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

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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.