Remarks on propagation of discontinuities in stationary radiative transfer

Abstract

We consider the stationary transport equation with the incoming boundary condition. We are interested in discontinuities of the solution. Under the generalized convexity condition, it is known that it has only boundary-induced discontinuities, which are discontinuities arising from discontinuous boundary data, they propagate along positive characteristic lines, and we can reconstruct the attenuation coefficient from boundary measurements by the inverse X-ray transform. In this article, we observe that coefficient-induced discontinuities, discontinuities of the solution arising from discontinuous coefficients, would also appear without the generalized convexity condition. If the set of discontinuous points of the coefficients contains at most finite number of flat parts, coefficient-induced discontinuities do not affect the inverse X-ray transform. We also remark that, under the generalized convexity condition, a three dimensional inverse problem can be reduced to the two dimensional one. A numerical experiment is exhibited.