Analysis of singular structures in elliptic and parabolic PDE with curvature effects

这个项目的重点是研究奇异结构的特点，解决各种偏微分方程和变分问题，出现在重要的物理模型。这里要研究的奇异结构有两种：涡（即复值阶参数的零点）和相边界（即实值阶参数的节点集）。大部分工作都是关于物理模型的设置——Ginzburg-Landau， Gross-Pitaevskii， Allen-Cahn或Kawasaki-Ohta——被考虑在曲面上。总的目标是进一步了解曲率在稳定或不稳定漩涡和相边界中所起的作用。首席研究员计划通过一系列技术来解决这些问题，包括伽玛收敛、重整化能量方法、二次变分参数、约束最小化和抛物线非线性偏微分方程方法。在这个项目中要进行的研究源于超导体和超流体的数学模型、合金的晶界和二嵌段共聚物。这项研究的主要目的是更好地理解曲率在影响这些物理系统的行为方面可能发挥的作用。例如，近年来，物理学家已经成功地制造出了球形的超导体，而不仅仅是扁平的，但是在这一点上，很少有理论工作致力于理解这种形状如何影响超导体对外加磁场的响应。超导表面的曲率可能会带来隐藏的好处或令人惊讶的行为，这些可以通过对潜在模型的仔细数学分析来揭示。首席研究员将与博士生合作开展这项研究，博士生将获得研究这些数学应用的宝贵经验，作为他们论文工作的一部分。

This project focuses on the investigation of singular structures characterizing the solutions to various partial differential equations and variational problems that arise in significant physical models. There are two types of singular structures to be studied here: vortices (i.e., zeros of complex-valued order parameters) and phase boundaries (i.e., nodal sets of real-valued order parameters). Much of the work concerns the setting where the physical model--Ginzburg-Landau, Gross-Pitaevskii, Allen-Cahn, or Kawasaki-Ohta--is considered on a curved surface. The general goal is to further our understanding of what role curvature plays in the stabilization or destabilization of vortices and phase boundaries. The principal investigator plans to attack these problems through a combination of techniques including gamma-convergence, renormalized energy methods, second-variation arguments, constrained minimization, and parabolic nonlinear partial differential equations methods.The investigations to be pursued in this project arise from mathematical models for superconductors and superfluids, grain boundaries in alloys, and di-block copolymers. The main thrust of the research is to gain a better understanding of how curvature may play a role in affecting the behavior of these physical systems. For example, in recent years physicists have succeeded in producing superconductors in the shape of spherical shells, rather than just being flat, but at this point there is little theoretical work devoted to understanding how this shape may affect the response of the superconductor to an applied magnetic field. There may be hidden benefits or surprising behavior induced by the curvature of the superconducting surface that can be revealed through a careful mathematical analysis of the underlying models. The principal investigator will carry out much of this research in collaboration with Ph.D. students who will gain valuable experience studying these applications of mathematics as part of their dissertation work.