The Radiation Transfer Equation with Reflection and Refraction Conditions. Continuous Dependence of Solutions on the Data and Limit Passage to the Problem with “Shooting Conditions”

Abstract

We study the boundary value problem for the radiation transfer equation with reflection and refraction conditions subject to the Fresnel laws which governs radiation transfer in a system of semitransparent bodies. We establish the continuous dependence of the solution on the data and prove that the solution of this problem converges to the solution of the radiation transfer equation with the “shooting condition” as the refraction exponent tends to 1. Under natural assumptions on the dependence of optic properties of bodies on the radiation frequency ν, we show that the radiation intensity I ν , regarded as a function of ν with values in , is a strongly measurable function, the Bochner integral (where is the absorption coefficient) is defined in L 1(D) and can be estimated in terms of the data of the problem. Bibliography: 9 titles.