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On the Lax-Phillips conjecture on the finite number of strictly convex obstacles. (Sur la conjecture de Lax et Phillips pour un nombre fini d’obstacles strictement convexes.) (French) Zbl 0884.35084
The author considers the boundary-value problem \[ (\partial^2_t- \Delta_x)u=0 \text{ in } \mathbb{R}\times \Omega, \quad Bu=0 \text{ on } \mathbb{R}\times \partial\Omega, \quad u(0,x)= f_1(x),\;\partial_t u(0,x) =f_2(x), \] where \(B=Id\) or \(B=\partial/ \partial\nu\), while \(\Omega= \mathbb{R}^3 \setminus K\). It is assumed that \(K= \cup^Q_{j=1} K_j\) with \(Q\geq 3\) and \(K_i \cap K_j =\emptyset\) for \(i\neq j\). Here \(K_j\) stands for given strictly convex bounded domains. It is known that the diffusion operator \(S(\lambda)\) associated with the above-mentioned boundary-value problem is a meromorphic function of \(\lambda\) with poles \(\lambda_j\) in the upper half-plane \(\text{Im} \lambda_j>0\). The main aim of the author is to give a lower limit to the number of the poles in the domain \(0<\text{Im} \lambda \leq\delta\), \(|\text{Re} \lambda |\leq r\). It is assumed that for every pair \(1\leq i\), \(j\leq Q\), \(i\neq j\), the convex hull of \(\overline K_i \cup \overline K_j\) has no common points with \(\overline K_\ell (\ell\neq i\), \(\ell\neq j)\). The paper involves also some results concerning the singularities of the distribution \[ U(t)= \begin{cases} \sum_j\exp (i\lambda_jt) \quad & \text{for } t>0,\\ \sum_j \exp(-i \lambda_jt) \quad & \text{for } t<0. \end{cases}. \]
35L20 Initial-boundary value problems for second-order hyperbolic equations
35A20 Analyticity in context of PDEs
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