Variational Problems with Singularities

本研究的目的是对物理学中的变分问题以及描述这些模型中的物理状态的偏微分方程（PDE）系统进行严格的数学分析。对于许多物理系统，系统的可观测状态被表征为相关联的能量泛函的极小化器。这些泛函的临界点通常求解一个偏微分方程系统，即欧拉-拉格朗日方程。未知的序参数（或相场）必须平衡能量中的竞争势和梯度项，这通常会导致有趣的几何图案的形成。一个经典的例子是金兹伯格-朗道模型，它最初是在超导的背景下引入的。 本提案中讨论的数学模型都来自物理文献或与物理学家和材料科学家的个人交流，致力于解释液晶，嵌段共聚物，微磁系统或原子和分子结合的基本特性。 这些都是物理科学研究中的当前问题，虽然所提出的问题处于基础理论水平，但它们与设计光学器件，医学传感器或共聚物溶液中的纳米级自组装等更多应用问题有关 解决这些问题将需要发展新的技术研究系统的非线性（和非局部）偏微分方程，通过发展新的和完善的想法和方法，在非线性和几何分析，包括尖锐的能量界限，单调性和η-椭圆方法，和测量理论技术的伽玛收敛和浓度紧。这些数学上的进步将部分地由物理洞察力和正式计算提出，但将以非线性分析和PDE正则性理论的基础为基础。所获得的分析结果将使人们对物理模型及其所描述的现象有一个更完整和可靠的理解，同时为分析、几何和物理之间的丰富相互作用提供了新的视角。

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.