Homogenization of biharmonic equations in domains perforated with tiny holes

Abstract

This paper deals with the homogenization of the biharmonic equation Δ²u=f in a domain containing randomly distributed tiny holes, with the Dirichlet boundary conditions. The size σ of the holes is assumed to be much smaller compared to the average distance ε between any two adjacent holes. We prove that as ε,σ→0, the solutions of the biharmonic equation converge to the solution of Δ²u+κu=f, where κ depends on the shape of the holes and relative order of σ with respect to ε.