Defects and patterns in the Calculus of Variations

点击翻译按钮获取中文摘要

The object of this research proposal is the rigorous mathematical analysis of variational problems arising in physics, and of the solutions of the associated systems of partial differential equations (PDE). For many physical systems the realizable configurations are those which minimize (globally or locally) an energy functional. Among the functionals we will study are: Ginzburg-Landau models, introduced in the context of superconductivity, but also a paradigm for many phenomena in condensed matter physics and materials; the Landau-de Gennes model for nematic liquid crystals; and the Ohta-Kawasaki energy for di-block copolymers. These models have many fundamental features which reveal themselves in singular perturbation limits, when one or more parameters in the model are sent to zero or infinity. In these limiting regime the solutions are observed to develop geometrical singularities, such as vortices, disclinations, or domain walls, and these defects give the most salient features of the system.******The overall goal is to develop new analytical tools to study singularly perturbed variational problems, to shed light on physical phenomena observed in experiments and simulations. The emphasis will be on variational problems for vector-valued functions, for which the Euler-Lagrange equations will be systems of nonlinear PDE. Many tools normally employed in studying a single PDE do not extend to systems, and so the analytical challenges are considerable but the mathematical interest is great. For instance, in the Landau-de Gennes theory of liquid crystals the states are described by Q-tensor fields (symmetric traceless matrix-valued functions,) and the singularities often occur because of eigenvalue crossing. This presents analytical hurdles in identifying and localizing singularities (line and ring defects, or "boojums" lying on boundary surfaces) which have subtle energy signatures. Further, mathematical models of block copolymers incorporate essential nonlocal interactions, which adds an extra layer of complexity to the PDEs describing these materials.******Solving these problems will entail the development of new methods for systems of nonlinear and nonlocal PDE. I will blend my own ideas with innovations coming from nonlinear and geometric analysis, including sharp energy bounds (via vortex-ball constructions); monotonicity and eta-ellipticity methods (developed for harmonic maps); bifurcation techniques; Gamma-convergence techniques (giving limiting energies characterizing the shape and interactions of singularities); and concentration-compactness methods (which break down minimizers with complex structure into component pieces.) The analytical results obtained will give a more complete and reliable understanding of these models and the phenomena they describe, while providing new perspectives on the rich interplay between analysis, geometry, and physics.