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The stationary translation-invariant Peierls equation of the theory of radiation transport in the space of tempered distributions and some properties of the Peierls potential. I. (English. Russian original) Zbl 0870.31006

Sib. Math. J. 38, No. 3, 455-470 (1997); translation from Sib. Mat. Zh. 38, No. 3, 533-550 (1997).
This article is the first part of a series devoted to studying the questions of uniqueness, existence, and integral representation for solutions to the stationary Peierls equation of the theory of radiation transport in homogeneous media.
The stationary Peierls equation in the case of constant isotropic cross-sections of absorption \(\varepsilon\geq 0\) and scattering \(b\geq 0\) with isotropic sources can be written as \[ w=b\widehat\varkappa_\alpha w+\widehat\varkappa_\alpha q, \] where \(\varkappa_\alpha\) is the operator of convolution with the distribution \(e^{-\alpha(x)}/4\pi|x|^2\), \(\alpha=\varepsilon+b\), and \(w\) and \(q\) are respectively the sought (field) function and the source function of a variable \(x\in\mathbb R^3\).
With this equation, the author associates the Peierls potential and the Peierls kernel. In this article, he establishes the main potential-theoretic properties of the Peierls potential such as the continuity principle, the classical and complete maximum principles, the positive maximum principle in the statement of K. Yosida, the balayage principle with respect to compact subsets, and the domination and balance principles. Also, he proves assertions about solvability of the Peierls equation in the class of tempered distributions on \(\mathbb R^3\).

MSC:

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
78A05 Geometric optics
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References:

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