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Conditionally stable unique continuation and applications to thermoacoustic tomography

  • We prove a conditional Hölder stability estimate for the Cauchy problem on the lateral boundary for the wave equation under a strictly convex foliation condition. We apply this estimate for the problem in multiwave tomography with partial data.

    Citation: Plamen Stefanov. Conditionally stable unique continuation and applications to thermoacoustic tomography[J]. Mathematics in Engineering, 2019, 1(4): 789-799. doi: 10.3934/mine.2019.4.789

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  • We prove a conditional Hölder stability estimate for the Cauchy problem on the lateral boundary for the wave equation under a strictly convex foliation condition. We apply this estimate for the problem in multiwave tomography with partial data.


    Let ΩRn be a bounded domain with a smooth boundary. The purpose of this note is to formulate geometric conditions for conditionally stable unique continuation of solutions of wave type of equations from Cauchy data on a part S of R×Ω into some subset (which we call Qε below) of R×Ω. We want to emphasize that the stability is of Hölder type, local, and conditional. Such type of stability holds even for elliptic second order equations; while in classical sense, that problem is unstable, of course. On the other hand, unique continuation is not conditionally Hölder stable in general: One has weak logarithmic stability only [2,3]. The behavior of the geodesic flow plays a critical role. The subdomain Qε of R×Ω where the conditional estimate holds is smaller that the domain of influence where we have unique continuation, and in particular has the property that every zero bicharacteristic (projected to the base) through it hits S. It remains an open question if that characterizes the optimal Qε. The proof is based on an estimate of Carleman type established in [7,10].

    We consider the case where a subdomain of Ω can be foliated by strictly convex hypersurfaces covering in particular a part Γ of Ω, see Figure 1. This is connected to the existence of a strictly convex function ρ as shown in [13]. Such assumptions have been found to be useful in Control Theory, see, e.g., [26] but they become more important with the recent progress in the local inversion of the geodesic X-ray transform in dimensions n3 [20,27] and the boundary and the lens rigidity problems [18,19]. We choose a suitable pseudo-convex function in R×Ω related to ρ which would allow us to apply the results in [7,10].

    Figure 1.  Left: R×Ω and ϕ1(0) in time-space. Right: The foliation in Ω.

    We present an application to multiwave tomography proving conditional local Hölder stability from partial measurements in Section 4.

    We start with a short review of the elliptic case. Let Δg be the Laplacian associated with a smooth Riemannian metric g in ˉΩ. Let ΓΩ be a relatively open subset of the boundary. We are interested in the stability of the following boundary value problem

    Δgu=fin Ω,u|Γ=h0,νu|Γ=h1, (1)

    where ν is the unit (in the metric) external normal. Note that (1) may not be solvable with any prescribed f, h0 and h1, even if f=0. On the other hand, the possible solution is unique in Ω, by classical unique continuation, which also follows from the theorem below.

    Then we have the following conditional stability theorem.

    Theorem 2.1. [10,Theorem 3.3.1] For every domain Ω0 with ˉΩ0ΩΓ, there exists κ(0,1), C>0, so that for any solution uH1(Ω) of (1) satisfies

    uH1(Ω0)C(F+u1κH1(Ω)Fκ) (2)

    with

    F:=fL2(Ω)+h0H1(Γ)+h1L2(Γ). (3)

    This theorem implies unique continuation in particular: If f, h0 and h1 all vanish, then u=0. It also implies conditional stability in the following sense: If u is a priori bounded in H1(Ω), and when F is uniformly bounded (the real interest in (3) is when F1), then

    uH1(Ω0)CFκ,

    where C depend on those a priori bounds. The result holds if we add lower order terms to Δg with real coefficients. Note that the geometry of g does not play an apparent role.

    The estimate is obtained with Carleman estimates and the existence of a positive constant κ comes from an optimization of the large parameter in the Carleman estimate and gets small, as Ω0 gets closer to Ω. This means decreased stability.

    Such an estimate may look unexpected since the Cauchy problem for elliptic equations is a classical example of a unstable problem. The reason it holds is the a priori assumption that u is bounded in the larger Ω where u can grow even more. The following example illustrates this fact.

    Example 1. Let Ω=(0,π)×(0,1) and consider the following family of harmonic functions

    u=sin(mx)sinh(my)

    with m=1,2,. Let Γ be the lower side of the square. Then u vanishes on Γ (and on the lateral sides of the square) and h1:=uy|y=0=msin(mx). Therefore, h0=0, h1L(Γ)=m. It is straightforward to show that uH1(Ω)Cem, therefore,

    m1emh1H1(Γ)CuH1(Ω),

    and there is no stability since m can be arbitrary large. It is also clear that even if we use different Sobolev norms, the conclusion would be the same.

    On the other hand, if Ωε=(0,π)×(0,1ε) with 0<ε<1, then it is easy to see that uH1(Ωε)Ce(1ε)m, as m. Then in (2),

    FCm,uH1(Ωε)Ce(1ε)m,uH1(Ω)Cem.

    Therefore, (2) holds for Ω0=Ωε with 0<κε and C=C(ε).

    Similar result holds for analytic continuation as well (analytic functions solve the elliptic ˉ equation). Even though analytic continuation is unstable, there is a a conditional stability in the complex plane, for example by the Hadamard three-circle theorem. In other words, there is conditional stability from one domain to another, bounded one, if there is an a priori estimate in a third even larger one but that third has to be complex.

    Let g be a Riemannian metric in Rn. Consider the wave equation

    Pu=fin [0,T]×Ω, (4)

    where P=2tL, and L=Δg+ajxj+q is a first order perturbation of Δg with smooth coefficients, and T>0 is fixed. Let ΓΩ be a relatively open subset. In time-space, the underlying metric is the Lorentzian one dt2+g.

    Let f=0 for now. Then the Cauchy data of u on [0,T]×Γ determine u uniquely in the domain of influence:

    QΓ,T={(t,x)[0,T]×Ω;|t|+dist(x,Γ)<T}, (5)

    by Tataru's uniqueness continuation theorem [23,24], see also [15,16] for a formulation in this setting. This argument does not prove Hölder type of stability, even a conditional one but there are weaker logarithmic estimates [2,3]. Indeed, if there are unit speed geodesics γ(t) so that (t,γ(t)) is contained in QΓ,T but does not hit [0,T]×Γ (and this is possible for some Γ's), one can concentrate solutions having singularities that do not reach the observation domain, see, e.g., [14] and then those singularities of u would be invisible. On the other hand, if that cannot happen, the singularities of u over Qγ,T (they are on the light cone) would be stably recoverable. That argument alone however does not imply that u restricted to Qγ,T or any subcompact set is stably recoverable. If we knew that u were supported in Qγ,T then one could apply the argument that a priori uniqueness and stable recovery of singularities implies actual Lipschitz stability [14]. On the other hand if u has such a support it would be zero by unique continuation. Note that for those arguments, u needs to solve the wave equation in QΓ,T only, up to its lateral boundary [0,T]×Ω.

    We recall the basic notions pseudo-convexity related to Carleman estimates and also recall the link between them and geometric convexity w.r.t. the geodesic flow. For the definitions below, see, e.g., [7]. Let P be a differential operator P with real principal symbol p(x,ξ). The oriented level hypersurface Σ={ψ=0} of a smooth function ψ with dψ0 is called pseudo-convex w.r.t. P if H2pψ>0 whenever ψ=p=Hpψ=0, ξ0, where Hp is the Hamiltonian vector field of p. This definition is independent of the choice of ψ defining Σ. In our case, p=τ2+|ξ|2, where |ξ| is again the norm of the covector ξ in the metric g. In geometric terms, those conditions say that Σ is strictly convex w.r.t. the lightlike geodesics of dt2+g.

    The smooth function ψ is pseudo-convex w.r.t. P if H2pψ>0 on p=0, ξ0. The function ψ is strongly pseudo-convex if it is pseudo-convex and {pψ,pψ}>0 for pψ=0, ξ0, τ>0, where pψ=p(x,ξ+iτψ). The level hypersurface Σ={ψ=0} of a smooth function ψ with dψ0 is called strongly pseudo-convex w.r.t. P if it is pseudo-convex and {pψ,pψ}>0 for pψ={pψ,ψ}=0, ξ0, τ>0.

    For second order operators P, strong pseudo-convexity of non-characteristic hypersurfaces is equivalent to pseudo-convexity.

    Given such a defining function ψ of a (strongly) pseudo-convex hypersurface, the function ϕ=exp(λψ)1 is a (strongly) pseudo-convex for λ1, see, e.g., [7,25]; and Σ={ϕ=0}, moreover, {ϕ>0}={ψ>0}.

    If g is a Riemannian or a Lorentzian metric, ψ is called strictly convex if the Hessian Hess(ψ)=ψ, where is the covariant derivative, is positive as a quadratic form, i.e.,

    2cH|v|2Hess(ρ)(v,v)with some cH>0.

    We recall a conditional stability theorem proven in [7,10]. The version below is as in [7,Theorem 3.3], see also [6,17]. Given a smooth domain Q and a smooth function ϕ, set

    Qε=Q{ε<ϕ},Qε=Q{ε<ϕ}. (6)

    In particular, Q0=Qε with ε=0.

    In [10,Theorem 3.4.1], a Hölder type conditional stability estimate for the wave, and for more general hyperbolic problems, see also [7], similar to (2) is proven assuming existence of a suitable pseudo-convex function. The goal of the next theorem is to connect the foliation condition of a subdomain of Ω with the existence of a pseudo-convex function and as a result to get conditional stability.

    Theorem 3.1. [7,10] Let Q be a bounded domain with R1+n with C1 boundary Q. Let SQ be relatively open. Assume that ϕC is pseudo-convex w.r.t. P on ˉQ and that ϕ<0 on QS. Then there exists κ(0,1), so that for any solution uH1(Q) of Pu=f in Q with

    u|S=h0,νu|S=h1 (7)

    satisfies, for 0<ε1,

    uH1(Qε)C(ε)(F+u1κH1(Q))Fκ), (8)

    where

    F:=fL2(Q)+h0H1(S)+h1L2(S). (9)

    The theorem actually holds for second order real principal type of operators with C1 coefficients in the real part and locally L lower order coefficients.

    We say that Ω0Ω is foliated by strictly convex hypersurfaces if there exists c>0 and a smooth ρ:M[c,0] so that dρ0 on M, the level sets of ρ=C are strictly convex and Ω0ρ1([c,0]) when viewed from ρC. The strict convexity property is equivalent to G2ρ>0 when ρ=c and Gρ=0, where G is the generator of the geodesic flow. In applications, it is also required ρ1(0)Ω0=ρ1(0)Ω0. It was shown in [13] that the existence of a strictly convex foliation is equivalent to the existence of a strictly convex function (of the kind ϕ(ρ) with some ϕ) if Ω0 is connected at least.

    Set

    cd:=inf

    see also the definition of \kappa in [26].

    Corollary 3.1. Let \rho\in C^\infty({\bf R}^n) be such that \rho\le0 in \bar\Omega , and | \mathrm{d}\rho| > 0 , {\rm Hess}(\rho)\ge 2c_H > 0 when -c\le\rho\le 0 . Let

    \begin{equation} \label{{T0}} T \gt T_0: = \max\left(\sqrt{c/c_H}, \sqrt{c/c_d} \right). \end{equation} (10)

    Let \phi = c+\rho-\alpha t^2 with 0 < \alpha < \min(c_H, c_d) . Then for 0 < \varepsilon\ll1 , there exists \kappa\in(0, 1) , C(\varepsilon) > 0 , so that for any solution u\in H^1(Q) , with Q = (0, T)\times \Omega , of (4) satisfying (7) we have (8).

    Proof. Clearly, \phi > 0 implies \rho > \alpha t^2-c\ge -c . On the other hand, \phi\le c in {\bf R}\times\bar\Omega , see also Figure 1. Therefore, \phi takes values in [0, c] in Q_0 . We show first that the levels \phi = C\in [0, c] are timelike when \phi > 0 , and in particular, non-characteristic. We have

    2\alpha|t| = 2\sqrt\alpha \sqrt{c-C+\rho} \le 2\sqrt\alpha \sqrt{c+\rho} \lt | \mathrm{d}\rho|.

    Since \mathrm{d}\phi = (-2\alpha t, \mathrm{d}\rho) , this proves the claim.

    We will show that \phi is pseudo-convex when \phi \in[0, c] . It is enough to do this in the tangent instead of the cotangent bundle, see, e.g., [17]. The generator of the lightlike geodesics of - \mathrm{d} t^2+g is proportional to \partial_t\pm G . We have (\partial_t\pm G)^2\phi = -2\alpha+G^2\rho . Since G^2\rho = {\rm Hess}(\rho)(v, v)\ge 2c_H , we get (\partial_t \pm G)^2\phi\ge 2c_H-2\alpha > 0 . Note that at this point of the proof we needed G^2\rho > 0 not just along directions tangent to the level sets of \rho . Therefore, \phi is pseudo-convex. We can take \psi = e^{\lambda\phi}-1 with \lambda\gg1 to get a strongly pseudo-convex function \psi with \{\phi > 0\} = \{\psi > 0\} . Then we apply Theorem 3.1.

    To estimate the largest |t| on S\cap \{\phi > 0\} , we write c+\rho-\alpha t^2 > 0 ; therefore |t| < \sqrt{(c+\rho)/\alpha} and since the maximal value of \rho in \bar\Omega is 0 , we get |t| < \sqrt{c/\alpha} . Since \alpha < \min(c_H, c_d) , taking \alpha closer and closer to its upper bound, we get that the supremum of |t| for such \alpha and \varepsilon > 0 is T_0 .

    The conditional estimate (8) would not hold if there is a lightlike geodesic in Q_ \varepsilon which does not hit S . We can see directly that this does not happen. Indeed, for every lighlike geodesic \gamma(t) , since 0 < \alpha < c_H ,

    \begin{equation} \label{{Hess}} \frac{ \mathrm{d}^2}{ \mathrm{d} t^2}\phi\circ\gamma(t) = {\rm Hess}(\phi)(\dot\gamma, \dot\gamma) \gt 0 \end{equation} (11)

    as long as \phi is pseudo-convex. Indeed, for every vector v = (v^0, v')\in {\bf R} \times {\bf R}^n , we have {\rm Hess}(\phi)(v, v) = -2\alpha |v^0|^2 +{\rm Hess}(\rho)(v', v') , and if v is lightlike, then |v^0|^2 = |v'|^2 , therefore, {\rm Hess}(\phi)(v, v)\ge (2c_H-2\alpha)|v'|^2 > 0 . Therefore, \phi\circ\gamma(t) is a strictly convex function and as such, it increases either strictly for t > 0 or for t < 0 in negative direction. Then \gamma(t) hits {\bf R}\times \partial \Omega in \{\phi > 0\} which is contained in S where we have "measurements". The singularities of u are lightlike only; and regarded as vectors (rather than covectors), they are lightlike as well and occupy entire lightlike geodesics (\gamma, \dot\gamma) . Therefore, all singularities of u are "measured".

    It is interesting to estimate how optimal the set \{\phi > 0\}\subset S is w.r.t. the requirement that each singular bicharacteristic of u reaches S . Let g be Euclidean and let \Omega = B(0, R) with R > 1 . One can take \rho = |x|^2-R^2 . Then -c < \rho < 0 defines the annuls \Omega_0: = \{\sqrt{R^2-c} < |x| < R\} , and we assume c < R^2 . We have {\rm Hess}(\rho)(v, v) = 2|v|^2 and \phi = c+|x|^2-R^2-\alpha t^2 . Then c_H = 1 and | \mathrm{d}\rho| = 2|x| ; therefore,

    \frac{| \mathrm{d}\rho|^2}{4(c+\rho)}\ge \frac{|x|^2}{|x|^2-R^2+c}\ge1.

    Thus c_d\ge1 . Then the condition for \alpha reduces to 0 < \alpha < 1 and T_0 = \sqrt{c} .

    On the timelike geodesics \gamma(t) = (t_0+t/\sqrt\alpha, x_0+t\theta) , |\theta| = 1 , (11) vanishes. This characterizes \{\phi = 0\} as the union of all such spacelike geodesics tangent to that hypersurface at some point (for uniqueness, one can take points on t = 0 ). For each fixed t , those are the geodesics starting from some x\in\bar\Omega with \phi(t, x)\ge0 maximizing the escape time (with speed \sqrt\alpha ) to the boundary. Taking \alpha < 1 closer and closer to 1 , we see that Q_0 is, asymptotically, the smallest domain for which there would be no lightlike geodesics through it not hitting S\cap\{\phi\ge0\} . Therefore, at least in this case, Q_ \varepsilon is a sharp domain.

    Let us say that we are interested in u|_{t = 0}: = f as in the next section. Note that to have unique continuation of f on some subdomain \Omega_0 in \bar\Omega from (-T, T)\times\Gamma with \Gamma\subset \partial \Omega , we need T such that every point x there can be reached from \Gamma at time T or less, see (5). This is a condition on the shortest geodesic from x (assuming convexity). In contrast, the conditional stability requires T to exceed the length of the longest escape time from every such x and every direction; and it is sharp at least in the special case above. To be more precise, denote by \gamma_{x, v}(\tau) the unit speed geodesic starting from (x, v)\in S\Omega . Set

    T_{\rm esc} = \sup\limits_{(x, v)\in S\Omega_0}\min (\tau(x, v), \tau(x, -v)),

    where \tau(x, v) is the shortest time \tau when \gamma_{x, v}(\tau) hits \Gamma , if it does; if it does not, we set \tau(x, v) = \infty . Then we need T > T_{\rm esc} . In the Euclidean example above, T_{\rm esc} is maximized when |x|^2 = R^2-c and v is tangent to that ball; then T_{\rm esc}\to T_0 as \alpha\to1- , see Figure 2.

    Figure 2.  The Euclidean case.

    Consider the multiwave tomography model in a closed domain modeling waves reflecting from the boundary, see [4,9,11,12,22] and the UCL photoacoustic imaging group experimental setup in [5]. In thermo- and photo-acoustic imaging, one sends a short microwave/optical impulse into a tissue which triggers heat expansion and generates ultrasound waves by the thermo-acoustic or the photo-acoustic effect. Those waves are measured on the boundary, very often just a subset of the whole boundary. The goal is to recover the acoustic source which would tell us more about the properties of the tissue. The model with reflecting waves is described below, following [22].

    Let \Omega\subset {\bf R}^n be a bounded smooth domain and fix T > 0 . The acoustic pressure u solves the problem

    \begin{equation} \left\{ \begin{array}{rcll} (\partial_t^2 -c^2\Delta)u & = &0 & \mbox{in $(0, T)\times \Omega$}, \\ \partial_\nu u|_{(0, T)\times \partial \Omega}& = &0, \\ u|_{t = 0} & = & f, \\ \quad \partial_t u|_{t = 0}& = &0. \end{array} \right. \end{equation} (12)

    where c(x) > 0 is a smooth wave speed in \bar\Omega , and \nu is the unit outward normal. One can replace \Delta by the Laplace-Beltrami operator related to a fixed Riemannian metric, see [21]. The measurements operator \Lambda is given by

    \begin{equation} \label{{TAT0}} \Lambda f = u|_{(0, T)\times \partial \Omega}. \end{equation} (13)

    The energy

    E_\Omega(t, u) = \int_\Omega\left( |\nabla u|^2 +c^{-2}|u_t|^2 \right) \mathrm{d} x

    is preserved. The natural space for f is the Dirichlet one with the norm \|f\|_{H_D(\Omega)} = \|\nabla f\|_{L^2(\Omega)} , which is really a norm on functions in H_0^1(\Omega) and equivalent to the norm in the latter space by the Poincaré inequality. We also assume that {\rm supp}~ f\in K with some smooth compact set K\Subset\Omega ; then f\in H_D(K) and that \partial \Omega is convex w.r.t. the metric c^{-2} \mathrm{d} x . Then all singularities of f hit \partial \Omega transversely and \Lambda: H_0^1(K)\to H^1((0, T)\times \partial \Omega) is continuous. Next, the zero initial condition for u_t at t = 0 implies that the even extension of u in the t variable solves the wave equation as well and we may assume that \Lambda is given on (-T, T)\times \partial \Omega .

    Assume that we know \Lambda f on (0, T)\times\Gamma , where \Gamma\subset \partial \Omega as before. The following sharp uniqueness theorem follows directly from unique continuation, see [22].

    Theorem 4.1. [22] \Lambda f|_{(0, T)\times\Gamma} determines uniquely f in the set

    \Omega_T : = \{x;\; {\rm dist}(x, \Gamma) \lt T\}

    and f restricted to \Omega\setminus\bar\Omega_T can be arbitrary.

    In [15], we also showed that one can get recover all visible singularities in a stable way (by time reversal, which is an FIO of order zero associated with a canonical graph). The point (x, \xi)\in T^*\Omega\setminus 0 is called visible singularity for this problem if either the geodesic issued from (x, \xi/|\xi|) or that issued from (x, -\xi/|\xi|) reaches \partial \Omega for time < T . Here, |\xi| is the norm of the covector \xi at x in the metric, identified with a vector by the metric. Note that the there might be singularities over \Omega_T that are invisible.

    Even when T^*\bar\Omega_0\setminus 0 contains visible singularities only (then \Omega_0\subset \Omega_T ), one cannot prove stability of recovery of f in a set \Omega_0 based on the uniqueness theorem above and combined with the microlocal stability statement because the latter only applies under the a priori assumption that {\rm supp}~ f\subset\Omega_0 as in [15]. Using the methods above, we get the following.

    Theorem 4.2. Let \rho and T_0 be as in Corollary 3.1 and let \Omega_ \varepsilon = \Omega\cap \rho^{-1}(-c+ \varepsilon, 0) , \varepsilon\ge0 . Let \Gamma = \partial \Omega \cap \rho^{-1}[-c, 0] . Then, if T > T_0 , for every \varepsilon\in (0, 1) we have

    \begin{equation} \label{{TAT2}} \|f\|_{H^{1}(\Omega_ \varepsilon)}\le C\|f\|_{H^s(\Omega)}^{1-\kappa} \|\Lambda f\|_{ H^1((0, T)\times\Gamma) }^\kappa \end{equation} (14)

    with some C > 0 , \kappa\in(0, 1) depending on \Omega_ \varepsilon .

    Proof. We apply Corollary 3.1. Set Q = [-T, T]\times\Omega with T > 0 large enough so that \bar Q\supset \bar Q_0 , with Q_ \varepsilon defined in (6). Even though Q does not have smooth boundary, one can always make it smooth if needed by a small modification and this would not affect Corollary 3.1. Note that \Omega_ \varepsilon = Q_ \varepsilon\cap \{t = 0\} . Then

    \begin{equation} \label{{TAT1}} \|u\|_{H^{1}(Q_ \varepsilon)}\le C( \varepsilon)\left( \|\Lambda f\|_{H^1(S)}+ \|u\|_{H^1(Q)}^{1-\kappa} \|\Lambda f\|_{H^1(S)}^\kappa \right) . \end{equation} (15)

    To estimate the u -term on the right, write

    \begin{split} \|u\|^2_{H^1(Q)}& = \int_{-T}^T\int_\Omega \left( |\nabla_x u|^2+c^{-2} |u_t|^2+|u|^2 \right) \mathrm{d} x\, \mathrm{d} t\\ & \le 2T \max\limits_{t\in [-T, T]} \int_\Omega \left( |\nabla_x u|^2+c^{-2}|u_t|^2+|u|^2 \right) \mathrm{d} x\\ & \le C \|f\|_{H^1}^2. \end{split}

    We also have \|\Lambda f\|_{H^1(S)}\le C\|f\|_{H^1} . Those arguments show that the r.h.s. of (15) can be estimated as in (14).

    To estimate the l.h.s. of (15) from below, notice first that for 0 < \delta\ll1 , [-\delta, \delta]\times \Omega_{2 \varepsilon}\subset Q_ \varepsilon , see also Figure 3.

    Figure 3.  Illustration for the proof of Theorem 4.2.

    Therefore,

    \|u\|^2_{H^1(Q_ \varepsilon)}\ge \|u\|^2_{H^1([-\delta, \delta]\times \Omega_{2 \varepsilon})} \ge \int_{-\delta}^\delta\int_{\Omega_{2 \varepsilon}} \left( |\nabla_x u|^2+c^{-2} |u_t|^2 \right) \mathrm{d} x\, \mathrm{d} t.

    By standard energy estimates used to prove for domain of influence/domain of influence results, E_{\Omega_{2 \varepsilon}}(t, u)\ge E_{\Omega_{3 \varepsilon}}(0, u) when \delta\ll1 , therefore,

    \|u\|^2_{H^1(Q_ \varepsilon)}\ge 2\delta \int_{\Omega_{3 \varepsilon}} |\nabla f|^2\, \mathrm{d} x.

    One the other hand,

    \|u\|_{H^1(Q_ \varepsilon)}\ge \|f\|_{H^{1/2}(\Omega_{3 \varepsilon)}}/C_1\ge \|f\|_{L^2(\Omega_{3 \varepsilon)}}/C_2

    by the trace theorem. Those two estimates allow is to estimate the l.h.s. of (15) by \|f\|_{H^1(\Omega_{3 \varepsilon})} .

    Therefore, we proved (14) with \Omega_0 replaced by \Omega_{3 \varepsilon} . We can extend \rho a bit so that it has the same properties as before on -1-3 \varepsilon\le \rho\le0 for 0 < \varepsilon\ll1 and then apply what we proved above.

    If \Gamma has the property that every geodesic of c^{-2} \mathrm{d} x^2 through K , reflecting from the boundary according to the law of geometric optics eventually intersects \Gamma either for positive and negative time \pm T , then one has even a Lipshitz non-conditional estimate on f in the whole K , as it follows form control theory, see [1,22], also [26]. On the other hand, in Theorem 4.2, \Gamma does not need to satisfy that condition and even of it does, one needs T to be larger in general that what we require. Also, on the rest of the boundary one can impose diferent boundary conditions, even absorbing ones, as long as the boundary value problem is well posed. One can have an infinite \partial \Omega with an infinite \Gamma as well (with \partial \Omega not closed in particular) and by finite speed of propagation, \Lambda f would be supported on a compact set depending on T .

    One can also apply those arguments to the classical model where the waves propagate into the whole space but measurements are still taken on (0, T)\times \partial \Omega . For that, one needs to recover, in a Hölder stable way, the normal derivative \partial_\nu on (-T, T)\times\Gamma from the knowledge of u there. This is a non-trivial step and uniqueness was first proved in [8] when c = 1 and in [15] for a variable c . First that we can extend the data to t\in (-T, -T) as an even function as noted above. Then we recover the Neumann data on a part of \Lambda f|_{(-T, T)\times\Gamma} first, then we use unique continuation to recover f near \Gamma only; that allows us to recover the Neumann (and therefore, the Cauchy) data on a large part of \Lambda f|_{(0, T)\times\Gamma} ; and in finitely many step, we recover f in \Omega_T . We will not pursue this further.

    The author thanks Lauri Oksanen for his remarks which helped improve the exposition. The work was partly supported by NSF Grant DMS-1600327.

    The author declares that there is no conflict of interest in this manuscript.



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