Author ORCID Identifier

Date Available


Year of Publication


Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation


Arts and Sciences



First Advisor

Dr. Zhongwei Shen


We establish W1,p estimates for solutions uε to the Laplace equation with Dirichlet boundary conditions in a bounded C1 domain Ωε, η perforated by small holes in ℝd. The bounding constants will depend explicitly on epsilon and eta, where epsilon is the order of the minimal distance between holes, and eta denotes the ratio between the size of the holes and epsilon. The proof relies on a large-scale Lp estimate for ∇uε, whose proof is divided into two main parts. First, we show that solutions of an intermediate problem for a Schr¨odinger operator in Ω can be used to approximate harmonic functions in Ωε, η as epsilon and eta approach zero. We then use a real-variable method to establish the large-scale Lp estimate for ∇uε. Sharpness is established for these results in all cases except when d ≥ 3 with p = d or d'.

Digital Object Identifier (DOI)

Funding Information

This study was supported by the National Science Foundation Division of Mathematical Sciences grants 1856235 and 2153585.

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