ASYMPTOTIC BEHAVIOR OF THE TRANSMISSION EIGENVALUES (Spectral and Scattering Theory and Related Topics)

全文

(1)61. 数理解析研究所講究録第2045巻 2017年 61-66. ASYMPTOTIC BEHAVIOR OF THE TRANSMISSION EIGENVALUES. GEORGI VODEV. 1. DEFINITION OF THE TRANSMISSION EIGENVALUES. $\Omega$\subset \mathrm{R}^{d}, d\geq 2 be a bounded, connected domain with a C^{\infty} smooth boundary $\Gamma$=\partial $\Omega$. complex number $\lambda$ \neq 0 with {\rm Re} $\lambda$ \geq 0 will be said to be a transmission eigenvalue if the following problem has a non‐trivial solution: Let. ,. A. \left{bginary}{l (\nablc_{1}(x)\nabl+$\ambd$^{2}n_1(x)u{}=0&\mathr{i}\mathr{n}$\Omega$,\ (nablc_{2}(x)\nabl+$\ambd$^{2}n_ (x)u{2}=0&\mathr{i}\mathr{n}$\Omega$,\ u_{1}= 2,c_{1}\partil_{\mathr{v}u_1=c{2}\partil_{$\nu}_{2&\mathr{o}\mathr{n}$\Gam $, \end{ary}\ight.. (1). 1 2 are strictly $\nu$ denotes the Euclidean inner unit normal to $\Gamma$, c_{j}, n_{j} \in C^{\infty}(\overline{ $\Omega$}) j positive real‐valued functions. The transmission eigenvalues can be viewed as the eigenvalues of the non‐symmetric operator A defined by. where. ,. =. \mathcal{A}\left(\begin{ar y}{l u_{1}\ u_{2} \end{ar y}\right)=\left(\begin{ar y}{l -\frac{1}n_{1}(x)\nabl c_{1}(x)\nabl u_{1}\ -\frac{1}n2(x)}\nabl c_{2}(x)\nabl u_{2} \end{ar y}\right). with domain. D(\mathcal{A})=\{(u_{1}, u_{2})\in \mathcal{H} \nabla c_{1}(x)\nabla u_{1}\in L^{2}( $\Omega$) \nabla c_{2}(x)\nabla u_{2}\in L^{2}( $\Omega$) :. u_{1}=u_{2},. where. ,. c_{1}\partial_{ $\nu$}u_{1}=c_{2}\partial_{ $\nu$}u_{2}. defined. by. .. mult. ,. $\Gamma$\}. on. H_{j}=L^{2}( $\Omega$, n_{j}(x)dx) (A-$\lambda$^{2})^{-1} (if it forms a meromorphic family). \mathcal{H}=H_{1}\oplus H_{2},. the resolvent. ,. Then the transmission. eigenvalues. are. and the multiplicity of. poles of pole $\lambda$_{k} is. the a. ($\lambda$_{k})=\displaystyle\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(2$\pi$i)^{-1}\int_{|$\lambda-\lambda$_{k}|=$\epsilon$}($\lambda$^{2}-A)^{-1}2$\lambda$d$\lambda$ =\mathrm{t}\mathrm{r}(2 $\pi$ i)^{-1}\'{I}_{|$\lambda-\lambda$_{k}|= $\epsilon$}($\lambda$^{2}-A)^{-1}2 $\lambda$ d$\lambda$.. Our goal is to study the asymptotic behavior of the counting function N(r)=\#\{ $\lambda$ ‐trans. eig. : | $\lambda$| \leq r\}, r > 1 We will see that it is closely related to the localization of the transmission eigenvalues on the complex plane. .. 2. THE DIRICHLET‐TO‐NEUMANN MAP. The Dirichlet -\mathrm{t} $\sigma$‐Neumann map, defined. by. N_{j}( $\lambda$). :. H^{1}( $\Gamma$). \rightarrow. L^{2}( $\Gamma$). ,. associated to the pair. (c_{j}, n_{j}). is. N_{j}( $\lambda$)f=\partial_{ $\nu$}u_{j}|_{ $\Gamma$}, where u_{j} solves the. equation. \left\{ begin{ar y}{l (\nabl c_{j}(x)\nabl +$\lambda$^{2}n_{j}(x)u_{j}=0&\mathrm{i}\mathrm{n} $\Omega$,\ u_{j}=f&\mathrm{o}\mathrm{n} $\Gam a$. \end{ar y}\right.. (2).

(2) 62 G. VODEV. Denote. by G_{\mathrm{j} , j=1 2, ,. the Dirichlet. .. \sqrt{G_{j}}. realization of the operator. self‐adjoint. Hilbert space H_{j} It is well‐known that Introduce the operator. N_{j}( $\lambda$). is. meromorphic. -n_{j}^{-1}\nabla c_{j}\nabla the. on. the. eigenvalues. of. .. T( $\lambda$)=c_{1}N_{1}( $\lambda$)-c_{2}N_{2}( $\lambda$) We have the. following. .. trace formula.. Suppose that the inverse T( $\lambda$)^{-1} of the operator A is meromorphic, too,. Lemma 1. vent. poles. with. exists. and. we. as a. meromorphic function. formula. Then the resol‐. have the. M($\gam a$)=M_{1}($\gam a$)+M_{2}($\gam a$)+\displaystyle\mathrm{t}\mathrm{r}(2$\pi$i)^{-1}\int_{$\gam a$}\frac{dT($\lambda$)}{d$\lambda$}T($\lambda$)^{-1}d$\lambda$. (3). where $\gamma$ \dot{u} a simple, positively orientied, piecewise smooth, closed curve in the complex plane, which avoids the poles of T( $\lambda$)^{-1} and the eigenvalues of \sqrt{G_{1}} and \sqrt{G_{2}}, M( $\gamma$) is the number. of. the transmission. operator. \sqrt{G_{\mathrm{j} }. eigenvalues. WEYL. 3.. The. following. Theorem 1.. result is. Suppose. M_{j}( $\gamma$). \dot{u} the number. the. of. eigenvalues of. the. ASYMPTOTICS FOR THE COUNTING FUNCTION. proved. in. [9].. either the condition. c_{1}(x)\equiv c_{2}(x)\equiv 1 or. inside $\gamma$ , and. inside $\gamma$.. in. n_{1}(x)\neq n_{2}(x). $\Omega$,. ,. \forall x\in $\Gamma$ ,. (isotropic case). (4). the condition. c_{1}(x)\neq c_{2}(x) Suppose. also that the operator. T( $\lambda$). ,. \forall x\in $\Gamma$. .. is invertible in. (5). (anisotropic case) a. region of. the. form. \{ $\lambda$\in \mathrm{C}: {\rm Re} $\lambda$>1, |{\rm Im} $\lambda$|\geq C({\rm Re} $\lambda$)^{1- $\kappa$}\}, C>0, 0< $\kappa$\leq 1 and. satisfies. ,. (6). there the bound. \Vert T( $\lambda$)^{-1}\Vert \leq C_{0}| $\lambda$|^{M_{0}}, C_{0}, M_{0}>0. Then. we. have the asymptotics. N(r)=($\tau$_{1}+$\tau$_{2})r^{d}+O_{ $\epsilon$}(r^{d- $\kappa$+ $\epsilon$}) , \forall 0< $\epsilon$\ll 1. (7). ,. where. $\omega$ d. being. the volume. $\tau$_{\mathrm{j}=\displaystyle\frac{$\omega$_{d}{(2$\pi$)^{d}\int_{$\Omega$}(\frac{n_{j}(x)}{c_{j}(x)}^{d/2}dx, of. the unit ball in. \mathrm{R}^{d}.. Known results. In the isotropic case when n_{2}\equiv 1, n_{1}(x)>1 on $\Omega$ , the asymptotic for a remainder term o(r^{d}) is proved by M. Faierman [3] and by L. Robbiano [12].. N(r). with. proof. It is inspired by the paper [1] where Weyl type asymptotics have proved for the counting function of the resonances associated to an exterior transmission problem. We can get an asymptotic for N(r)-N(r/2) by using the trace formula (3), the Weyl asymptotics for the counting functions of the eigenvalues of G_{1} and G_{2} and the Theorems of Caratheodory and Jensen. We use in an essential way that \dim $\Gamma$=d-1. Idea of the. been. ,.

(3) 63 ASYMPTOTIC BEHAVIOR OF THE TRANSMISSION EIGENVALUES. 4.. PARABOLIC. EIGENVALUE‐FREE REGIONS. Thus, the problem of proving Weyl asymptotics for the counting function N(r) is one of proving parabolic eigenvalue‐free regions. The following result is proved concerns the isotropic case. that. reduced to in. [14]. and. Theorem 2. Assume the condition. \mathrm{c}_{1}(x)\equiv c_{2}(x)\equiv 1 Then there. are no. transmission. n_{1}(x)\neq n_{2}(x). $\Omega$,. in. eigenvalues. ,. \forall x\in $\Gamma$. (8). .. in. \{ $\lambda$\in \mathrm{C} : {\rm Re} $\lambda$\geq 0, |{\rm Im} $\lambda$|\geq C_{ $\epsilon$}({\rm Re} $\lambda$+1)^{\frac{1}{2}+ $\epsilon$}\}, \foral 0< $\epsilon$\ll 1. In this. case. In the cases.. asymptotic (7) holds with $\kappa$=1/2.. the. anisotropic. following. The. case. the situation is. result is. proved. in. more. interesting. and. one. has to. distinguish. two sub‐. [14].. Theorem 3. Assume the condition. (c_{1}(x)-c_{2}(x))(c_{1}(x)n_{1}(x)-c_{2}(x)n_{2}(x))<0, \forall x\in $\Gamma$ Then there. are no. transmission. eigenvalues. in the union. of. (9). .. the sets. \{ $\lambda$\in \mathrm{C} : 0\leq{\rm Re} $\lambda$\leq 1, {\rm Re} $\lambda$\geq C_{N}(|{\rm Im} $\lambda$|+1)^{-N}\}, \foral N\gg 1, and. In this. case. the. \{ $\lambda$\in \mathrm{C} : {\rm Re} $\lambda$\geq 1, |{\rm Im} $\lambda$|\geq C_{ $\epsilon$}({\rm Re} $\lambda$)^{\frac{1}{2}+ $\epsilon$}\}, \foral 0< $\epsilon$\l 1.. asymptotic (7) holds with $\kappa$=1/2.. Assume the condition. (c_{1}(x)-c_{2}(x))(c_{1}(x)n_{1}(x)-c_{2}(x)n_{2}(x))>0, \forall x\in $\Gamma$ Then there. are no. transmission. eigenvalues. (10). .. in. \{ $\lambda$\in \mathrm{C} : {\rm Re} $\lambda$\geq 0, |{\rm Im} $\lambda$|\geq C({\rm Re} $\lambda$+1)^{\frac{3}{5} \}. In this. the asymptotic either the condition. or. case. (7). the condition. then there. are no. transmission. holds with. $\kappa$=2/5 Moreover, if in .. \displaystyle \frac{n_{1}(x)}{c_{1}(x)}\neq\frac{n_{2}(x)}{c_{2}(x)}, \foral x\in $\Gam a$. ,. \displaystyle \frac{n_{1}(x)}{c_{1}(x)}=\frac{n_{2}(x)}{c_{2}(x)}, \foral x\in $\Gam a$. ,. eigenvalues. addition to. (10). we assume. (11) (12). in. \{ $\lambda$\in \mathrm{C} : {\rm Re} $\lambda$\geq 0, |{\rm Im} $\lambda$|\geq C_{ $\epsilon$}({\rm Re} $\lambda$+1)^{\frac{1}{2}+ $\epsilon$}\}, \foral 0< $\epsilon$\ll 1. Remark.. One. eigenvalues. in. can. show that under the condition. (9). there. are. infinitely. \{ $\lambda$\in \mathrm{C} : 0\leq{\rm Re} $\lambda$\leq C_{N}(|{\rm Im} $\lambda$|+1)^{-N}\} and that their. counting function, N^{-}(r) satisfies ,. an. asymptotic of the form. N^{-}(r)=$\tau$_{0}r^{d-1}+O(r^{d-2}). .. many transmission.

(4) 64 G. VODEV. isotropic case when c_{1} \equiv c_{2} \equiv 1, n_{2} \equiv 1, n_{1}(x) > 1 Krupchyk, P. Ola and L. Päiväxinta [4] that there are no. Known results. In the. proved by M. Hitrik, eigenvalues in. K.. on. $\Omega$ , it. was. transmission. \{ $\lambda$\in \mathrm{C} : {\rm Re} $\lambda$\geq 0, |{\rm Im} $\lambda$|\geq C({\rm Re} $\lambda$+1)^{\frac{23}{25}}\}. z. To prove the above theorems we make our problem semi‐classical h^{2}$\lambda$^{2} \pm 1+i{\rm Im} z , if |{\rm Re}$\lambda$^{2}| \geq |{\rm Im}$\lambda$^{2}| , and h =. =. =. |{\rm Re}$\lambda$^{2}|. \leq. |{\rm Im}$\lambda$^{2}| Clearly, h\sim| $\lambda$|^{-1} .. The. .. proof. 0<h\ll 1, |{\rm Im} z|. OPS_{1/2- $\epsilon$}^{1}( $\Gamma$). with. a. \geq h^{1/2- $\epsilon$}. where m_{\mathrm{j} denotes the restriction on $\Gamma$ of the function the Laplace Beltrami operator -$\Delta$_{ $\Gamma$}, $\Gamma$ being considered. Recall that. by. the Euclidean. a\in S_{ $\delta$}^{k}( $\Gamma$) 0\leq $\delta$<1/2 ,. ,. if. the. ,. following. N_{j}(z, h)=-ihN_{j}( $\lambda$) (see [14]). ,. the Direchlet‐to‐Neumann map. with. \rightarrow. =. on. principal symbol. $\rho$_{j}(x, $\xi$)=\sqrt{-r_{0}(x, $\xi$)+m_{j}(x)z} the Riemannian metric induced. =. of Theorems 2 and 3 is based. semi‐classical properties of the Dirichlet‐to‐Neumann map Theorem 4. For every 0< $\epsilon$\ll 1, N_{j}(z, h) is an h- $\Psi$ DO of class. by putting h=|{\rm Re}$\lambda$^{2}|^{-1/2}, h^{2}$\lambda$^{2} z {\rm Re} z+i if. |{\rm Im}$\lambda$^{2}|^{-1/2},. n_{j}/c_{j} as a. {\rm Im}$\rho$_{j}>0, principal symbol of manifold equipped with. and r_{0} is the. ,. Riemannian. one.. a\in C^{\infty}(T^{*} $\Gamma$). satisfies the bounds. |\partial_{x}^{ $\alpha$}\partial_{ $\xi$}^{ $\beta$}a(x, $\xi$)| \leq C_{ $\alpha,\ \beta$}h^{- $\delta$(| $\alpha$|+ $\beta$|)}\langle $\xi$\rangle^{k-| $\beta$|}. It is well‐known that for h- $\Psi$ DOs with such. [2]).. Thus. getting eigenvalue‐free regions. symbols. is reduced to. there is. inverting. a. very nice calculus. (e.g.. see. the operator. T(z, h)=c_{1}N_{1}(z, h)-c_{2}N_{2}(z, h) with. a. principal symbol. c_{1}$\rho$_{1}-c_{2}$\rho$_{2}=\displaystyle\frac{\overline{}c(x)(\mathrm{c}_{0}(x)r_{0}(x,$\xi$)-z)}{\mathcal{C}_{1$\rho$_{1}+C2$\rho$_{2} where \overline{c} and c_{0}. are. the restrictions. on. $\Gamma$ of the functions. c_{1}n_{1}-c_{2}n_{2}. respectively. In c_{0}(x) \neq 0, \forall x condition The. (10). (13). \displayst le\frac{ _{1}^{2}-c_{2}^{2}{c_{1}n_{1}-c_{2}n_{2}. and. the isotropic case we have c_{0}\equiv 0 on $\Gamma$ , while in the anisotropic case we have Under the condition (9) we have c_{0}(x) < 0, \forall x \in $\Gamma$ , while under the. \in $\Gamma$. we. .. have. parametrix of. c_{0}(x)>0,. N_{j}(z, h). \forall x\in $\Gamma$.. is bad when {\rm Re} z=1. near. the. glancing region. $\Sigma$_{j}=\{(x, $\xi$)\in T^{*} $\Gamma$ : r_{0}(x, $\xi$)-m_{j}(x)=0\}. Therefore, to improve the above results one has to improve the parametrix construction in the glancing region. Indeed, a better parametrix has been constructed in [15] for strictly concave domains valid for. |{\rm Im} z|\geq h^{1- $\epsilon$}. ,. which led to. some. improvements. in this. case..

(5) 65 ASYMPTOTIC BEHAVIOR OF THE TRANSMISSION EIGENVALUES. 5. OPTIMAL. We. EIGENVALUE‐FREE REGIONS. improve the above eigenvalue‐free regions if $\Sigma$_{1}\cap$\Sigma$_{2}=\emptyset More precisely,. can. we. .. have the. following (see [16]). Theorem 5. Assume either the condition mission. In this. eigenvalues. the asymptotic. case. (8). or. (9).. the condition. Then there. are no. trans‐. in. \{ $\lambda$\in \mathrm{C}:{\rm Re} $\lambda$\geq 1, |{\rm Im} $\lambda$|\geq C>0\} (7) holds with $\kappa$=1.. (14). .. eigenvalue‐free region (14) has been previously proved in [10] in the case of a ball and by Leung and Colton [6] that in the isotropic case when $\Omega$ is a ball and the refraction indices n_{1} and n_{2} constants, the eigenvalue‐free region (14) is optimal. In the anisotropic case we also have the following (see [16]). Tthe. constant coefficients. It is shown. Theorem 6. Assume the conditions. (10). and. (11).. Then there. are no. transmission. eigenvalues. in. In this. the. case. \{ $\lambda$\in \mathrm{C} : {\rm Re} $\lambda$\geq 0, |{\rm Im} $\lambda$| \geq C\log({\rm Re} $\lambda$+2)\}, C>0 asymptotic (7) holds with $\kappa$=1.. Define the cut‐off function. where 0. <. $\phi$(t)=1. $\delta$\ll 1 is. $\chi$_{j}^{0}\in C_{0}^{\infty}(T^{*} $\Gamma$) by $\chi$_{j}^{0}(x, $\xi$)= $\phi$((r_{0}(x, $\xi$)-m_{j}(x))$\delta$^{-2}). |t|\leq 1, $\phi$(t)=0 for |t|\geq 2 following (see [16]).. for. from the. independent of h and z and $\phi$\in C_{0}^{\infty}(\mathrm{R}) independent of h and z Theorems 5. small parameter. a. ,. ,. is also. .. Theorem 7. Let {\rm Re} z=1 and let 0< $\epsilon$<1 be constants. C_{ $\delta$}>1. and. (15). .. 0<h_{0}( $\epsilon$, $\delta$)\ll 1. arbitrary. Then, for. such that. we. ,. 0 \leq. $\phi$\leq 1,. and 6 follow. 0< $\delta$\ll 1 there. every. \Vert N_{j}(z, h)-\mathrm{O}\mathrm{p}_{h}($\rho$_{j}(1-$\chi$_{j}^{0})+hb_{j})\Vert_{L^{2}( $\Gamma$)\rightar ow H_{h}^{1}( $\Gamma$)}\leq C $\delta$ for C\'{o} h\leq |{\rm Im} z| \leq h^{ $\epsilon$}, 0<h\leq h_{0}( $\epsilon$, $\delta$) where C>0 is a and b_{j}\in S_{0}^{0}( $\Gamma$) is independent of h, z and the function n_{j}. ,. Here. H_{h}^{1}( $\Gamma$). denotes the Sobolev space. are. have. constant. (16). independent of h,. equipped with the semi‐classical. z. and. $\delta$,. norm.. 6. THE DEGENERATE ISOTROPIC CASE. We will consider the. case. when. c_{1}(x)\equiv c_{2}(x)\equiv 1 We have the. in. n_{1}(x)=n_{2}(x). $\Omega$,. ,. \forall x\in $\Gamma$.. following (see [17]).. Theorem 8. Assume that there \dot{u}. an. integer j\geq 1 such that. \partial_{ $\nu$}^{s}(n_{1}(x)-n_{2}(x))=0, \forall x\in $\Gamma$, 0\leq s\leq j-1. (17). ,. and. \partial_{ $\nu$}^{j}(n_{1}(x)-n_{2}(x))\neq 0, \forall x\in $\Gamma$ Then there. are no. transmission. eigenvalues. (18). .. in. \{ $\lambda$\in \mathrm{C} : {\rm Re} $\lambda$\geq 0, |{\rm Im} $\lambda$|\geq C({\rm Re} $\lambda$+1)^{1- $\kappa$}j\}, where. $\kappa$_{j}=2(3j+2)^{-1}. It has been. and. (18). .. In this. case. previously proved by. there. are no. transmission. the asymptotic. Lakshtanov and. eigenvalues. in. (7). holds with $\kappa$=$\kappa$_{j}.. Vainberg [5]. that under the conditions. |\mathrm{a}x\mathrm{g} $\lambda$|\geq $\epsilon$, | $\lambda$|\geq C_{ $\epsilon$}\gg 1,. \forall 0< $\epsilon$\ll 1.. (17).

(6) 66 G. VODEV. OPEN. 7.. Conjecture 1. For an arbitrary domain ues satisfies the Weyl asymptotics. PROBLEMS. $\Omega$ , the. counting function of the transmission eigenval‐. N(r)=($\tau$_{1}+$\tau$_{2})r^{d}+O(r^{d-1}). (19). .. REFERENCES. [1]. $\Gamma$ CARDOSO, G. Popov AND G. VODEV, Asymptotics of the problem, Commun. Partial Dffi. Equations 26 (2001), 1811‐1859. .. [2]. number. of. resonances. in the transmission. J. SJöSTRAND, Spectral asymptotics in semi‐classical limit, London Mathematical Society, Series, 268, Cambridge University Press, 1999. FAIERMAN, The interior transmission problem: spectral theory, SIAM J. Math. Anal. 46 (1) (2014), 803‐. M. DIMASSI. AND. Lecture Notes. [3]. M. 819.. [4]. M.. [5]. KRUPCHYK,. eigenvalues,. E. LAKSHTANOV. value. [6]. AND. B.. P. OLA AND L.. Math. Res. Lett. 18. Y.‐J. LEUNG. AND. (2012),. D.. PÄIVÄRINTA,. (2011),. The intenor transmission. (2013),. and bounds. of. to the. zsotropic. interior transmission. eigen‐. 104003.. COLTON, Complex transmission eigenvalues for spherically stratified media, Inverse. 075005.. H. PHAM AND P. STEFANOV, Weyl asymptotics of the transmzssion eigenvalues for tion, Inverse problems and imaging 8(3) (2014), 795810.. [8]. problem. 279‐293.. VAINBERG, Apphcation of elliptic theory. Inverse Problems 29. problem,. Problems 28. [7]. K.. HITRIK,. transmission. a. constant index. of refrac‐. PETKOV, Location of eigenvalues for the wave equation with dissipative boundary conditions, Inverse Prob‐ imaging 10(4) (2016), 1111‐1139. [9] V. PETKOV AND G. VODEV, Asymptotics of the number of the intenor transmission eigenvalues, J. Spectral V.. lems and. Theory 7(1) (2017),. [10]. V. PETKOV. and imaging,. [11] [12]. L. L.. to appear.. G.. VODEV,. 11(2) (2017),. Localization. of the. interior transmission. eigenvalues for. a. ball,. Inverse Problems. to appear.. ROBBIANO, Spectral analysis of interior transmission eigenvalues, Inverse Problems 29 (2013), 104001. ROBBIANO, Counting function for interior transmission eigenvalues, Mathematical Control and Related. Fields. [13] [14] [15] [16]. AND. 6(1) (2016),. 167‐183.. SYLVESTER, 7ransmission eigenvalues in one dimension, Inverse Problems 29 (2013), 104009. G. VODEV, Transmission eigenvalue‐free regions, Comm. Math. Phys. 336 (2015), 1141‐1166. G. VODEV, Thunsmission eigenvalues for strictly concave domains, Math. Ann. 366 (2016), 301‐336. G. VODEV, High‐frequency approximation of the interior Dirichlet‐to‐Neumann map and applications to the transmission eigenvalues, preprint 2017. [17] G. VODEV, Parabohc transmission eigenvalue‐free regions in the degenerate isotropic case, preprint 2017. J.. UNIVERSrTÉ. 92208,. DE. NANTES, LABORATOIRE DE MATHÉMATrQCIES JEAN LERAY, 03, FHANCE Georgi. Vodev\mathfrak{G}\mathrm{m}\mathrm{i}\mathrm{V} ‐nantes. fr. 44322 NANTES CEDEX. E ‐mail address:. 2 RUE DE LA. HOUSSrNIÈRE,. BP.

(7)