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Stability estimates in the three-dimensional inverse problem for the transport equation. (English) Zbl 0894.35130

Let \(D= \{x\in\mathbb{R}^3\mid| x|<1\}\), and \(\partial D={\mathbf S}^2\) be the boundary of this ball. The transport equation \[ \nabla v\cdot\nu+ \sigma v+ Sv=u,\quad (x,\nu)\in G\equiv D\times{\mathbf S}^2 \] is considered for the function \(v= v(x,\nu)\). Here \[ \sigma= \sigma(x),\;u=u(x),\;\nu= (\sin\theta\cos\varphi,\sin\theta \sin\varphi, \cos\theta),\;Sv\equiv \int_{{\mathbf S}^2}K(x,\nu, \nu')v(x,\nu')d\nu', \] where \(d\nu'= \sin\theta'd\theta'd\varphi'\) is the element of area. The kernel \(K\) of this operator \(S\) is said to be scattering indicatrix. It is assumed that \(K\) is a known function.
We study the problem of recovering the functions \(\sigma(x)\), \(u(x)\) from two observations. Upon the condition that the attenuation and scattering are sufficiently small, stability estimates are found.

MSC:

35R30 Inverse problems for PDEs
45K05 Integro-partial differential equations
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