ u k (cid:1) . (137)Hence the result in the neighborhood of the boundary.Following (133), we can write k (1 − χ ) u k ≤ C (cid:0) k f k + k∇ u k (cid:1) . (138)Adding (137) and (138), we obtain k u k ≤ C (cid:0) k f k + k∇ u k (cid:1) . Hence the result. 35 ppendix B: Proof of Lemma 3.1
This appendix is devoted to prove Lemma 3.1.Let χ ∈ C ∞ ( R n ) such that χ = 1 in the support of u . We want to show that op (Λ s ) e µϕ χop (Λ − s ) is bounded in L . Recalling that for all u and v ∈ S ( R n ), wehave F ( uv )( ξ ′ ) = ( 12 π ) n − F ( u ) ∗ F ( v )( ξ ′ ) , ∀ ξ ′ ∈ R n − . Then F ( op (Λ s ) e µϕ χop (Λ − s ) v )( ξ ′ , µ ) = h ξ ′ , µ i s F ( e µϕ χop (Λ − s ) v )( ξ ′ , µ )= ( 12 π ) n − h ξ ′ , µ i s ( g ( ξ ′ , µ ) ∗ h ξ ′ , µ i − s F ( v ))( ξ ′ , µ ) , where g ( ξ ′ , µ ) = F ( e µϕ χ )( ξ ′ , µ ). Then we have F ( op (Λ s ) e µϕ χop (Λ − s ) v )( ξ ′ , µ ) = Z g ( ξ ′ − η ′ , µ ) h ξ ′ , µ i s h η ′ , µ i − s F ( v )( η ′ , µ ) dη ′ . Let k ( ξ ′ , η ′ ) = g ( ξ ′ − η ′ , µ ) h ξ ′ , µ i s h η ′ , µ i − s . Our goal is to show that R K ( ξ ′ , η ′ ) F ( v )( η ′ , µ ) dη ′ is bounded in L . To do it, we will use Lemma of Schur. It suffices to prove thatthere exists M >
N > Z | K ( ξ ′ , η ′ ) | dξ ′ ≤ M and Z | K ( ξ ′ , η ′ ) | dη ′ ≤ N. In the sequel, we suppose s ≥ s < R >
0, we have h ξ ′ , µ i R g ( ξ ′ , µ ) = Z h ξ ′ , µ i R e − ix ′ ξ ′ ξ ( x ) e µϕ ( x ) dx ′ = Z (1 − ∆ + µ ) R ( e − ix ′ ξ ′ ) χ ( x ) e µϕ ( x ) dx ′ = Z e − ix ′ ξ ′ (1 − ∆ + µ ) R ( χ ( x ) e µϕ ( x ) ) dx ′ . Then there exists
C >
0, such that (cid:12)(cid:12) h ξ ′ , µ i R g ( ξ ′ , µ ) (cid:12)(cid:12) ≤ Ce Cµ . (139)Moreover, we can write Z | K ( ξ ′ , η ′ ) | dξ ′ = Z (cid:12)(cid:12)(cid:12)(cid:12) g ( ξ ′ − η ′ , µ ) h ξ ′ − η ′ , µ i R h ξ ′ , µ i s h η ′ , µ i − s h ξ ′ − η ′ , µ i R (cid:12)(cid:12)(cid:12)(cid:12) dξ ′ . Using (139), we obtain Z | K ( ξ ′ , η ′ ) | dξ ′ ≤ Ce Cµ Z h ξ ′ , µ i s h η ′ , µ i − s h ξ ′ − η ′ , µ i R dξ ′ . Z h ξ ′ , µ i s h η ′ , µ i − s h ξ ′ − η ′ , µ i R dξ ′ = Z | ξ ′ |≤ ǫ | η ′ | h ξ ′ , µ i s h η ′ , µ i − s h ξ ′ − η ′ , µ i R dξ ′ + Z | η ′ |≤ ǫ | ξ ′ | h ξ ′ , µ i s h + η ′ , µ i − s h ξ ′ − η ′ , µ i R dξ ′ , ǫ > . If | ξ ′ | ≤ ǫ | η ′ | , we have h ξ ′ , µ i s h η ′ , µ i − s h ξ ′ − η ′ , µ i R ≤ C h η ′ , µ i s h η ′ , µ i − s h ξ ′ − η ′ , µ i R ≤ C h ξ ′ − η ′ , µ i R ∈ L if 2 R > n − . If | η ′ | ≤ ǫ | ξ ′ | , i.e h ξ ′ − η ′ , µ i ≥ δ h ξ ′ , µ i , δ >
0, we have h ξ ′ , µ i s h η ′ , µ i − s h ξ ′ − η ′ , µ i R ≤ C h ξ ′ − η ′ , µ i R − s ∈ L if 2 R − s > n − . Then there exists
M >
0, such that Z | K ( ξ ′ , η ′ ) | dξ ′ ≤ M e Cµ . By the same way, we show that there exists
N >
0, such that Z | K ( ξ ′ , η ′ ) | dη ′ ≤ N e Cµ . Using Lemma of Schur, we have ( op (Λ s ) e µϕ χop (Λ − s )) is bounded in L and (cid:13)(cid:13) op (Λ s ) e µϕ χop (Λ − s ) (cid:13)(cid:13) L ( L ) ≤ Ce Cµ . Applying in op (Λ s ) u , we obtain the result. References [1] C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the obser-vation, control, and stabilization of waves from the boundary.
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