Of concern to quantum chemists and solid state physicists is the approximate numerical computation of the ground state wave function, and the ground state energy and density for molecular and other quantum mechanical systems. Since the number of molecules in bulk matter is of the order of 10e26 , direct computation is too cumbersome or impossible in many situations. In 1927, L. Thomas and E. Fermi proposed replacing the ground state wave function by the ground state density, which is a function of only three variables. Independently, each found an approximate expansion for the energy associated with a density. (The wave function uniquely determines the density, but not conversely.)

A computationally better approximate expansion was found in the 1960’s by W. Kohn and his collaborators; for this work Kohn got the Nobel Prize in Chemistry in 1998. A successful attempt to put Thomas-Fermi theory into a rigorous mathematical framework was begun in the 1970’s by E. Lieb and B. Simon and was continued and expanded by Ph. Benilan, H. Brezis and others. Very little rigorous mathematics supporting Kohn density functional theory is known. In this talk I will present a survey of rigorous Thomas-Fermi theory, including recent developments and open problems (in the calculus of variations and semilinear elliptic systems).

In 1985, J. Ericksen derived a model for uniaxial liquid crystals to allow for disclinations (i.e. line defects or curve singularities). It involved not only a unit orientation vectorfield on a region of R^3  but also a scalar order parmeter quantify- ing the expected inner product between this vector and the molecular orientation. FH.Lin, in several papers, related this model, for certain material constants, to harmonic maps to a metric cone over S^2. He showed that a minimizer would be continuous everywhere but would have higher regularity fail on the singular de- fect set s^{-1}(0). The optimal partial regularity result of R.Hardt-FH.Lin in 1993, for this model, led to regularity away from isolated points, which unfortunately still excluded line singularities. This paper accordingly also introduced a modified model involving maps to a metric cone over RP^2, the real projective plane. Here the nontrivial homotopy leads to the optimal estimate of the singular set being 1 dimensional. In 2010, J. Ball and A.Zarnescu discussed a derivation from the de Gennes Q tensor and interesting orientability questions using RP2. In recent ongo- ing work with FH.Lin and O. Alper, we see that the singular set with this model necessarily consists of Holder continuous curves. We will also survey some of the many more elaborate liquid crystal PDE’s involving a general director functional, the full Q tensor model, and possible coupling with fluid velocity. 

In this talk I discuss the different types of models which originate from a lubrication approximation of viscous coating flow dynamics on the outer surface of a rotating cylinder, that is in the presence of a gravitational field. Analytical and numerical results related to existence, uniqueness and stability of solutions will be presented.

The existence and uniqueness of Gevrey regularity solution for a class of nonlinear
bistable gradient flows, with the energy decomposed into purely convex and concave parts,
such as epitaxial thin film growth and square phase field crystal models, are discussed in this talk.
The polynomial pattern of the nonlinear terms in the chemical potential enables one to derive a
local in time solution with Gevrey regularity, with the existence time interval length dependent
on certain functional norms of the initial data. Moreover, a detailed Sobolev estimate for the gradient
equations results in a uniform in time bound, which in turn establishes a global in
time solution with Gevrey regularity. An extension to a system of gradient flows,
such as the three-component Cahn-Hilliard equations, is also addressed in this talk.