Model of stationary diffusion with absorption in domains with fine-grained random boundaries

Abstract

UDC 517.95, 519.21
We consider a boundary-value problem for the equation of stationary diffusion in a porous medium filled with small ball inclusions with absorbing surfaces. Absorption is described by a Robin’s nonlinear boundary condition. The locations and radii of the inclusions are randomly distributed and described by a set of finite-dimensional distribution functions. We study the asymptotic behavior of solutions to the problem when the number of balls increases and their radii decrease. We derive a homogenized equation for the main term of the asymptotics, and determine sufficient conditions for the distribution functions under which the solutions converge to the solutions of the homogenized problem in probability.