Dirichlet Problems in Perforated Domains

2024/02/20 by Righi, Robert, Shen, Zhongwei
#35B27 #35Q35 #76D07 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.13021

Abstract

In this paper we establish W^1,p estimates for solutions u_ε to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, C¹ domain Ω_{ε, η} in ℝ^d. 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 L^p estimate for ∇ u_ε, whose proof is divided into two parts. In the first part, we show that as ε, η approach zero, harmonic functions in Ω_{ε, η} may be approximated by solutions of an intermediate problem for a Schrödinger operator in Ω. In the second part, a real-variable method is employed to establish the large-scale L^p estimate for ∇ u_ε by using the approximation at scales above ε. The results are sharp except in the case d≥ 3 and p=d or d^′.

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