A three-year continuing grant is recommended in support of mathematical models and methods related to problems in continuum physics. The systems of differential equations which occur and the associated variational principles used to derive the equations are often unstable, incompletely posed or not well-posed. Investigations of several aspects of these problems, requiring an assortment of methods, will be carried out. Objectives, from the viewpoint of continuum theory, concern stability, phase transitions and defects which occur in ordered materials. Primary examples appropriate to this work are liquid crystals and crystalline solids. Mathematically, one wishes to obtain analytic interpretations of certain continuum phenomena, expressed in terms of constraints and invariance groups and the behavior of solutions for typical problems. Of special interest is the change of constraints or groups, leading them to become the mechanisms of phase transitions. Singular behavior may be exhibited as well in these circumstances. Solutions of the static equilibrium configuration correspond to stationary points of the energy density. They take the form of harmonic mappings into the sphere in three dimensions. When boundary conditions are placed on the model, singularities can arise in solutions (called defects in the literature). Defects play a central role in understanding the behavior of the material. Major new theoretical work and initial numerical procedures have been developed by a number of researchers in the past two years. The present work will continue analyzing the theory of defects and establishing validation of numerical procedures. Additional work will focus on uniqueness of solutions and measurements of the size (Hausdorff dimension) of the singular set. This research, part of a larger effort dedicated to the study of crystal structures, represents an impressive combination of mathematically sophisticated ideas and models of the physical world. The steps taken by the principal investigator in clarifying some of the difficulties has been impressive. Others are now being attracted to this area. Funding should be continued without question.
