Robert Hardt

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.

Stratification of real analytic mappings and images

Robert M. Hardt* (Minneapolis) Contents w 1. Introduction 193 w Stratification and Mappings 194 w Semianalytic Sets 198 w Semianalytic Shadows 200 w 5. Semianalytic Shadow Chains 207 1. Introduction Here it is shown that the image I of a semianalytic set under a proper (real) analytic mapping of (real) analytic manifolds admits a locally-finite partition into connected submanifolds P such that QcClosP and dimQ

Triangulation of subanalytic sets and proper light subanalytic maps

The smallest nonempty class ~c4 of subsets of a real analytic space M which is closed under the formation of locally finite unions, intersections, complements, and connected components and which contains Ac~g ~t {0} for any A in ,~ and realvalued function g analytic in a neighborhood of Clos A is the class of semianalytic subsets of M (see [6, w 1] for a list of references, the most basic being [11]). The smallest such class also containing all proper real analytic images of semianalytic sets is the (strictly larger, for dim M > 3 ) class of subanalytic subsets of M ([6, 8, 14, 15]). A continuous function from a subset of real analytic space M into a real analytic space N is called a subanalytic map if its graph is a subanalytic subset of M • N. Here triangulations of subanalytic sets are constructed using only subanalytic maps throughout. A map is proper (respectively, light) if its inverse image preserves compact (respectively, discrete) sets. Our main results are:

Minimal surfaces with isolated singularities

For n≥3, there exists an embedded minimal hypersurface in Rn+1 which has an isolated singularity but which is not a cone. Each example constructed here is asymptotic to a given, completely arbitrary, nonplanar minimal cone and is stable in case the cone satisfies a strict stability inequality.

A remark on H1 mappings

AbstractWith

Mathematical Questions of Liquid Crystal Theory

\mathbb{B} = \left\{ {\varepsilon \mathbb{R}^3 :\left| x \right|< 1} \right\}

Elastic plastic deformation

, we here construct, for each positive integer N, a smooth function