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In this paper we establish W1,p estimates for solutions uε to Laplace’s equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, C1 domain Ωε,η in Rd. The bounding constants depend explicitly on two small parameters ε and η, where ε represents the scale of the minimal distance between holes, and η denotes the ratio between the size of the holes and ε. The proof relies on a large-scale Lp estimate for ∇uε, whose proof is divided into two parts. In the first part, we show that as ε,ηapproach zero, harmonic functions in Ωε,η may be approximated by solutions of an intermediate problem for a Schr¨odinger operator in Ω. In the second part, a real-variable method is employed to establish the large-scale Lp estimate for ∇uε by using the approximation at scales above ε. The results are shown to be sharp except in one particular case d≥3 and p= d or d′.
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