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 L² 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 W^1,p space for p>1. More precisely, a uniform W^1,p estimate in ε for p∈(1, ∞] and a W^1,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 L^p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in ε.