David Kinderlehrer

Existence and Partial Regularity of Static Liquid Crystal Configurations

We establish the existence and partial regularity for solutions of some boundary-value problems for the static theory of liquid crystals. Some related problems involving magnetic or electric fields are also discussed.

Frustration in ferromagnetic materials

We examine the theory of micromagnetics developed by W. F. Brown. We show that in the case often considered, with exchange energy omitted, the minimum of the free energy is not attained for uniaxial materials but is attained for cubic materials. A study of the minimizing sequences reveals that these accurately model many features of observed domain structure. Finally, we reexamine the so-called “coercivity paradox” from the viewpoint of nonlinear stability theory.

Numerical approximation of the solution of variational problem with a double well potential

Variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value. Rather, the gradients of minimizing sequences do not converge pointwise and can have oscillations. However, the gradients do converge in the weak topology, i.e., their local spatial averages converge.Such functionals arise in the description of equilibria of crystals or other ordered materials. Stable configurations for solid crystals which have symmetry-related (martensitic) energy wells have a fine-scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit. An analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals.

Mathematical Questions of Liquid Crystal Theory

A liquid crystal is a mesomorphic phase of a material which occurs between its liquid and solid phases. Frequently the material is composed of rod-like molecules which display orientational order, unlike a liquid, but lacking the lattice structure of a solid. It may flow easily and so may also be thought of as an anisotropic fluid. This anisotropy is evident in the way it transmits light; for example, a nematic liquid crystal is optically uniaxial. We take the opportunity of this note to discuss a few of the analytical issues which arise in the attempt to study static equilibrium configurations. One attractive feature of this subject is that it has a well developed continuum description in the Ericksen-Leslie [E1,[L1] theory. Some of the questions have significance in the context of harmonic mappings into spheres and we shall attempt to clarify these connections. We refer to the articles by F. Leslie [L2],[L3] in these proceedings both for other aspects of the static theory and for an introduction to flow problems.

Smoothness of Linear Laminates

Linearly elastic laminates are examples of materials whose equilibrium equations may have only bounded measurable coefficients, yet whose solutions may be fairly smooth. This is quite different from general experience, where regularity of solutions is determined by the closeness of the system to a diagonal one. A particular situation where a laminate may appear is a highly twinned elastic or ferroelectric crystal, and there are questions related to these materials which make it useful to know some properties of these special systems of equations.

Minimum Energy Configurations for Liquid Crystals: Computational Results

Two numerical algorithms which have been successfully employed to compute minimum energy configurations for liquid crystals are given. The results of computational experiments using these algorithms show that several critical points of the energy functional are not local minima.

Elastic plastic deformation

The equilibrium configuration of an elastic perfectly plastic body may be described by its stress or its strain. By use of a first variation formula, a description of the strain tensor, not necessarily unique, is obtained from the stress, which is unique.Most aspects of this work extend to more general elastic plastic models, in particular ones which lack convexity.

Existence, uniqueness, and regularity results for the two-body contact problem

The problem of contact between two elastic bodies is studied under the assumption of nonzero initial gap in the potential contact region. The related variational inequality is stated and existence, uniqueness, and local regularity results are proved for its solution.

The Variety of Configurations of Static Liquid Crystals

Here we make several observations on a variety of special classes of harmonic maps from domains in IR3 to S 2. Such maps are relevant for the study of liquid crystals; see e.g. [HK1], [HKL1], [HKLu]. Part of our work is motivated by questions about the nonuniqueness and the number of harmonic maps having fixed boundary data. Here we show that the number may actually be infinite. For example, by Corollary 3.2 below there exists a one parameter family of distinct energy-minimizers each having the same Dirichlet boundary data.

Twinning of Crystals (II)

Certain properties of a crystalline substance may appear only below, for instance, a certain critical temperature and are frequently accompanied by equilibrium states exhibiting a marked decrease in symmetry. One such example of this is the appearance of twinned crystals in what sometimes may be regarded as an austenite/martensite transition. In the higher temperature austenite the crystal is cubic, while in the lower temperature martensite it is tetragonal. Another example is the appearance of spontaneous polarization in a ferroelectric, like Rochelle salt. Also in this instance, the crystal structure is more symmetric in the absence of spontaneous polarization.