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Wave bases in multidimensional inverse problems. (English. Russian original) Zbl 0698.35164

Math. USSR, Sb. 67, No. 1, 23-42 (1990); translation from Mat. Sb. 180, No. 5, 584-602 (1989).
Let \(\Omega \subset {\mathbb{R}}^ n\) (n\(\geq 1)\) be a bounded domain with boundary \(\partial \Omega =\Gamma \in C^ 2\). Consider the problem \[ \rho (x)u_{tt}-\Delta u=0,\quad (x,t)\in \Omega \times [0,T];\quad u|_{t<0}=0,\quad \partial_{\nu}u|_{\Gamma \times [0,T]}=f(\gamma,t). \] Here \(\rho \in C^ 2(\Omega)\), \(0<\rho_ 1\leq \rho (x)\leq \rho_ 2\), \(\nu =\nu (\gamma)\) (\(\gamma\in \Gamma)\) is the outer normal to \(\Gamma\). Consider the operator \(R: f(\gamma,t)\to u^ f(\gamma,t),\) where \(u^ f\) is the solution of the above problem, \(\gamma\in \Gamma\). The inverse problem considered in the paper is to recover \(\rho\) (x) given the operator R. The author proposes a procedure for recovering \(\rho\) (x) for x near the boundary and a suitable modification of that procedure when x is arbitrary. Some ideas and results from boundary value control theory are used. The main tools in the recovering procedure are the bases of waves coming from sources located on the boundary.
Reviewer: P.Stefanov

MSC:

35R30 Inverse problems for PDEs
35L05 Wave equation
93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
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