Zhongwei Shen

Education:
Ph.D., University of Chicago, 1989
M.S., Chinese Academy of Sciences, 1985
B.S., Peking University, 1982
Biography:

 A native of China, Zhongwei Shen received his B.S. in Mathematics from Peking University at the age of eighteen. After completing his M.S. at the Institute of Mathematics of the Chinese Academy of Sciences, he came to the USA and entered the University of Chicago in 1985, where he received his Ph.D. in Mathematics in 1989 under the supervision of Carlos Kenig. Shen joined the faculty at the University of Kentucky in 1995, where he was promoted to professor in 2003. He served as chair of the Mathematics Department from 2007 to 2011.

Research Interests:
analysis and partial differential equations
Research

Shen's research interests lie at the interface of harmonic analysis and partial differential equations. He is particularly interested in boundary value problems in non-smooth domains, the homogenization theory, spectrum properties of Schrodinger operators, the Navier-Stokes equations, unique continuation problems, and Riesz transforms associated with general second-order elliptic operators. His recent work, supportetd in part by the National Science Foundation and by the Simons Foundation (Simons Fellow 2021),  has focused on quantitative homogenization of partial differential equations.

Zhongwei Shen is a member of the inaugural class of the American Mathematical Society (AMS) Fellows (2012). He was awarded a Centennial Fellowship by the AMS in 1997. Shen held the Ralph E. and Norma L. Edwards Research Professorship in Mathematics from 2007 to 2010 and was awarded a University of Kentucky Research Professorship for 2013-2014. In 2016 he was awarded the title of College of Arts and Sciences Distinguished Professor.

A list of Shen's publications is available on MathSciNet http://www.ams.org/msnmain?fn=130&fmt=hl&pg1=IID&s1=227185&v1

 

Graduate Training

Harmonic Analysis; Partial Differential Equations.

Selected Publications:

                                                                                                                                                 

  • Sharp Convergence Rates for Darcy's Law, Communications in PDEs 47 (6), pp.1098-1123 (2022).
  • Compactness and Large-scale Regularity for Darcy's Law, Journal de Mathematiques Pures et Appliquees 163 (9), pp.673-701 (2022).
  • Uniform Boundary Controllability and Homogenization of Wave Equations, (with Fanghua Lin), Journal of European Mathematical Society 24, pp.3031-3053 (2022). Erratum, JEMS 24, pp.3055-3057 (2022).
  • Combined Effects of Homogenization and Singular Perturbations: Quantitative Estimates, (with Weisheng Niu), Asymptotic Analysis 128 (3), pp.351-384 (2022).
  • Large-Scale Lipschitz Estimates for Elliptic Systems with Periodic High-Contrast Coefficients, Communications in PDEs 46 (6), pp.1027-1057 (2021).
  • Quantitative Estimates in Reiterated Homogenization, (with Weisheng Niu and Yao Xu), Journal of Functional Analysis 279 (11), 108759 (2020).
  • Regularity of Homogenized Boundary Data in Periodic Homogenization of Elliptic Systems, (with Jinping Zhuge), Journal of European Mathamatical Society 22 (9), pp.2751-2776 (2020).
  • Homogenization of Parabolic Equations with Non-self-similar Scales, (with Jun Geng), Archive for Rational Mechanics and Analysis 236 (1), pp.145-188 (2020).
  • Asymptotic Expansions of Fundamental Solutions in Parabolic Homogenization, (with Jun Geng), Analysis & PDE 13 (1), pp.147-170 (2020).
  • Extrapolation for the L^p Dirichlet Problem in Lipschitz Domains, Acta Math Sinica (English Series) 35 (6), pp.1074-1084 (2019).
  • Nodal Sets and Doubling Conditions in Elliptic Homogeization, (with Fanghua Lin), Acta Math Sinica (English Series) 35 (6), pp.815-831 (2019).
  • Periodic Homogenization of Elliptic Systems, 291 pages, Birkhauser (2018).
  • Boundary Layers in Periodic Homogenization of Neumann Problems, (with Jinping Zhuge), Communications in Pure and Applied Mathematics 71 (11), pp.2163-2219 (2018).
  • Convergence Rates and Interior Estimates in Homogenization of Higher Order Elliptic Systems, (with Weisheng Niu and Yao Xu), Journal of Functional Analysis 274, pp.2356-2398 (2018).
  • Approximate Correctors and Convergence Rates in Almost-Periodic Homogenization, (with Jinping Zhuge), Journal de Mathematiques Pures et Appliquees 110, pp.187-238 (2018).
  • Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity, (with Jun Geng and Liang Song), Archive for Rational Mechanics and Analysis 224, pp.1205-1236 (2017).
  • Boundary Estimates in Elliptic Homogenization, Analysis & PDE 10, pp.653-694 (2017).
  • Convergence Rates in Parabolic Homogenization with Time-dependent Coefficients, (with Jun Geng), Journal of Functional Analysis 272, pp.2092-2113 (2017).
  • Convergence Rates in Periodic Homogenization of Systems of Elasticity, (with Jinping Zhuge),  Proceedings of the American Mathematical Society 145(3), pp. 1187-1202 (2017).
  • Lipschitz Estimates in Almost-Periodic Homogenization, (with Scott N. Armstrong), Communications in Pure and Applied Mathematics 69(10), pp. 1882-1923 (2016).
  • Homogenization of Stokes Systems and Uniform Regularity Estimates, (with Shu Gu), SIAM Journal on Mathematical Analysis 47(5), pp.4025-4057 (2015).
  • Convergence Rates and Holder Estimates in Almost-Periodic Homogenization of Elliptic Systems, Analysis & PDE 8, pp.1565-1601 (2015).
  • Uniform Regularity Estimates in Parabolic Homogenization, (with Jun Geng), Indiana University Mathematics Journal 64, pp.697-733 (2015).
  • Periodic Homogenization of Green and Neumann Functions, (with Carlos Kenig and Fanghua Lin), Communications in Pure and Applied Mathematics 67, pp.1219-1262 (2014).
  • On L^p Estimates in Homogenization of Elliptic Equations of Maxwell's Type, (with Liang Song), Advances in Mathematics 252, pp.7-21 (2014).
  • Homogenization of Elliptic Systems with Neumann Boundary Conditions, (with Carlos Kenig and Fanghua Lin), Journal of the American Mathematical Society 26, pp.901-937 (2013).
  • Estimates of Eigenvalues and Eigenfunctions in Periodic Homogenization, (with Carlos Kenig and Fanghua Lin), Journal of the European Mathematical Society 15, pp.1901-1925 (2013).
  • Resolvent Estimates in L^p for the Stokes Operator in Lipschitz Domains, Archive for Rational Mechanics and Analysis 205, pp.395-424 (2012).
  • Convergence Rates in L^2 for Elliptic Homogenization Problems, (with Carlos Kenig and Fanghua Lin), Archive for Rational Mechanics and Analysis 203, pp.1009-1036 (2012).
  • Layer Potential Methods for Elliptic Homogenization Problems, (with Carlos Kenig), Communications in Pure and Applied Mathematics 64, pp.1-44 (2011).
  • Homogenization of Elliptic Boundary Value Problems in Lipschitz Domains, (with Carlos Kenig), Mathematische Annalen 350, pp.867-917 (2011).
  • The L^p Boundary Value Problems on Lipschitz Domains, Advances in Mathematics 216, pp.212-254 (2007).
  • Necessary and Sufficient Conditions for the Solvability of the L^p Dirichlet Problem on Lipschitz Domains, Mathematische Annalen 336, pp.697-725 (2006).
  • On Absolute Continuity of the Periodic Schrodinger Operators, International Mathematics Research Notices 2001, pp.1-32 (2001). 
  • On Bounds of N(\lambda) for a Magnetic Schrodinger Operator, Duke Mathematical Journal 94, pp.479-507 (1998). 
  • Eigenvalue Asymptotics and Exponential Decay of Eigenfunctions for Schr\"odinger Operators with Magnetic Fields, Transactions of the American Mathematical Society 348, pp.4465-4488 (1996).
  •  L^p Estimates for Schrodinger Operators with Certain Potentials, Annales de L'Institut Fourier (Grenoble) 45, pp.513-546 (1995). 
  • Boundary Value Problems for Parabolic Lam\'e Systems and a Nonstationary Linearized System of Navier-Stokes Equations in Lipschitz Cylinders, American Journal of Mathematics 113, pp.293-373 (1991).