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An inverse problem of determining a source in the stationary transport equation for a medium with refraction. (English. Russian original) Zbl 0851.53046

Sib. Math. J. 35, No. 4, 835-843 (1994); translation from Sib. Mat. Zh. 35, No. 4, 937-945 (1994).
Summary: We consider the problem of finding the source distribution for particles (or radiation) in a bounded domain \(D\) from the outgoing flow through the boundary of \(D\). The domain \(D\) is filled with a medium that influences the particles trifold: first, the medium absorbs the particles; second, it scatters them; third, it bends the particle trajectories so that, between two scattering acts, a particle moves with unit velocity along a geodesic of some Riemannian metric. The medium is assumed to be known; i.e., the absorption, scattering diagram, and Riemannian metric are given. We obtain uniqueness for a solution to the problem and a stability estimate under certain assumptions of smallness of the absorption, scattering diagram, and curvature of the metric.

MSC:

53Z05 Applications of differential geometry to physics
82C70 Transport processes in time-dependent statistical mechanics
85A25 Radiative transfer in astronomy and astrophysics
45G10 Other nonlinear integral equations
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References:

[1] V. A. Sharafutdinov, ”A tomography emission problem for inhomogeneous media,” Dokl. RAN,326, No. 3, 446–448 (1992).
[2] V. A. Sharafutdinov, Integral Geometry of Tensor Fields [in Russian], Nauka, Novosibirsk (1993). · Zbl 0795.53070
[3] L. P. Einsenhart, Riemannian Geometry [Russian translation], Izdat. Inostr. Lit., Moscow (1948).
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