Mathematical Sciences: Mathematical Theory of Liquid Crystals and Free Boundaries

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9401546 Lin This award supports mathematical research on problems in nonlinear partial differential equations, variational calculus and geometric measure theory. In the first part, work will be done analyzing singularities which contain the generalized version of Reifenberg's theorem and its applications, the p- harmonic mapping approach to Ginzburg-Landau vertices and the study of dynamic and deformation of defects in nematics. In addition, an investigation defects in biaxial liquid crystals will be initiated. The second part of the project concerns the evolution of liquid crystals. It is represented by a nonlinear system which can be viewed as a nonlinear coupling between flow of harmonic maps and a Navier-Stokes system. Of particular interest is the determination of global existence, uniqueness and stability of classical or weak solutions. Work will also be done in studying the partial regularity of such systems by exploring free interface problems arising in elasticity, optimal design and liquid crystal droplets. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***