Convergence for elliptic equations in periodic perforated domains

Convergence for the solutions of elliptic equations in periodic perforated domains is concerned. Let ϵ denote the size ratio of the holes of a periodic perforated domain to the whole domain. It is known that, by energy method, the gradient of the solutions of elliptic equations is bounded uniformly in ϵ   in L2L2 space. Also, when ϵ   approaches 0, the elliptic solutions converge to a solution of some simple homogenized elliptic equation. In this work, above results are extended to general W1,pW1,p space for p>1p>1. More precisely, a uniform W1,pW1,p estimate in ϵ   for p∈(1,∞]p∈(1,∞] and a W1,pW1,p convergence result for p∈(nn−2,∞] for the elliptic solutions in periodic perforated domains are derived. Here n   is the dimension of the space domain. One also notes that the LpLp norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in ϵ.