Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map
aa r X i v : . [ m a t h . A P ] M a y STABLE DETERMINATION OUTSIDE A CLOAKING REGION OF TWOTIME-DEPENDENT COEFFICIENTS IN AN HYPERBOLIC EQUATION FROM DIRICHLETTO NEUMANN MAP
MOURAD BELLASSOUED AND IBTISSEM BEN AÏCHAA
BSTRACT . In this paper, we treat the inverse problem of determining two time-dependent coefficients appear-ing in a dissipative wave equation, from measured Neumann boundary observations. We establish in dimension n ě , stability estimates with respect to the Dirichlet-to-Neumann map of these coefficients provided that areknown outside a cloaking regions. Moreover, we prove that it can be stably recovered in larger subsets of thedomain by enlarging the set of data. Keywords:
Inverse problems, Dissipative wave equation, Time-dependent coefficients, Stability estimates.
1. I
NTRODUCTION
Statement of the problem.
This paper deals with the inverse problem of determining two time-dependent coefficients in a dissipative wave equation from boundary observations. Let Ω be a boundeddomain of R n , n ě , with C boundary Γ “ B Ω . Given T ą , we introduce the following dissipativewave equation(1.1) $’’’’&’’’’% B t u ´ ∆ u ` a p x, t qB t u ` b p x, t q u “ in Q “ Ω ˆ p , T q ,u p x, q “ u p x q , B t u p x, q “ u p x q in Ω ,u p x, t q “ f p x, t q on Σ “ Γ ˆ p , T q , where f P H p Σ q , u P H p Ω q , u P L p Ω q , and the coefficients a P C p Q q and b P C p Q q are assumedto be real valued. It is well known (see [14]) that if f p¨ , q “ u | Γ , there exists a unique solution u to theequation (1.1) satisfying u P C pr , T s , H p Ω qq X C pr , T s , L p Ω qq . Moreover, there exists C ą , such that(1.2) }B ν u } L p Σ q ` } u p¨ , t q} H p Ω q ` }B t u p¨ , t q} L p Ω q ď C ` } f } H p Σ q ` } u } H p Ω q ` } u } L p Ω q ˘ . Here ν denotes the unit outward normal to Γ at x and B ν u stands for ∇ u ¨ ν .In the present paper, we address the uniqueness and the stability issues in the study of an inverse problemfor the dissipative wave equation (1.1), in the presence of an absorbing coefficient a and a potential b thatdepend on both space and time variables. We consider three different sets of data and we aim to show that a and b can be recovered in some specific subsets of the domain, by probing it with disturbances generated onthe boundary. The Dirichlet data f is considered as a disturbance that is used to probe the medium which isassumed to be quiet initially.The problem of identifying coefficients appearing in hyperbolic boundary value problems was treatedvery well and there are many works that are relevant to this topic. In the case where the unknown coefficient is depending only on the spatial variable, Rakesh and Symes [22] proved by means of geometric opticssolutions, a uniqueness result in recovering a time-independent potential in a wave equation from globalNeumann data. The uniqueness by local Neumann data, was considered by Eskin [13] and Isakov [16]. In[5], Bellassoued, Choulli and Yamamoto proved a log-type stability estimate, in the case where the Neumanndata are observed on any arbitrary subset of the boundary. Isakov and Sun [18] proved that the knowledgeof local Dirichlet-to-Neumann map yields a stability result of Hölder type in determining a coefficient in asubdomain. As for the stability obtained from global Neumann data, one can see Sun [29], Cipolatti andLopez [11]. The case of Riemannian manifold was considered by Bellassoued and Dos Santos Ferreira [6],Stefanov and Uhlmann [28].All the mentioned papers are concerned only with time-independent coefficients. In the case where thecoefficient is also depending on the time variable, There is a uniqueness result proved by Ramm and Rakesh[23], in which they showed that a time-dependent coefficient appearing in a wave equation with zero initialconditions, can be uniquely determined from the knowledge of global Neumann data, but only in a precisesubset of the cylindrical domain Q that is made of lines making an angle of ˝ with the t -axis and meetingthe planes t “ and t “ T outside Q . However, inspired by the work of [30], Isakov proved in [17], that thetime-dependent coefficient can be recovered from the responses of the medium for all possible initial data,over the whole domain Q .It is clear that with zero initial data, there is no hope to recover a time-dependent coefficient appearingin a hyperbolic equation over the whole cylindrical domain, even from the knowledge of global Neumanndata, because the value of the solution can be effected by the value of the initial conditions, which is actuallydue to a fundamental concept concerning hyperbolic equations called the domain of dependence (see [14]).Moreover, we can prove that the backward light-cone with base Ω is a cloaking region, that is we can notuniquely recover the coefficients in this region.As for uniqueness results, we have also the paper of Stefanov [26], in which he proved that a time-dependent potential appearing in a wave equation can be uniquely recovered from scattering data and thepaper of Ramm and Sjöstrand [24], in which they proved a uniqueness result on an infinite time-spacecylindrical domain Ω ˆ R t .The stability in this case, was considered by Salazar [25], who extended the result of Ramm and Sjöstrand[24] to more general coefficients and he established a stability result for compactly supported coefficientsprovided T is sufficiently large. We also refer to the works of Kian [19, 20] who followed techniques usedby Bellassoued, Jellali and Yamamoto [7, 8] and proved uniqueness and a log - log type stability estimatefrom the knowledge of partial Neumann data. As for stability results from global Neumann data we referto Ben Aïcha [10] who proved recently a stability of log -type in recovering a zeroth order time-dependentcoefficient in different regions of the cylindrical domain by considering different sets of data. We alsorefer to Waters [32] who derived, in Riemannian case, conditional Hölder stability estimates for the X-raytransform of the time-dependent potential appearing in the wave equation.As for results of hyperbolic inverse problems dealing with single measurement data, one can see [2, 3, 9,12, 15, 27] and the references therein.Inspired by the work of Bellassoued [4] and following the same strategy as in Ben Aïcha [10], we provein this paper stability estimates in the recovery of the unknown coefficients a and b via different types ofmeasurements and over different subsets of the domain Q . AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 3
Main results.
Before stating our main results we first introduce the following notations.Let r ą be such that T ą r and Ω Ď B p , r { q “ ! x P R n , | x | ă r { ) . We set Q r “ B p , r { q ˆp , T q and we consider the annular region around the domain Ω , A r “ ! x P R n , r ă | x | ă T ´ r ) , and the forward and backward light cone: C ` r “ ! p x, t q P Q r , | x | ă t ´ r , t ą r ) , C ´ r “ ! p x, t q P Q r , | x | ă T ´ r ´ t, T ´ r ą t ) . Finally, we denote Q ˚ r “ C ` r X C ´ r Q r, ˚ “ Q X Q ˚ r , and quadQ r, “ Q X C ` r . We remark that the open subset Q r, ˚ is made of lines making an angle of ˝ with the t -axis and meetingthe planes t “ and t “ T outside Q r and Q r, is made of lines making an angle of ˝ with the t -axis andmeeting only the planes t “ outside Q r . We notice that Q r, ˚ Ă Q r, Ă Q . Note, that in the particular casewhere Ω “ B p , r { q , we have Q r, ˚ “ Q ˚ r and Q r, “ C ` r (see Figure 1 in [10]).In the present paper, we will prove stability estimates for the inverse problem under consideration in threecases. We will consider three different sets of data and we will prove that the coefficients a and b can bestably determined in three different regions of the cylindrical domain Q . Given M , M ą , we considerthe set of admissible coefficients a and b : A p M , M q “ tp a, b q P C p Q r q ˆ C p Q r q ; } a } C p Q q ď M , } b } C p Q q ď M u . Finally, we define the following space H “ ! f P H p Σ q , f p¨ , q “ on Γ ) , equipped with the norm of H p Σ q . Determination of coefficients from boundary measurements:
In the first case, we will assume thatthe initial conditions p u , u q are zero and our set of data will be given only by boundary measurements enclosed by the Dirichlet-to-Neumann map Λ a,b defined as follows Λ a,b : H p Σ q ÝÑ L p Σ q f ÞÝÑ B ν u. Note that from (1.2) we have Λ a,b is continuous from H p Σ q to L p Σ q . We denote by } Λ a,b } its norm in L p H p Σ q , L p Σ qq . We can prove that it is hopeless to uniquely determine a and b in the case where thesecoefficients are supported in the cloaking region C “ ! p x, t q P Q r ; | x | ă r ´ t, t ă r ) . The first result of this paper can be stated as follows
Theorem 1.1.
Let T ą Diam p Ω q . There exist constants C ą and µ , µ P p , q such that we have } a ´ a } L p Q r, ˚ q ď C ´ } Λ a ,b ´ Λ a ,b } µ ` | log } Λ a ,b ´ Λ a ,b }| ´ ¯ µ , for any p a i , b i q P A p M , M q such that } a i } H p p Q q ď M , for some p ą n { ` { , p a , b q “ p a , b q in Q r z Q r, ˚ and p ∇ a , ∇ b q “ p ∇ a , ∇ b q on B Q r X B Q r, ˚ . Here C depends only on Ω , M , M , T and n . M. BELLASSOUED AND I. BEN AÏCHA
The above statement claims stable determination of the time-dependent coefficient a from the Neumannboundary measurements Λ a,b in Q r, ˚ Ă Q , provided a is known outside Q r, ˚ . By Theorem 1.1, we canreadily derive the following result Theorem 1.2.
Let T ą Diam p Ω q . There exist two constants C ą and µ P p , q such that we have } b ´ b } H ´ p Q r, ˚ q ď C ´ log | log } Λ a ,b ´ Λ a ,b }| µ ¯ ´ , for any p a i , b i q P A p M , M q such that } a i } H p p Q q ď M , for some p ą n { ` { , p a , b q “ p a , b q in Q r z Q r, ˚ and p ∇ a , ∇ b q “ p ∇ a , ∇ b q on B Q r X B Q r, ˚ . Here C depends only on Ω , M , M , T and n . This mentioned result shows that the time-dependent potential b can also be stably determined, from theknowledge of the boundary measurements Λ a,b in the same subset Q r, ˚ Ă Q , provided it is known outsidethis region. As a consequence, we have the following existence result Corollary 1.3.
Under the same assumptions of Theorem 1.1 and Theorem 1.2, we have that Λ a ,b “ Λ a ,b implies a “ a and b “ b in Q r, ˚ . Determination of coefficients from boundary measurements and final data:
In order to extend theabove results to a larger region Q r, Ą Q r, ˚ , we require more information about the solution u of the waveequation 1.1. So, in this case we will add the final data of the solution u . This leads to defining the followingboundary operator (response operator): R a,b : H p Σ q ÝÑ K : “ L p Σ q ˆ H p Ω q ˆ L p Ω q f ÞÝÑ pB ν u, u p¨ , T q , B t u p¨ , T qq . We conclude from (1.2), that R a,b is a continuous operator form H p Σ q to K . We denote by } R a,b } its normin L p H p Σ q , K q . Theorem 1.4.
Let T ą Diam p Ω q . There exist constants C ą and µ , µ P p , q such that we have } a ´ a } L p Q r, q ď C ´ } R a ,b ´ R a ,b } µ ` | log } R a ,b ´ R a ,b }| ´ ¯ µ , for any p a i , b i q P A p M , M q such that } a i } H p p Q q ď M , for some p ą n { ` { , p a , b q “ p a , b q in Q r z Q r, and p ∇ a , ∇ b q “ p ∇ a , ∇ b q on B Q r X B Q r, . Here C depends only on Ω , M , M , T and n . From the above statement, we can readily derive the following consequence
Theorem 1.5.
Let T ą Diam p Ω q . There exist two constants C ą and µ P p , q such that we have } b ´ b } H ´ p Q r, q ď C ´ log | log } R a ,b ´ R a ,b }| µ ¯ ´ , for any p a i , b i q P A p M , M q such that } a i } H p p Q q ď M , for some p ą n { ` { , p a , b q “ p a , b q in Q r z Q r, and p ∇ a , ∇ b q “ p ∇ a , ∇ b q on B Q r X B Q r, . Here C depends only on Ω , M , M , T and n . As a consequence, we have the following uniqueness result
Corollary 1.6.
Under the same assumptions of Theorem 1.4 and Theorem 1.5, we have that R a ,b “ R a ,b implies a “ a and b “ b in Q r, . AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 5
Determination of coefficients from boundary measurements and final data by varying the initial data:
In the first and the second case, we can see that there is no hope to recover the unknown coefficients a and b over the whole domain, since the initial data p u , u q are zero. However, we shall prove that this is nolonger the case by considering all possible initial data.We define the following space F “ H p Σ q ˆ H p Ω q ˆ L p Ω q . In this case we will consider observationsgiven by the following operator: I a,b : F ÝÑ K p f, u , u q ÞÝÑ pB ν u, u p¨ , T q , B t u p¨ , T qq . By (1.2), we deduce that I a,b is continuous from F into K , we denote by } I a,b } its norm in L p F , K q . Havingsaid that, we are now in position to state the last main result. Theorem 1.7.
There exist constants C ą and µ , µ P p , q such that the following estimate holds } a ´ a } L p Q q ď C ´ } I a ,b ´ I a ,b } µ ` | log } I a ,b ´ I a ,b }| ´ ¯ µ , for any p a i , b i q P C p Q q ˆ C p Q q , such that } a i } C p Q q ` } a i } H p p Q q ď M for some p ą n { ` { , } b i } C p Q q ď M and p ∇ a , ∇ b q “ p ∇ a , ∇ b q on Σ . Here C depends only on Ω , M , M , T and n . As a consequence, we have the following result
Theorem 1.8.
There exist two constants C ą and µ P p , q such that the following estimate holds } b ´ b } H ´ p Q q ď C ´ log | log } I a ,b ´ I a ,b }| µ ¯ ´ , for any p a i , b i q P C p Q q ˆ C p Q q , such that } a i } C p Q q ` } a i } H p p Q q ď M for some p ą n { ` { , } b i } C p Q q ď M and p ∇ a , ∇ b q “ p ∇ a , ∇ b q on Σ . Here C depends only on Ω , M , M , T and n . Corollary 1.9.
Under the same assumptions of Theorem 1.7 and Theorem 1.8, we have that I a ,b “ I a ,b implies that a “ a and b “ b everywhere in Q . The outline of this paper is as follows. Section 2 is devoted to the construction of geometric opticssolutions to the equation (1.1). Using these particular solutions, we establish in Section 3 stability estimatesfor the absorbing coefficient a and the potential b . Section 4 and 5 are devoted to the proof of the results ofthe second and third case respectively. In appendix A, we develop the proof of an analytic technical result.2. C ONSTRUCTION OF GEOMETRIC OPTICS SOLUTIONS
The present section is devoted to the construction of suitable geometrical optics solutions for the dissipa-tive wave equation (1.1), which are key ingredients to the proof of our main results. The construction here isa modification of a similar result in [10]. We shall first state the following lemma which is needed to provethe main statement of this section.
Lemma 2.1. (see [14] ) Let
T, M , M ą , a P L p Q q and b P L p Q q , such that } a } L p Q q ď M and } b } L p Q q ď M . Assume that F P L p , T ; L p Ω qq . Then, there exists a unique solution u to the followingequation (2.3) $’’’’&’’’’% B t u ´ ∆ u ` a p x, t qB t u ` b p x, t q u p x, t q “ F p x, t q in Q,u p x, q “ “ B t u p x, q in Ω ,u p x, t q “ on Σ , M. BELLASSOUED AND I. BEN AÏCHA such that u P C pr , T s ; H p Ω qq X C pr , T s ; L p Ω qq , Moreover, there exists a constant C ą such that (2.4) }B t u p¨ , t q} L p Ω q ` } ∇ u p¨ , t q} L p Ω q ď C } F } L p ,T ; L p Ω qq . Armed with the above lemma, we may now construct suitable geometrical optics solutions to the dissi-pative wave equation (1.1) and to its retrograde problem. For this purpose, we consider ϕ P C p R n q andnotice that for all ω P S n ´ the function φ given by(2.5) φ p x, t q “ ϕ p x ` tω q , solves the following transport equation(2.6) pB t ´ ω ¨ ∇ q φ p x, t q “ . We are now in position to prove the following statement
Lemma 2.2.
Let p a i , b i q P A p M , M q , i “ , . Given ω P S n ´ and ϕ P C p R n q , we consider thefunction φ defined by (2.5). Then, for any λ ą , the following equation (2.7) B t u ´ ∆ u ` a p x, t qB t u ` b p x, t q u “ in Q, admits a unique solution u ` P C pr , T s ; H p Ω qq X C pr , T s ; L p Ω qq , of the following form (2.8) u ` p x, t q “ φ p x, t q A ` p x, t q e iλ p x ¨ ω ` t q ` r ` λ p x, t q , where A ` p x, t q is given by (2.9) A ` p x, t q “ exp ´ ´ ż t a p x ` p t ´ s q ω, s q ds ¯ , and r ` λ p x, t q satisfies (2.10) r ` λ p x, q “ B t r ` λ p x, q “ , in Ω , r ` λ p x, t q “ on Σ . Moreover, there exists a positive constant C ą such that (2.11) λ } r ` λ p¨ , t q} L p Ω q ` }B t r ` λ p¨ , t q} L p Ω q ď C } ϕ } H p R n q . Proof.
We proceed as in the proof of a similar result in [10]. We put g p x, t q “ ´ ´ B t ´ ∆ ` a p x, t qB t ` b p x, t q ¯´ φ p x, t q A ` p x, t q e iλ p x ¨ ω ` t q ¯ . In light of (2.7) and (2.8), it will be enough to prove the existence of r ` “ r ` λ satisfying(2.12) $’’’’’&’’’’’% ´ B t ´ ∆ ` a p x, t qB t ` b p x, t q ¯ r ` “ g p x, t q ,r ` p x, q “ B t r ` p x, q “ ,r ` p x, t q “ , and obeying the estimate (2.11). From (2.6) and using the fact that A ` p x, t q solves the following equation pB t ´ ω ¨ ∇ q A ` p x, t q “ ´ a p x, t q A ` p x, t q , AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 7 we obtain the following identity g p x, t q “ ´ e iλ p x ¨ ω ` t q ´ B t ´ ∆ ` a p x, t qB t ` b p x, t q ¯´ φ p x, t q A ` p x, t q ¯ “ ´ e iλ p x ¨ ω ` t q g p x, t q , where g P L p , T, L p Ω qq . Thus, in view of Lemma 2.1, there exists a unique solution r ` P C pr , T s ; H p Ω qq X C pr , T s ; L p Ω qq , satisfying (2.12). Let us now define by w the following function(2.13) w p x, t q “ ż t r ` p x, s q ds. We integrate the equation (2.12) over r , t s , for t P p , T q . Then, in view of (2.13), we have ´ B t ´ ∆ ` a p x, t qB t ` b p x, t q ¯ w p x, t q “ ż t g p x, s q ds ` ż t ´ b p x, t q ´ b p x, s q ¯ r ` p x, s q ds ` ż t B s a p x, s q r ` p x, s q ds. Therefore, w is a solution to the following equation $’’’’’&’’’’’% ´ B t ´ ∆ ` a p x, t qB t ` b p x, t q ¯ w p x, t q “ F p x, t q ` F p x, t q in Q,w p x, q “ “ B t w p x, q in Ω ,w p x, t q “ on Σ , where F and F are given by(2.14) F p x, t q “ ż t g p x, s q ds, and F p x, t q “ ż t ´ b p x, t q ´ b p x, s q ¯ r ` p x, s q ds ` ż t B s a p x, s q r ` p x, s q ds. Let τ P r , T s . Applying Lemma 2.1 on the interval r , τ s , we get }B t w p¨ , τ q} L p Ω q ď C ´ } F } L p ,T ; L p Ω qq ` T ` M ` M ˘ ż τ ż Ω ż t | r ` p x, s q| ds dx dt ¯ . From (2.13), we get }B t w p¨ , τ q} L p Ω q ď C ´ } F } L p ,T ; L p Ω qq ` ż τ ż t }B s w p¨ , s q} L p Ω q ds dt ¯ ď C ´ } F } L p ,T ; L p Ω qq ` T ż τ }B s w p¨ , s q} L p Ω q ds ¯ . Therefore, from Gronwall’s Lemma, we find out that }B t w p¨ , τ q} L p Ω q ď C } F } L p ,T ; L p Ω qq . As a consequence, in light of (2.13), we conclude that } r ` p¨ , t q} L p Ω q ď C } F } L p ,T ; L p Ω qq . Further, ac-cording to (2.14), F can be written as follows F p x, t q “ ż t g p x, s q ds “ iλ ż t g p x, s qB s p e iλ p x ¨ ω ` s q q ds. M. BELLASSOUED AND I. BEN AÏCHA
Integrating by parts with respect to s , we conclude that there exists a positive constant C ą such that } r ` p¨ , t q} L p Ω q ď Cλ } ϕ } H p R n q . Finally, since } g } L p ,T ; L p Ω qq ď C } ϕ } H p R n q , the energy estimate (2.4) associated to the problem (2.12)yields }B t r ` p¨ , t q} L p Ω q ` } ∇ r ` p¨ , t q} L p Ω q ď C } ϕ } H p R n q . This completes the proof of the lemma. (cid:3)
As a consequence we have the following lemma
Lemma 2.3.
Let p a i , b i q P A p M , M q , i “ , . Given ω P S n ´ and ϕ P C p R n q , we consider thefunction φ defined by (2.5). Then, the following equation (2.15) B t u ´ ∆ u ´ a p x, t qB t u ` b p x, t q u “ in Q, admits a unique solution u ´ P C pr , T s ; H p Ω qq X C pr , T s ; L p Ω qq , of the following form (2.16) u ´ p x, t q “ ϕ p x ` tω q A ´ p x, t q e ´ iλ p x ¨ ω ` t q ` r ´ λ p x, t q , where A ´ p x, t q is given by (2.17) A ´ p x, t q “ exp ´ ż t a p x ` p t ´ s q ω, s q ds ¯ , and r ´ λ p x, t q satisfies (2.18) r ´ λ p x, T q “ B t r ´ λ p x, T q “ , in Ω , r ´ λ p x, t q “ on Σ . Moreover, there exists a constant C ą such that (2.19) λ } r ´ λ p¨ , t q} L p Ω q ` }B t r ´ λ p¨ , t q} L p Ω q ď C } ϕ } H p R n q . Proof.
We prove this result by proceeding as in the proof of Lemma 2.2. Putting r g p x, t q “ ´ ´ B t ´ ∆ ´ a p x, t qB t ` b p x, t q ¯´ φ p x, t q A ´ p x, t q e ´ iλ p x ¨ ω ` t q ¯ . Then, it would be enough to see that if r ´ “ r ´ λ is solution to the following system $’’’’’&’’’’’% ´ B t ´ ∆ ´ a p x, t qB t ` b p x, t q ¯ r ´ p x, t q “ r g p x, t q in Q,r ´ p x, T q “ “ B t r ´ p x, T q in Ω ,r ´ p x, t q “ on Σ , then, r ` p x, t q “ r ´ p x, T ´ t q is a solution to (2.12) with g p x, t q “ r g p x, T ´ t q and a p x, t q , b p x, t q arereplaced by a p x, T ´ t q and b p x, T ´ t q . (cid:3) AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 9
3. D
ETERMINATION OF COEFFICIENTS FROM BOUNDARY MEASUREMENTS
In this section we prove stability estimates for the absorbing coefficient a and the potential b appearing inthe initial boundary value problem (1.1) by the use of the geometrical optics solutions constructed in section2 and the light-ray transform. We assume that Supp ϕ Ă A r , in such a way we haveSupp ϕ X Ω “ H and p Supp ϕ ˘ T ω q X Ω “ H , @ ω P S n ´ . Stability for the absorbing coefficient.
The present section is devoted to the proof of Theorem 1.1.Our goal here is to show that the time dependent coefficient a depends stably on the Dirichlet-to-Neumannmap Λ a,b . In the rest of this section, we define a in R n ` by a “ a ´ a in Q r and a “ on R n ` z Q r . Westart by collecting a preliminary estimate which is needed to prove the main statement of this section.3.1.1. Preliminary estimate.
The main purpose of this section is to give a preliminary estimate, whichrelates the difference of the absorbing coefficients to the Dirichlet-to-Neumann map. Let ω P S n ´ , and p a i , b i q P A such that p a , b q “ p a , b q in Q r z Q r, ˚ . We set a “ a ´ a , b “ b ´ b , and A p x, t q “ p A ´ A ` qp x, t q “ exp ´ ´ ż t a p x ` p t ´ s q ω, s q ds ¯ . Here, we recall the definition of A ´ and A ` A ´ p x, t q “ exp ´ ż t a p x ` p t ´ s q ω, s q ds ¯ , A ` p x, t q “ exp ´ ´ ż t a p x ` p t ´ s q ω, s q ds ¯ . The main result of this section can be stated as follows
Lemma 3.1.
Let p a i , b i q P A p M , M q , i “ , . There exists C ą such that for any ω P S n ´ and ϕ P C p A r q , the following estimate holds true ˇˇˇ ż R n ϕ p y q ” exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ ı dy ˇˇˇ ď C ´ λ } Λ a ,b ´ Λ a ,b } ` λ ¯ } ϕ } H p R n q , for any sufficiently large λ ą . Here C depends only on Ω , T , M and M .Proof. In view of Lemma 2.2, and using the fact that Supp ϕ X Ω “ H , there exists a geometrical opticssolution u ` to the equation $&% B t u ` ´ ∆ u ` ` a p x, t qB t u ` ` b p x, t q u ` “ in Q,u ` p x, q “ B t u ` p x, q “ in Ω , in the following form(3.20) u ` p x, t q “ ϕ p x ` tω q A ` p x, t q e iλ p x ¨ ω ` t q ` r ` λ p x, t q , corresponding to the coefficients a and b , where r ` p x, t q satisfies (2.10), (2.11). Next, let us denote by f λ the function f λ p x, t q “ u ` p x, t q | Σ “ ϕ p x ` tω q A ` p x, t q e iλ p x ¨ ω ` t q . We denote by u the solution of $’’’’&’’’’% B t u ´ ∆ u ` a p x, t qB t u ` b p x, t q u “ in Q,u p x, q “ B t u p x, q “ in Ω ,u “ f λ on Σ . Putting u “ u ´ u ` . Then, u is a solution to the following system(3.21) $’’’’&’’’’% B t u ´ ∆ u ` a p x, t qB t u ` b p x, t q u “ a p x, t qB t u ` ` b p x, t q u ` in Q,u p x, q “ B t u p x, q “ in Ω ,u p x, t q “ on Σ . where a “ a ´ a and b “ b ´ b . On the other hand Lemma 2.3 and the fact that (Supp ϕ ˘ T ω qX Ω “ H ,guarantee the existence of a geometrical optic solution u ´ to the backward problem of (1.1) $&% B t u ´ ´ ∆ u ´ ´ a p x, t qB t u ´ ` p b p x, t q ´ B t a p x, t qq u ´ “ in Q,u ´ p x, T q “ “ B t u ´ p x, T q in Ω , corresponding to the coefficients a and p´B t a ` b q , in the form(3.22) u ´ p x, t q “ ϕ p x ` tω q e ´ iλ p x ¨ ω ` t q A ´ p x, t q ` r ´ λ p x, t q , where r ´ λ p x, t q satisfies (2.18), (2.19). Multiplying the first equation of (3.21) by u ´ , integrating by partsand using Green’s formula, we obtain ż T ż Ω a p x, t qB t u ` u ´ dx dt ` ż T ż Ω b p x, t q u ` u ´ dx dt “ ż T ż Γ p Λ a ,b ´ Λ a ,b qp f λ q u ´ dσ dt. (3.23)On the other hand, by replacing u ` and u ´ by their expressions, we have ż T ż Ω a p x, t qB t u ` u ´ dx dt “ ż T ż Ω a p x, t qB t ϕ p x ` tω q e iλ p x ¨ ω ` t q A ` r ´ λ dx dt ` ż T ż Ω a p x, t q ϕ p x ` tω q e iλ p x ¨ ω ` t q B t A ` r ´ λ dx dt ` ż T ż Ω a p x, t qB t ϕ p x ` tω q ϕ p x ` tω qp A ` A ´ q dx dt ` ż T ż Ω a p x, t q ϕ p x ` tω qB t A ` A ´ dx dt ` iλ ż T ż Ω a p x, t q ϕ p x ` tω q e iλ p x ¨ ω ` t q A ` r ´ λ dx dt ` ż T ż Ω a p x, t q ϕ p x ` tω q e ´ iλ p x.ω ` t q A ´ B t r ` λ dx dt ` iλ ż T ż Ω a p x, t q ϕ p x ` tω qp A ` A ´ q dx dt ` ż T ż Ω a p x, t qB t r ` λ r ´ λ dx dt “ iλ ż T ż Ω a p x, t q ϕ p x ` tω q A dx dt ` I λ , where A “ A ` A ´ . In light of (3.23), we have(3.24) iλ ż T ż Ω a p x, t q ϕ p x ` tω q A p x, t q dx dt “ ż T ż Γ p Λ a ,b ´ Λ a ,b qp f λ q u ´ dσ dt ´ ż T ż Ω b p x, t q u ` u ´ dx dt ´ I λ . Note that for λ sufficiently large, we have(3.25) } u ` u ´ } L p Q q ď C } ϕ } H p R n q , and | I λ | ď C } ϕ } H p R n q . On the other hand, since on Σ , we have u ` “ f λ and r ´ λ “ r ` λ “ , then, we get the following estimate ˇˇˇ ż T ż Γ p Λ a ,b ´ Λ a ,b qp f λ q u ´ dσ dt ˇˇˇ ď } Λ a ,b ´ Λ a ,b } } f λ } H p Σ q } u ´ } L p Σ q ď } Λ a ,b ´ Λ a ,b } } u ` ´ r ` λ } H p Q q } u ´ ´ r ´ λ } H p Q q ď Cλ } Λ a ,b ´ Λ a ,b } } ϕ } H p R n q . (3.26) AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 11
Consequently, by (3.24), (3.25) and (3.26), we obtain ˇˇˇ ż T ż Ω a p x, t q ϕ p x ` tω q A p x, t q dx dt ˇˇˇ ď C ´ λ } Λ a ,b ´ Λ a ,b } ` λ ¯ } ϕ } H p R n q , where A p x, t q “ exp ´ ´ ż t a p x ` p t ´ s q ω, s q ds ¯ . Then, using the fact a p x, t q “ outside Q r, ˚ andmaking this change of variables y “ x ` tω , one gets the following estimation ˇˇˇ ż T ż R n a p y ´ tω, t q ϕ p y q exp ´ ´ ż t a p y ´ sω, s q ds ¯ dy dt ˇˇˇ ď C ´ λ } Λ a ,b ´ Λ a ,b } ` λ ¯ } ϕ } H p R n q . Bearing in mind that ż T ż R n a p y ´ tω, t q ϕ p y q exp ´ ´ ż t a p y ´ sω, s q ds ¯ dy dt “ ´ ż T ż R n ϕ p y q ddt ” exp ´ ´ ż t a p y ´ sω, s q ds ¯ı dy dt “ ´ ż R n ϕ p y q ” exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ ı dy, we conclude the desired estimate given by ˇˇˇ ż R n ϕ p y q ” exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ ı dy ˇˇˇ ď C ´ λ } Λ a ,b ´ Λ a ,b } ` λ ¯ } ϕ } H p R n q . This completes the proof of the lemma. (cid:3)
Stability for the light-ray transform.
The light-ray transform R maps a function f P L p R n ` q defined in R n ` into the set of its line integrals. More precisely, if ω P S n ´ and p x, t q P R n ` , the function R p f qp x, ω q : “ ż R f p x ´ tω, t q dt, is the integral of f over the lines tp x ´ tω, t q , t P R u . The goal in this section is to obtain an estimate thatlinks the light-ray transform of the absorbing coefficient a “ a ´ a to the measurement Λ a ,b ´ Λ a ,b on a precise set. Using the above lemma, we can control the light-ray transform of a as follows: Lemma 3.2.
Let p a i , b i q P A p M , M q , i “ , . There exist C ą , δ ą , β ą and λ ą such thatfor all ω P S n ´ , we have | R p a qp y, ω q| ď C ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ β ¯ , a.e. y P R n , for any λ ě λ . Here C depends only on Ω , T , M and M .Proof. Let ψ P C p R n q be a positive function which is supported in the unit ball B p , q and such that } ψ } L p R n q “ . Define(3.27) ϕ h p x q “ h ´ n { ψ ´ x ´ yh ¯ , where y P A r . Then, for h ą sufficiently small we can verify thatSupp ϕ h X Ω “ H , and Supp ϕ h ˘ T ω X Ω “ H . Moreover, we have ˇˇˇ exp ” ´ ż T a p y ´ sω, s q ds ı ´ ˇˇˇ “ ˇˇˇ ż R n ϕ h p x q ” exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ ı dx ˇˇˇ ď ˇˇˇ ż R n ϕ h p x q ” exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ exp ´ ´ ż T a p x ´ sω, s q ds ¯ı dx ˇˇˇ ` ˇˇˇ ż R n ϕ h p x q ” exp ´ ´ ż T a p x ´ sω, s q ds ¯ ´ ı dx ˇˇˇ . (3.28)Therefore, since we have ˇˇˇ exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ exp ´ ´ ż T a p x ´ sω, s q ds ¯ˇˇˇ ď C ˇˇˇ ż T a p y ´ sω, s q´ a p x ´ sω, s q ds ˇˇˇ , and using the fact that ˇˇˇ ż T ´ a p y ´ sω, s q´ a p x ´ sω, s q ¯ ds ˇˇˇ ď C | y ´ x | , we deduce upon applying Lemma 2.3 with ϕ “ ϕ h the following estimation ˇˇˇ exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ ˇˇˇ ď C ż R n ϕ h p x q | y ´ x | dx ` C ´ λ } Λ a ,b ´ Λ a ,b } ` λ ¯ } ϕ h } H p R n q . On the other hand, we have } ϕ h } H p R n q ď Ch ´ and ż R n ϕ h p x q| y ´ x | dx ď Ch.
So that we end up getting the following inequality ˇˇˇ exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ ˇˇˇ ď C h ` C ´ λ } Λ a ,b ´ Λ a ,b } ` λ ¯ h ´ . Selecting h small such that h “ { λh , that is h “ λ ´ { , we find two constants δ ą and β ą such that ˇˇˇ exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ ˇˇˇ ď C ” λ δ } Λ a ,b ´ Λ a ,b } ` λ β ı . Now, using the fact that | X | ď e M | e X ´ | for any | X | ď M , we deduce that ˇˇˇ ´ ż T a p y ´ sω, s q ds ˇˇˇ ď e M T ˇˇˇ exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ ˇˇˇ . Hence, we conclude that for all y P A r and ω P S n ´ we have ˇˇˇ ż T a p y ´ sω, s q ds ˇˇˇ ď C ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ β ¯ . Since a “ a ´ a “ outside Q r, ˚ , this entails that for all y P A r , and ω P S n ´ , we have(3.29) ˇˇˇˇż R a p y ´ tω, t q dt ˇˇˇˇ ď C ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ β ¯ . Moreover, if y P B p , r { q , we have | y ´ tω | ě | t | ´ | y | ě | t | ´ r . Hence, one can see that p y ´ tω, t q R C ` r if t ą r { . On the other hand, we have p y ´ tω, t q R C ` r if t ď r . Thus, we conclude that p y ´ tω, t q R C ` r Ą Q r, ˚ for t P R . This and the fact that a “ a ´ a “ outside Q r, ˚ , entails that for all y P B p , r { q and ω P S n ´ , we have a p y ´ tω, t q “ , @ t P R . AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 13
By a similar way, we prove for | y | ě T ´ r { , that p y ´ tω, t q R C ´ r Ą Q r, ˚ for t P R and then a p y ´ tω, t q “ . Hence, we conclude that(3.30) ż R a p y ´ tω, t q dt “ , a.e. y R A r , ω P S n ´ . Thus, by (3.29) and (3.30) we finish the proof of the lemma by getting | R p a qp y, ω q| “ ˇˇˇˇż R a p t, y ´ tω q dt ˇˇˇˇ ď C ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ β ¯ , a.e . y P R n , ω P S n ´ . The proof of Lemma 3.2 is complete. (cid:3)
Our goal is to obtain an estimate linking the Fourier transform with respect to p x, t q of the absorbingcoefficient a “ a ´ a to the measurement Λ a ,b ´ Λ a ,b in this set(3.31) E “ tp ξ, τ q P p R n zt R n uq ˆ R , | τ | ă | ξ |u . We denote by p F the Fourier transform of a function F P L p R n ` q with respect to p x, t q : p F p ξ, τ q “ ż R ż R n F p x, t q e ´ ix ¨ ξ e ´ itτ dx dt. We aim for proving that the Fourier transform of a is bounded as follows: Lemma 3.3.
Let p a i , b i q P A p M , M q , i “ , . There exist C ą , δ ą , β ą and λ ą , such thatthe following estimate | p a p ξ, τ q| ď C ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ β ¯ , holds for any p ξ, τ q P E and λ ě λ .Proof. Let p ξ, τ q P E and ζ P S n ´ be such that ξ ¨ ζ “ . Setting ω “ τ | ξ | ¨ ξ ` d ´ τ | ξ | ¨ ζ. Then, one can see that ω P S n ´ and ω ¨ ξ “ τ. On the other hand by the change of variable x “ y ´ tω , wehave for all ξ P R n and ω P S n ´ the following identity ż R n R p a qp y, ω q e ´ iy ¨ ξ dy “ ż R n ´ ż R a p y ´ tω, t q dt ¯ e ´ iy ¨ ξ dy “ ż R ż R n a p x, t q e ´ ix ¨ ξ e ´ itω ¨ ξ dx dt “ p a p ξ, ω ¨ ξ q “ p a p ξ, τ q , where we have set p ξ, τ q “ p ξ, ω ¨ ξ q P E. Bearing in mind that for any t P R , Supp a p¨ , t q Ă Ω Ă B p , r { q ,we deduce that ż R n X B p , r ` T q R p a qp ω, y q e ´ iy ¨ ξ dy “ p a p ξ, τ q . Then, in view of Lemma 3.2, we finish the proof of this lemma. (cid:3)
End of the proof of Theorem 1.1.
We are now in position to complete the proof of Theorem 1.1, usingthe result we have already obtained and an analytic argument that is inspired by [1] and adapted for our case. For ρ ą and κ P p N Y t uq n ` , we put | κ | “ κ ` ... ` κ n ` , B p , ρ q “ t x P R n ` , | x | ă ρ u . We state the following result which is proved in Appendix A (see also [31]).
Lemma 3.4.
Let O be a non empty open set of the unit ball B p , q Ă R d , d ě , and let F be an analyticfunction in B p , q , that satisfy }B κ F } L p B p , qq ď M | κ | ! p ρ q | κ | , @ κ P p N Y t uq d , for some M ą , ρ ą and N “ N p ρ q . Then, we have } F } L p B p , qq ď N M ´ γ } F } γL p O q , where γ P p , q depends on d , ρ and | O | . The above lemma claims conditional stability for the analytic continuation. For classical results forthis type, one can see Lavrent’ev, Romanov and Shishatskiˇı [21]. For a fixed α ą , we set F α p τ, ξ q “ p a p α p ξ, τ qq , for all p ξ, τ q P R n ` . It is easy to see that F α is analytic and we have |B κ F α p ξ, τ q| “ |B κ p a p α p ξ, τ qq| “ ˇˇˇˇ B κ ż R n ` a p x, t q e ´ iα p t,x q¨p ξ,τ q dx dt ˇˇˇˇ “ ˇˇˇˇż R n ` a p x, t qp´ i q | κ | α | κ | p x, t q κ e ´ iα p x,t q¨p ξ,τ q dx dt ˇˇˇˇ . This entails that |B κ F α p ξ, τ q| ď ż R n ` | a p x, t q| α | κ | p| x | ` t q | κ | dx dt ď } a } L p Q r, ˚ q α | κ | p T q | κ | ď C | κ | ! p T ´ q | κ | e α . The, upon applying Lemma 3.4 with M “ Ce α , ρ “ T ´ , and O “ ˚ E X B p , q , where ˚ E “ tp ξ, τ q P R ˆ p R n zt R n uq , | τ | ă | ξ |u , one may find a constant µ P p , q such that we have for all p ξ, τ q P B p , q , the following estimation | F α p ξ, τ q| “ | p a p α p ξ, τ qq| ď Ce α p ´ γ q } F α } γL p O q . Now the idea is to find an estimate for the Fourier transform of a in a suitable ball. Using the fact that α ˚ E “ t α p ξ, τ q , p ξ, τ q P ˚ E u “ ˚ E , we obtain for all p ξ, τ q P B p , α q| p a p ξ, τ q| “ | F α p α ´ p ξ, τ qq| ď Ce α p ´ γ q } F α } γL p O q ď Ce α p ´ γ q } p a } µL p B p ,α qX ˚ E q ď Ce α p ´ γ q } p a } γL p ˚ E q . (3.32)The next step in the proof is to deduce an estimate that links the unknown coefficient a to the measurement Λ a ,b ´ Λ a ,b . To obtain such estimate, we need first to decompose the H ´ p R n ` q norm of a into thefollowing way } a } { γH ´ p R n ` q “ ´ ż |p τ,ξ q|ă α p ` |p τ, ξ q| q ´ | p a p ξ, τ q| dξdτ ` ż |p ξ,τ q|ě α p ` |p τ, ξ q| q ´ | p a p ξ, τ q| dξdτ ¯ { γ ď C ´ α n ` } p a } L p B p ,α qq ` α ´ } a } L p R n ` q ¯ { γ . AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 15
Hence, in light of (3.32) and Lemma 3.3, we get } a } { γH ´ p R n ` q ď C ´ α n ` e α p ´ γ q p λ δ } Λ a ,b ´ Λ a ,b } ` λ β q γ ` α ´ ¯ { γ ď C ´ α n ` γ e α p ´ γ q γ λ β } Λ a ,b ´ Λ a ,b } ` α n ` γ e α p ´ γ q γ λ ´ β ` α ´ { γ ¯ . Let α ą be sufficiently large and assume that α ą α . Setting λ “ α n ` µγ e α p ´ µ q µγ , and using the fact that α ą α , one can see that λ ą λ and α n ` γ e α p ´ γ q γ λ ´ β “ α ´ { γ . This entails that } a } { γH ´ p R n ` q ď C ´ α β p n ` q` δ p n ` q βγ e α p β ` δ qp ´ γ q βγ } Λ a ,b ´ Λ a ,b } ` α ´ { γ ¯ ď C ´ e Nα } Λ a ,b ´ Λ a ,b } ` α ´ { γ ¯ , where N depends on δ, β, n, and γ . The next step is to minimize the right hand-side of the above inequalitywith respect to α . We need to take α sufficiently large. So, there exists a constant m ą such that if ă } Λ a ,b ´ Λ a ,b } ă m , and α “ N | log } Λ a ,b ´ Λ a ,b } | , then, we have the following estimation } a } H ´ p Q r, ˚ q ď } a } H ´ p R n ` q ď C ´ } Λ a ,b ´ Λ a ,b } ` | log } Λ a ,b ´ Λ a ,b } | ´ { γ ¯ γ { ď C ´ } Λ a ,b ´ Λ a ,b } γ { ` | log } Λ a ,b ´ Λ a ,b }| ´ ¯ . (3.33)Now if } Λ a ,b ´ Λ a ,b } ě m , we have } a } H ´ p Q r, ˚ q ď C } a } L p Q r, ˚ q ď CM c γ { m γ { ď CMm γ { } Λ a ,b ´ Λ a ,b } γ { , hence (3.33) holds. Let us now consider θ ą such that p : “ s ´ “ n ` ` θ. Use Sobolev’s embeddingtheorem we find by interpolating } a } L p Q r, ˚ q ď C } a } H n ` θ p Q r, ˚ q ď C } a } ´ ηH ´ p Q r, ˚ q } a } ηH s ´ p Q r, ˚ q ď C } a } ´ ηH ´ p Q r, ˚ q , for some η P p , q . This completes the proof of Theorem 1.1. This will be a key ingredient in proving theresult of the next section.3.2. Stability for the potential.
This section is devoted to the proof of Theorem 1.2. By means of thegeometrical optics solutions constructed in Section 2, we will show using the stability estimate we havealready obtained for the absorbing coefficient a , that the time dependent potential b depends stably on theDirichlet-to-Neumann map Λ a,b . As before, given ω P S n ´ , p a i , b i q P A p M , M q such that p a , b q “p a , b q in Q r z Q r, ˚ , we set a “ a ´ a , b “ b ´ b and A p x, t q “ p A ´ A ` qp x, t q “ exp ´ ´ ż t a p x ` p t ´ s q ω, s q ds ¯ , where A ´ and A ` are given by A ´ p x, t q “ exp ´ ż t a p x ` p t ´ s q ω, s q ds ¯ , A ` p x, t q “ exp ´ ´ ż t a p x ` p t ´ s q ω, s q ds ¯ . In the rest of this section, we define b in R n ` by b “ b ´ b in Q r and b “ on R n ` z Q r . We start bygiving a preliminary estimate that will be used to prove the main statement of this section. Lemma 3.5.
Let p a i , b i q P A p M , M q , i “ , . There exists C ą such that for any ω P S n ´ and ϕ P C p A r q , the following estimate holds ˇˇˇż T ż R n b p y ´ tω, t q ϕ p y q dy dt ˇˇˇ ď C ´ λ } Λ a ,b ´ Λ a ,b } ` λ } a } L p Q r, ˚ q ` λ ¯ } ϕ } H p R n q , for any λ ą sufficiently large. Here C depends only on Ω , M , M and T .Proof. We start with the identity (3.23), except this time we will isolate the electric potential ż T ż Ω b p x, t q u ` u ´ dx dt “ ż T ż Γ p Λ a ,b ´ Λ a ,b qp f λ q u ´ dσ dt ´ ż T ż Ω a p x, t qB t u ` u ´ dx dt. By replacing u ` and u ´ by their expressions we get ż T ż Ω b p x, t q ϕ p x ` tω q A p x, t q dx dt “ ż T ż Γ p Λ a ,b ´ Λ a ,b qp f λ q u ´ dσ dt ´ ż T ż Ω b p x, t q ϕ p x ` tω q A ´ p x, t q e ´ iλ p x ¨ ω ` t q r ` λ p x, t q dx dt ´ ż T ż Ω b p x, t q r ` λ p x, t q r ´ λ p x, t q dx dt ´ ż T ż Ω a p x, t qB t u ` u ´ dx dt ´ ż T ż Ω b p x, t q ϕ p x ` tω q A ` p x, t q e iλ p x ¨ ω ` t q r ´ p x, t q dx dt “ ż T ż Γ p Λ a ,b ´ Λ a ,b qp f λ q u ´ dσ dt ` I λ . (3.34)Then, in view of (3.34), we have ż T ż Ω b p x, t q ϕ p x ` tω q dxdt “ ż T ż Ω b p x, t q ϕ p x ` tω qp ´ A q dxdt ` ż T ż Γ p Λ a ,b ´ Λ a ,b qp f λ q u ´ dσdt ` I λ . From (2.11), (2.19) and using the fact that a “ a ´ a “ outside Q r, ˚ , we find(3.35) | I λ | ď C ´ λ } a } L p Q r, ˚ q ` λ ¯ } ϕ } H p R n q . By the trace theorem, we get ˇˇˇ ż T ż Γ p Λ a ,b ´ Λ a ,b qp f λ q u ´ dσ dt ˇˇˇ ď } Λ a ,b ´ Λ a ,b }} f λ } H p Σ q } u ´ } L p Σ q ď Cλ } Λ a ,b ´ Λ a ,b } } ϕ } H p R n q . (3.36)On the other hand, we have(3.37) ˇˇˇ ż T ż Ω b p x, t q ϕ p x ` tω qp ´ A q dx dt ˇˇˇ ď C } a } L p Q r, ˚ q } ϕ } H p R n q . Then, in light of (3.35)-(3.37), taking to account that b “ b ´ b “ outside Q r, ˚ and using the change ofvariables y “ x ` tω we get ˇˇˇ ż T ż R n b p y ´ tω, t q ϕ p y q dy dt ˇˇˇ ď C ´ λ } Λ a ,b ´ Λ a ,b } ` λ } a } L p Q r, ˚ q ` λ ¯ } ϕ } H p R n q . This completes the proof of the Lemma. (cid:3)
Now the idea is to deduce an estimate for the light ray transform of the time-dependent unknown coeffi-cient b in order to control thereafter its Fourier transform. AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 17
Lemma 3.6.
Let p a i , b i q P A p M , M q , i “ , . There exists C ą , δ ą , β ą and λ ą such thatfor all ω P S n ´ , the following estimate holds ˇˇˇ R p b qp y, ω q ˇˇˇ ď C ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ δ } a } L p Q r, ˚ q ` λ β ¯ , a. e y P R n , for any λ ą λ . Here C depends only on Ω , T , M and M .Proof. We proceed as in the proof of Lemma 3.2. We consider the sequence p ϕ h q h defined by (3.27) with y P A r . Since we have ˇˇˇ ż T b p y ´ tω, t q dt ˇˇˇ “ ˇˇˇ ż T ż R n b p y ´ tω, t q ϕ h p x q dx dt ˇˇˇ ď ˇˇˇ ż T ż R n b p x ´ tω, t q ϕ h p x q dx dt ˇˇˇ ` ˇˇˇ ż T ż R n ´ b p y ´ tω, t q ´ b p x ´ tω, t q ¯ ϕ h p x q dx dt ˇˇˇ . Then, by applying Lemma 3.5 with ϕ “ ϕ h , and since | b p y ´ tω, t q ´ b p x ´ tω, t q| ď C | y ´ x | , we obtain ˇˇˇ ż T b p y ´ tω, t q dt ˇˇˇ ď C ´ λ } Λ a ,b ´ Λ a ,b } ` λ } a } L p Q r, ˚ q ` λ ¯ } ϕ h } H p R n q ` C ż R n | x ´ y | ϕ h p x q dx. On the other hand, since } ϕ h } H p R n q ď Ch ´ and ż R n | x ´ y | ϕ h p x q dx ď C h , we conclude that ˇˇˇ ż T b p y ´ tω, t q dt ˇˇˇ ď C ´ λ } Λ a ,b ´ Λ a ,b } ` λ } a } L p Q r, ˚ q ` λ ¯ h ´ ` C h.
Selecting h small such that h “ h ´ { λ . Then, we find two constants δ ą and β ą such that ˇˇˇ ż T b p y ´ tω, t q ˇˇˇ ď C ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ δ } a } L p Q r, ˚ q ` λ β ¯ . Using the fact that b “ b ´ b “ outside Q r, ˚ , we then conclude that for all y P A r and ω P S n ´ , ˇˇˇ ż R b p y ´ tω, t q dt ˇˇˇ ď C ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ δ } a } L p Q r, ˚ q ` λ β ¯ . Next, by arguing as in the derivation of Lemma 3.2, we end up upper bounding the light-ray transform of b ,for all y P R n . (cid:3) At this point, it is convenient to recall that our goal is to obtain an estimate for the Fourier transform of b in a precise set. So, by proceeding by a similar way as in the previous section, we get this result. Lemma 3.7.
Let p a i , b i q P A p M , M q , i “ , . There exists C ą , δ ą , β ą and λ ą , such thatthe following estimate | p b p ξ, τ q| ď C ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ δ } a } L p Q r, ˚ q ` λ β ¯ , a. e , p ξ, τ q P E, for any λ ą λ . Here C depends only on Ω , T , M and M . Next, using the above estimation as well as the analytic continuation argument, that is Lemma 3.4, weupper bound the Fourier transform of b in a suitable ball B p , α q as follows(3.38) | p b p ξ, τ q| ď Ce α p ´ γ q ´ λ δ } Λ a ,b ´ Λ a ,b } ` λ δ } a } L p Q r, ˚ q ` λ β ¯ γ , for some γ P p , q and where α ą is assumed to be sufficiently large. Then, in order to deduce anestimate linking the unknown coefficient b to the measurement Λ a ,b ´ Λ a ,b , we control the H ´ p R n ` q norm of b as follows } b } γ H ´ p R n ` q ď C ” α n ` } p b } L p B p ,α qq ` α ´ } b } L p R n ` q ı γ . So, by the use of (3.38), we obtain the following inequality(3.39) } b } γ H ´ p R n ` q ď C ” α n ` γ e α p ´ γ q γ ´ λ δ ǫ ` λ δ } a } L p Q r, ˚ q ` λ ´ β ¯ ` α ´ γ ı , where we have set ǫ “ } Λ a ,b ´ Λ a ,b } . In light of Theorem 1.1, one gets } b } γ H ´ p R n ` q ď C ” α n ` γ e α p ´ γ q γ ´ λ δ ǫ ` λ δ ǫ µ µ ` λ δ | log ǫ | ´ µ ` λ ´ β ¯ ` α ´ γ ı , for some γ, µ , µ P p , q and δ, β ą . Let α ą be sufficiently large and we take α ą α . Setting λ “ α n ` γβ e α p ´ γ q γβ . By α ą α , we can assume λ ą λ . Therefore, the estimate (3.39) yields } b } γ H ´ p R n ` q ď C ” e Nα ` ǫ ` ǫ s ` | log ǫ | ´ µ ˘ ` α ´ γ ı , for some s, µ , µ P p , q , and where N is depending on n, γ, δ and β . Thus, if ǫ is small, we have(3.40) } b } γ H ´ p R n ` q ď C ´ e Nα | log ǫ | ´ µ ` α ´ γ ¯ . In order to minimize the right hand side of the above inequality with respect to α , we need to take α sufficiently large. So, we select α as follows α “ N log | log ǫ | µ , where we have assumed that ǫ ă c ď . Then, the estimate (3.40) yields } b } H ´ p Q r, ˚ q ď } b } H ´ p R n ` q ď C ´ log | log } Λ a ,b ´ Λ a ,b }| µ ¯ ´ . This completes the proof of Theorem 1.2.4. D
ETERMINATION OF COEFFICIENTS FROM BOUNDARY MEASUREMENTS AND FINAL DATA
In this section, we prove Theorem 1.4 and 1.5. We will extend the stability estimates obtained in the firstcase to a larger region Q r, Ą Q r, ˚ . We shall consider the geometric optics solutions constructed in Section2, associated with a function ϕ obeying supp ϕ X Ω “ H . Note that this time, we have more flexibility onthe support of the function ϕ and we don’t need to assume that supp ϕ ˘ T ω X Ω “ H anymore. We recallthat the observations in this case are given by the following operator R a,b : H p Σ q ÝÑ K f ÞÝÑ pB ν u, u p¨ , T q , B t u p¨ , T qq , associated to the problem (1.1) with p u , u q “ p , q . We denote by R a,b p f q “ B ν u, R a,b p f q “ u p¨ , T q , R a,b p f q “ B t u p¨ , T q . AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 19
Stability for the absorbing coefficient.
In this section we will prove that the absorbing coefficient a can be stably recovered in a larger region if we further know the final data of the solution u of the dissipativewave equation (1.1). In the rest of this section, we define a “ a ´ a in Q r and a “ on R n ` z Q r . Weshall first prove the following statement Lemma 4.1.
Let p a i , b i q P A p M , M q , i “ , . Let ϕ P C p R n q be such that supp ϕ X Ω “ H . Thereexists C ą , such that for any ω P S n ´ , the following estimate holds ˇˇˇ ż R n ϕ p y q ” exp ´ ´ ż T a p y ´ sω, s q ds ¯ ´ ı dy ˇˇˇ ď C ´ λ } R a ,b ´ R a ,b } ` λ ¯ } ϕ } H p R n q . Here C depends only on Ω , T , M and M .Proof. In view of Lemma 2.1 and using the fact that supp ϕ X Ω “ H , there exists a geometrical opticsolution u ` to the wave equation $’&’% ´ B t ´ ∆ ` a p x, t qB t ` b p x, t q ¯ u ` “ in Q,u ` p x, q “ B t u ` p x, q “ in Ω , in the following form(4.41) u ` p x, t q “ ϕ p x ` tω q A ` p x, t q e iλ p x ¨ ω ` t q ` r ` λ p x, t q , corresponding to the coefficients a and b , where r ` λ p x, t q satisfies (2.10) and (2.11). We denote f λ p x, t q “ u ` p x, t q | Σ “ ϕ p x ` tω q A ` p x, t q e iλ p x ¨ ω ` t q . Let u be the solution of $’’’’&’’’’% B t u ´ ∆ u ` a p x, t qB t u ` b p x, t q u “ in Q,u p x, q “ B t u p x, q “ in Ω ,u “ f λ on Σ . Putting u “ u ´ u ` . Then, u is a solution to the following system(4.42) $’’’’&’’’’% B t u ´ ∆ u ` a p x, t qB t u ` b p x, t q u “ a p x, t qB t u ` ` b p x, t q u ` in Q,u p x, q “ B t u p x, q “ in Ω ,u p x, t q “ on Σ , where a “ a ´ a and b “ b ´ b . On the other hand, Lemma 2.3 guarantees the existence of a geometricaloptic solution u ´ to the adjoint problem B t u ´ ´ ∆ u ´ ´ a p x, t qB t u ´ ` p b p x, t q ´ B t a p x, t qq u ´ “ in Q, corresponding to the coefficients a and p´B t a ` b q , in the form(4.43) u ´ p x, t q “ ϕ p x ` tω q e ´ iλ p x ¨ ω ` t q A ´ p x, t q ` r ´ λ p x, t q , where r ´ λ p x, t q satisfies (2.18) and (2.19). Multiplying the first equation of (4.42) by u ´ , integrating by partsand using Green’s formula, we get ż T ż Ω a p x, t qB t u ` u ´ dxdt “ ż T ż Γ p R a ,b ´ R a ,b qp f λ q u ´ p x, t q dσdt ´ ż Ω p R a ,b ´ R a ,b qp f λ q u ´ p x, T q dx ´ ż Ω p R a ,b ´ R a ,b qp f λ q ” a p x, T q u ´ p x, T q ´ B t u ´ p x, T q ı dx ´ ż T ż Ω b p x, t q u ` p x, t q u ´ p x, t q dx dt. (4.44)By replacing u ` and u ´ by their expressions, using (3.25) and the Cauchy-Schwartz inequality, we obtain ˇˇˇ ż T ż Ω a p x, t q ϕ p x ` tω q A p x, t q dx dt ˇˇˇ ď Cλ ”´ } u ´ } L p Σ q ` } u ´ p¨ , T q} L p Ω q ` }B t u ´ p¨ , T q} L p Ω q ¯ ´ }p R a ,b ´ R a ,b qp f λ q} L p Σ q `}p R a ,b ´ R a ,b qp f λ q} H p Ω q `}p R a ,b ´ R a ,b qp f λ q} L p Ω q ¯ `} ϕ } H p R n q ı . Then, by setting φ λ “ ´ u ´| Σ , u ´ p¨ , T q , B t u ´ p¨ , T q ¯ , one can see that ˇˇˇ ż T ż Ω a p x, t q ϕ p x ` tω q A p x, t q dx dt ˇˇˇ ď Cλ ´ } R a ,b ´ R a ,b }} f λ } H p Σ q } φ λ } K ` } ϕ } H p R n q ¯ . Therefore, by the trace theorem we get ˇˇˇ ż T ż Ω a p x, t q ϕ p x ` tω q A p x, t q dx dt ˇˇˇ ď C ´ λ } R a ,b ´ R a ,b } ` λ ¯ } ϕ } H p R n q . Finally, we use the fact that a “ a ´ a “ outside Q r, and we complete the proof of the lemma byarguing as in the proof of Lemma 3.1. (cid:3) Next, by considering the sequence ϕ h defined by (3.27) with y R Ω , taking to account that a “ a ´ a “ outside Q r, and arguing as in Section 3.1, we complete the proof of Theorem 1.4.4.2. Stability for the potential.
We are now in position to prove Theorem 1.5. We aim to show by the useof Theorem1.4, that the potential b can be stably recovered in the region Q r, , with respect to the operator R a,b . In the rest of this section, we define b in R n ` by b “ b ´ b in Q r and b “ on R n ` z Q r . Lemma 4.2.
Let p a i , b i q P A p M , M q , i “ , . There exists C ą such that for any ω P S n ´ and ϕ P C p R n q such that supp ϕ X Ω “ H , the following estimate holds ˇˇˇ ż T ż R n b p y ´ tω, t q ϕ p y q dy dt ˇˇˇ ď C ´ λ } R a ,b ´ R a ,b } ` λ } a } L p Q r, q ` λ ¯ } ϕ } H p R n q , where C depends only on Ω , M , M and T .Proof. We start with the identity (4.44), except this time we isolate the potential b , we get ż T ż Ω b p x, t q u ` u ´ dxdt “ ż T ż Γ p R a ,b ´ R a ,b qp f λ q u ´ p x, t q dσdt ´ ż Ω p R a ,b ´ R a ,b qp f λ q u ´ p x, T q dx ´ ż Ω p R a ,b ´ R a ,b qp f λ q ” a p x, T q u ´ p x, T q ´ B t u ´ p x, T q ı dx ´ ż T ż Ω a p x, t qB t u ` p x, t q u ´ p x, t q dx dt. So, by replacing u ` and u ´ by their expressions, taking to account (3.35), (3.37) and the fact that a “ a ´ a “ outside Q r, , and making the change of variables y “ x ` tω , we obtain ˇˇˇ ż T ż R n b p y ´ tω, t q ϕ p y q dy dt ˇˇˇ ď C ´ λ } R a ,b ´ R a ,b } ` λ } a } L p Q r, q ` λ ¯ } ϕ } H p R n q . AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 21
This completes the proof of the lemma. (cid:3)
In order to complete the proof of Theorem 1.5, it will be enough to consider the sequence p ϕ h q definedby (3.27), with, y R Ω , use the fact b “ b ´ b “ outside Q r, and repeat the same arguments of Section3.2 5. D ETERMINATION OF COEFFICIENTS FROM BOUNDARY MEASUREMENTS AND FINAL DATA BYVARYING THE INITIAL DATA
In the present section, we deal with the same inverse problem, except the set of data, in this case, is madeof the responses of the medium for all possible initial data. For p a i , b i q P C p Q q ˆ C p Q q , i “ , , wedefine p a, b q “ p a ´ a , b ´ b q in Q and p a, b q “ p , q on R n ` z Q . By proceeding as in the derivation ofTheorem 1.1 and Theorem 1.4, we prove a log -type stability estimate in the determination of the absorbingcoefficient a over the whole domain Q , from the knowledge of the measurement I a,b .To prove such estimate, we proceed as in Section 3.1 and 4.1, except this time, we have more flexibilityon the support of the function ϕ h defined by (3.27). Namely, we don’t need to impose any condition on itssupport anymore (we fix y P R n ).The same thing for the determination of the time-dependent potential b . we argue as in Section 3.2 and4.2 to prove a log - log -type stability estimate in recovering the time dependent coefficient b with respect tothe operator I a,b , over the whole domain Q .A PPENDIX
A. P
ROOF OF L EMMA
Lemma A.1.
Let J be an open interval in r´ , s , and g be an holomorphic function in the unit disc D p , q Ă C satisfying (A.45) | g p z q| ď , | z | ă . Then, there exist γ P p , q and N ą such that the following estimate holds } g } L p B p , qq ď N } g } γL p J q , where N and γ are depending only on | J | .Proof. We should first notice that for all n ě , there exist p n ` q points such that ´ ď x ă ... ă x n ď , with x i P J , i “ , .., n , and satisfying the following estimation(A.46) x i ´ x i ´ ě | J | n ` , for i “ , ..., n. Let z P C . We denote by P n p z q “ n ÿ i “ g p x i q ź j ‰ i p z ´ x j q ź j ‰ i p x i ´ x j q ´ . In order to prove this lemma, we need first to find an upper bound for | P n p z q| . To do that we first notice thatfor l ą l we have x l ´ x l “ l ÿ i “ l ` p x i ´ x i ´ q . Hence, (A.46) entails that $’&’% p x j ´ x i q ě p j ´ i q | J | n ` j ą i, p x i ´ x j q ě p i ´ j q | J | n ` j ă i. As a consequence we have the following estimation ź j ‰ i | x i ´ x j | ě i ´ ź j “ p i ´ j q | J |p n ` q n ź j “ i ` p j ´ i q | J |p n ` q ě i ! | J | i p n ` q i p n ´ i q ! | J | n ´ i p n ` q n ´ i . (A.47)On the other hand, it is easy to see that for | z | ď and x j P J , j “ , .., n , we have ź j ‰ i | z ´ x j | ď ź j ‰ i p| z | ` | x j |q ď , Putting this together with (A.47), we end up getting this result(A.48) | P n p z q| ď n ÿ i “ C in p n ` q n n ! | E | n } g } L p J q ď e ˆ | J | ˙ n } g } L p J q . The next step of the proof is to control | g p z q ´ P n p z q| . For this purpose, let us introduce the followingfunction: for all ξ P C , such that | ξ | “ , we denote by G p ξ q “ g p ξ qp ξ ´ z q ´ n ź j “ p ξ ´ x j q ´ . Applying the residue Theorem, one obtains the following identity iπ ż | ξ |“ G p ξ q dξ “ ˜ Res p G, z q ` n ÿ k “ Res p G, x k q ¸ “ ´ g p z q ´ P n p z q ¯ n ź j “ p z ´ x j q ´ . From this and the hypothesis (A.45), it follows that for | z | ď , and x i P J , we have(A.49) | g p z q ´ P n p z q| ď ˆ ` ˙ n ` ´ ´ ¯ ´p n ` q ď ˆ ˙ n . Combining (A.48) with (A.49), one gets || g || L p B p , { qq ď ˆ ˙ n ` e ˆ | J | ˙ n } g } L p J q , n ě . To complete the proof of the lemma, we need to minimize the right hand side of the last estimate with respectto n . To this end, let us define the following function ψ p x q “ e ´ x log p { q ` e } g } L p J q e x log p {| J |q , x P R . A simple calculation show that the function ψ reaches a minimum at this point x “ ” log ´ | J | ¯ı ´ log ” log p { q e } g } L p J q log p {| J |q ı . Then, we end up getting the desired result. (cid:3)
AVE EQUATION WITH TIME-DEPENDENT COEFFICIENTS 23
We move now to establish the second result by the use of Hadamard’s three-circle theorem and LemmaA.1.
Lemma A.2.
Let ϕ be an analytic function in r´ , s , and I an open interval in r´ , s . We assume thatthere exist positive constants M and ρ such that (A.50) | ϕ p k q p s q| ď M k ! p ρ q k , k ě , s P r´ , s . Then, there exist N “ N p ρ, | I |q and γ “ γ p ρ, | I |q such that we have (A.51) | ϕ p s q| ď N } ϕ } γL p I q M ´ γ , for any s P r´ , s . Proof.
In light of (A.50), we have for all s P r´ , s , ˇˇˇ ÿ k ě ϕ p k q p s q k ! p z ´ s q k ˇˇˇ ď ÿ k ě M p ρ q ´ k | z ´ s | k . This entails that for all s P r´ , s and for all z P B p s, ρ q , we have the following estimation ˇˇˇ ÿ k ě ϕ p k q p s q k ! p z ´ s q k ˇˇˇ ď M ÿ k ě p ρ q ´ k ρ k ď M, (A.52)which implies that ϕ can be extended to an holomorphic function in D ρ “ Y B p s, ρ q for ´ ď s ď . Weneed first to construct a specific open interval in r´ , s to apply Lemma A.1. To this end, we notice that(A.53) r´ , s Ă ď ď j ď n I j “ ď ď j ď n ” s j ´ ρ , s j ` ρ ” , where we have putted s j “ ´ ` p j ´ q ρ { , { ρ ď n ď { ρ ` { and assumed that I j X I j “ H ,for all j, j “ , ...n , j ‰ j . Therefore, the open interval I can be written as the meeting of p I j X I q , for ď j ď n where p I j X I q č j ‰ j p I j X I q “ H , for j, j “ , ..., n . Now, we fix j P t , ..., n u such that | I j X I | “ max ď j ď n | I j X I | . We define J s j ,ρ “ ρ p I j X I ´ s j q . In light of (A.53) , we deduce that J s j ,ρ is an open interval of r´ , s . Next, we consider the function g defined on D p , q as follows g p z q “ ϕ p s j ` ρz q M .
The estimate (A.52) entails that | g p z q| ď for | z | ď . Bearing in mind that the function g is holomorphicin the unit disc, we deduce from Lemma A.1 the existence of two constants N “ N p| I |q and γ “ γ p| I |q such that the following estimate holds } g } L p B p , { qq ď N } g } γL p J sj ,ρ q ď N p M q ´ γ } ϕ } L p I j X I q . This combined with the fact that } g } L p B p , { qq “ p M q ´ } ϕ } L p B p s j ,ρ { qq yield the following result(A.54) } ϕ } L p B p s j ,ρ { qq ď N } ϕ } γL p I q M ´ γ . Now, we aim to extend this result to the interval r´ , s . To this end, let r ą , satisfying(A.55) ρ ď r ď r ď ρ. Let p a j q j ě “ p s j q j ě be a sequence such that r´ , s Ă ď ď j ď n B p a j , r q and satisfying(A.56) $&% B p a j ` , r q Ă B p a j , r q for j P t j , ..., n u B p a j ´ , r q Ă B p a j , r q for j P t , ..., j u . In view of Hamdamard’s three-circle theorem, using (A.54) and (A.55) we get } ϕ } L p B p a j , r qq ď } ϕ } θL p B p a j , ρ qq } ϕ } ´ θL p B p a j ,ρ qq ď N } ϕ } γL p I q M ´ γ , (A.57)where θ “ log ρ { r log 2 . Then, using the fact that B p a j ` , r q Ă B p a j , r q for j P t j , ..., n u , we deduce } ϕ } L p B p a j ` ,r qq ď } ϕ } L p B p a j , r qq ď N } ϕ } γL p I q M ´ γ . From this and Hadamard’s three-circle theorem, we obtain } ϕ } L p B p a j ` , r qq ď } ϕ } θ L p B p a j ` ,r qq } ϕ } ´ θ L p B p a j ` ,ρ q ď N } ϕ } γL p I q M ´ γ , where θ “ log ρ { r log ρ { r . So, from (A.56) and a repeated application of Hadamard’s three circle theorem, we get } ϕ } L p B p a j , r qq ď N } ϕ } γL p I q M ´ γ , j P t j ` , ...n u . By a similar way, we prove that } ϕ } L p B p a j , r qq ď N } ϕ } γL p I q M ´ γ , j P t , ...j u . As a consequence, we obtain } ϕ } L pr´ , sqq ď n ÿ j “ } ϕ } L p B p a j , r qq ď N } ϕ } γL p I q M ´ γ . This completes the proof of the Lemma. (cid:3)
A.1.
Proof of Lemma 3.4.
Notice first that there exists a sequence of open intervals p I j q j such that E “ I ˆ ... ˆ I j ˆ ... ˆ I d Ă O Ă B p , q . Let x “ p x , x , ..., x d q be fixed in B(0,1). We consider the analytic function ϕ j defined as follows(A.58) ϕ j p s q “ F p x , ..., x j ´ , s, x j ` , ..., x d q , s P r´ , s . Assume that there exist positive constants M and ρ such that | ϕ j p s q p k q | ď M k ! p ρ q k , s P r´ , s . Then, in view of lemma A.2, we conclude the existence of of N “ N p ρ, | I j |q and γ “ γ p ρ, | I |q such thatwe have | ϕ j p s q| ď N } ϕ j } γ j L p I j q M ´ γ j , s P r´ , s , This and (A.58) yield(A.59) | F p x q| ď N j sup x j P I j | F p x q| γ j M ´ γ j . Therefore, by iterating (A.59), we get | F p x q| ď N N γ ...N γ ...γ d ´ d sup x P E | F p x q| γ ...γ d M ´ γ ...γ d . This completes the of the lemma.
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