Weakly singular integral equations with operator-valued kernels and an application to radiation transfer problems

Abstract

The standard problem of radiation transfer in a bounded regionG ⊂ ℝⁿ can be reformulated as a weakly singular integral equation with an unknown functionu: G→C(S ⁿ⁻¹) and a kernelK: ((G × G ∖ }x=y}, which is continuously differentiable with respect to the operator strong convergence topology. We take these observations into the basis of an abstract treatment of weakly singular integral equations with ℒ(E)-valued kernels, whereE is a Banach space. Our purpose is to characterize the smoothness of the solution by proving that it belongs to special weighted spaces of smooth functions. On the way, realizing the proof techniques, we establish the compactness of the integral operator or its square inL _p (G,E),BC(G,E), and other spaces of interest in numerical analysis as well as in weighted spaces of smooth functions. The smoothness results are specified for the standard problem of radiation transfer as well as for the corresponding eigenvalue problem.