Asymptotic Behavior of the Solution of the Integral Transport Equation in the Vicinity of a Curved Material Interface

Abstract

This paper deals with the differentiability properties of the solution $\phi $of a linear integral equation arising in transport theory. Two one-dimensional cases are considered, corresponding to either a spherically or a cylindrically symmetric domain in $R^3 $. For such domains, a singular representation is derived showing explicitly the behavior of $\phi $ and its derivatives near the boundary of the domain. The representation is derived from certain basic properties of the integral transport operator T in Sobolev spaces $W_p^m ( \Omega )$. In particular, it is shown that if $\phi \in W_p^m ( \Omega ),m \geqq 0,p < \infty $, then $T\phi \in W_p^{m + 1} ( \Omega ) \oplus S$. where S is finite-dimensional.