Variational methods for Ginzburg-Landau systems

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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). The Ginzburg-Landau model was originally introduced in the context of superconductivity, but mathematical models of a similar kind have become ubiquitous in the study of physical systems, including Bose-Einstein condensation, micromagnets, copolymers, and liquid crystals. In certain limiting regimes 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 in this research program is to develop new analytical tools to study singularly perturbed Ginzburg-Landau systems and their geometrical singularities. The proposed problems differ from previous work in that they concern vector-valued functions, yielding systems of nonlinear PDE. Many tools normally employed in studying a single PDE (such as explicit solutions, comparison principles, phase-plane analysis, Liouville theorems) do not extend easily to systems. The projects I propose for the grant period concern specific Ginzburg-Landau systems exhibiting singularities of two types, vortices and domain walls, and their resolution will yield insights into the nature of singularity formation in Ginzburg-Landau models in general. Solving them will entail the development of new techniques for studying systems of nonlinear PDE, by melding my own ideas and methods with innovations coming from various areas in nonlinear and geometric analysis. For example, this includes sharp energy bounds (via vortex-ball constructions or similar measures of concentration); monotonicity and eta-ellipticity methods (as developed in studying harmonic maps); bifurcation techniques; Gamma-convergence techniques (for identifying limiting energies which characterize singularity shape and interactions); and concentration-compactness methods. These mathematical advances will be suggested in part by physical insight and formal calculations, but will be based on methods of nonlinear analysis and PDE regularity theory. 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.