Variational Problems with Singularities

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 systems of partial differential equations (PDE) which describe physical states in these models. For many physical systems, the observable states of the system are characterized as minimizers of an associated energy functional. Critical points of these functionals typically solve a PDE system, the Euler-Lagrange equations. The unknown order parameter (or phase field) must balance competing potential and gradient terms in the energy, often leading to the formation of interesting geometrical patterns. A classical example is the Ginzburg-Landau model which was originally introduced in the context of superconductivity. The mathematical models discussed in this proposal all arise from the physical literature or from personal communication with physicists and materials scientists, working to explain the fundamental properties of liquid crystals, block co-polymers, micro-magnetic systems, or atomic and molecular binding. These are current issues in physical science research, and while the proposed problems are at a fundamental theoretical level they are connected to more applied questions of designing optical devices, medical sensors, or nano-scale self-assembly in co-polymer solutions Solving these problems will entail the development of new techniques for studying systems of nonlinear (and nonlocal) PDE, by developing new and refined ideas and methods in nonlinear and geometric analysis, including sharp energy bounds, monotonicity and eta-ellipticity methods, and measure-theoretic techniques of Gamma-convergence and concentration-compactness. These mathematical advances will be suggested in part by physical insight and formal calculations, but will be grounded in the fundamentals of nonlinear analysis and PDE regularity theory. The analytical results obtained will give a more complete and reliable understanding of the physical models and the phenomena they describe, while providing new perspectives on the rich interplay between analysis, geometry, and physics.