Research
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. He was an instructor at Princeton University from 1989 to 1991 and an assistant professor at Purdue University from 1991 to 1995. Shen joined the faculty at the University of Kentucky in 1995, where he was promoted to professor in 2003 and became chair of the Mathematics Department in 2007.
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, spectrum properties of Schrodinger operators, the Navier-Stokes equations, unique continuation problems, and Riesz transforms associated with elliptic operators. His most recent work has been centered on boundary value problems for elliptic systems and higher order elliptic equations in Lipschitz domains.
Shen was awarded a Centennial Fellowship by the American Mathematical Society in 1997. He delivered a one-hour invited address at the AMS Southeastern Section Meeting at Baton Rouge, Louisiana in March 2008. He currently holds the Ralph E. and Norma L. Edwards Research Professorship in Mathematics.
Selected Publications
The Neumann Problem in L^p on Lipschitz and Convex Domains, (with Aekyoung Shin Kim), J. of Functional Analysis 255, pp.1817-1830 (2008).
W^{1,p} Estimates for Elliptic Homogenization Problems in Nonsmooth Domains, Indiana Univ. Math. J. 57(5), pp.2283-2298 (2008).
Uniform Sobolev Inequalities and Absolute Continuity of Periodic Operators, (with Peihao Zhao), Transactions of AMS 360 (2008), pp.1741-1758.
The L^p Boundary Value Problems on Lipschitz Domains, Adv. in Math. 216 (2007), pp.212-254.
A Relationship between the Dirichlet and Regularity Problems for Elliptic Equations, Math. Res. Letters 14 (2007), pp.205-213.
On Estimates of Biharmonic Functions on Lipschitz and Convex Domains, J. Geometric Analysis 16 (2006), pp.721-734.
Necessary and Sufficient Conditions for the Solvability of the Lp Dirichlet Problem on Lipschitz Domains, Math. Ann. 336 (2006), pp.697-725.
The L^p Dirichlet Problem for Elliptic Systems on Lipschitz Domains, Math. Res. Letters 13 (2006), pp.143-159.
Weighted Estimates for Elliptic Systems in Lipschitz Domains, Indiana Univ. Math. J. 55 (2006), pp.1135-1154.
Bounds of Riesz Transforms on L^p Spaces for Second Order Elliptic Operators, Ann. Inst. Fourier (Grenoble) 55 (2005), pp.173-197.
Weighted Estimates in L^2 for Laplace's Equation in Lipschitz Domains, Transactions of AMS 357 (2005), pp.2843-2870.
On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domain, (with Loredana Lanzani), Comm. Partial Diff. Eqs. 29 (2004), pp.91-109.
The Spectrum of Schrodinger Operators with Positive Potentials in Riemannian Manifolds, Proceedings of AMS 131 (2003), pp.3447-3456.
Boundary Value Problems in Morrey Spaces for Elliptic Systems in Lipschitz Domains, Amer. J. Math. 125 (2003), pp.1079-1115.
On the Bethe-Sommerfeld Conjecture for Higher-Order Elliptic Operators, Math. Ann. 326 (2003), pp.19-41.
The Periodic Schrodinger Operators with Potentials in the Morrey Class, J. of Functional Analysis 193 (2002), pp.3145-345.
Absolute Continuity of Periodic Schrodinger Operators with Potentials in the Kato Class, Illinois J. Math. (2001) 45, 873-893.
On Absolute Continuity of the Periodic Schr\"odinger Operators, International Math. Res. Notices 2001 (1) (2001), pp.1-31.
On the Dimension of the Attractor for the Non-Homogeneous Navier-Stokes Equations in Non-Smooth Domains, (with R. Brown and P. Perry), Indiana Univ. Math. J. 49 (2000), pp.81-112.
On Fundamental Solutions of Generalized Schr\Odinger Operators, J. of Functional Analysis 167 (1999), pp.521-564.
On Moments of Negative Eigenvalues for the Pauli Operator, J. Diff. Eqs. 149 (1998), pp.292-327.
Regularity of Solutions to a Contact Problem, (with R. Brown and P. Shi), Transactions of AMS. 350 (1998), pp.4053-4063.
On Bounds of N(\lambda) for a Magnetic Schr\"odinger Operator, Duke Math. J. 94 (1998), pp.479-507.
The Additive Turbulent Decomposition for the Two-Dimensional Incompressible Navier-Stokes Equations:Convergence Theorems and Error Estimates, (with R. Brown and P. Perry), SIAM J. Appl. Math. 59 (1998), pp.139-155.