This talk is concerned with the study of Dirichlet problems in a domain perforated with a large number of tiny holes. The study is motivated by the homogenization theory of elliptic equations and systems in perforated domains, which are used to model perforated materials and fluid flows in porous media. In a joint project with R. Righi and J. Wallace, we consider Laplace's equation and establish W^{1,P} estimates with sharp bounding constants depending explicitly on the minimal distances between the holes and the size of holes. The proof relies on a large-scale estimate for harmonic functions in perforated domains. A key step involves an observation that a harmonic function in a perforated domain may be well approximated by solutions of an intermediate problem for a Schodinger operator in a fixed domain.

Biography

Zhongwei Shen received his B.S. in mathematics from Peking University, M.S. from the Institute of Math of the Chinese Academy of Sciences, and his Ph.D. from the University of Chicago. He joined the faculty at the University of Kentucky in 1995, where he was promoted to professor in 2003 and served as chair of the Math Department from 2007 to 2011. Shen's research lies at the interface of harmonic analysis and partial differential equations. He is a member of the inaugural class of the American Math Society Fellow (2012) and a Simons Fellow (2021). In 2016 he was awarded the title of College of Arts and Sciences Distinguished Professor at the University of Kentucky.