Homogenization For The Poisson’s Equation In Randomly Perforated Domains

We assume that D(?)R2 is a bounded open set,(Ω,F,P)is a probability space,Φ is a collection generated by a random point process,and R={ρi}zi∈Φ(?)R+ is a sequence of random radius.We denote for any ε>0 where rε=exp(-1/ε2ρi).Assuming that {ρi,zi} satisfy a certain random distribution,we will prove that when ε→0,the solution uε(ω,·)of the Poisson’s equation converges weakly in H1(D)to the solution u0 of where C0 is a determinate positive constant,and its value depends on the size and position of the balls Brε(ezi).The stochastic homogenization of(*)in Rn(n>2)has been considered by[1],while the problem on R2 was left over.We solved it here.