Partial differential equations with rapidly oscillating coefficients are used to model various processes in materials with rapidly oscillating microstructures, such as composite and perforated materials. The theory of homogenization shows that such strongly inhomogeneous material may be approximately described via a homogenized or effective homogeneous material. In this general talk I will introduce the theory of homogenization and describe some of the recent progress in the quantitative homogenization of elliptic equations and systems with periodic coefficients. The results to be presented includes optimal regularity estimates, up to the boundary and uniform with respect to the inhomogeneity scale, and sharp convergence rates.
A native of Hunan, Zhongwei Shen received his B.S. in Mathematics from Peking University at the age of eighteen. He received his Ph.D. from the University of Chicago under the supervision of Carlos Kenig. He joined the faculty at the University of Kentucky in 1995 and served as chair of its Mathematics Department from 2007 to 2011. Zhongwei Shen is a member of the inaugural class of the American Mathematical Society Fellows (2012).