Structure of Local Minimizers in Superconductivity and Models for Phase Transitions

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9705774 Sternberg In this proposal, the principal investigator decribes a research program in nonlinear partial differential equations and the calculus of variations. The research topics come from three areas: the Ginzburg-Landau model for superconductivity, the Cahn-Hilliard model for phase transitions, and the structure of stable solutions to volume-constrained least area problems. A common goal throughout the proposal is to study the existence, disappearance, and structure of local minimizers to problems arising in continuum mechanics when relevant parameters are altered. In the work centered on superconductivity, the P.I. will focus on problems concerning the complex relationship between the size of an applied magnetic field and the behavior of vortices which appear within the superconducting sample. The projects related to phase transitions concern the structure of transition layers bridging two (or more) phases within the context of the Cahn-Hilliard theory of phase transitions. This work is inextricably linked to the study of volume-constrained least area problems. These geometry problems are quite subtle, and their resolution should shed light on the structure of phase transitions in the Cahn-Hilliard setting as well. This proposal concerns research projects related to the study of superconductors and to the study of phase transitions in fluids or alloys. In the production of superconductors, a major concern is the emergence of vortices which are little "defects" in the sample leading to a loss of perfect conductivity. In this proposal, the P.I. will use the so-called Ginzburg-Landau theory as a model for the superconductor in order to investigate the emergence, disappearance, and placement of these "defects" when the sample is subjected to a magnetic field. In the research devoted to phase transitions, the P.I. will study mathematical models for binary alloys (Cahn-Hilliard theory) and for two-fluid systems (van der Waals the ory). The goal is to understand the possible shape of the boundaries between different phases of a medium. Particular attention will be paid to the effect of the boundary of the container holding the medium on the structure of these so-called transition layers. These studies are related to questions about the possible shape of bubbles confined to a container. Since these bubbles tend to hug the wall of the enclosing container, one wants to understand how the specific shape of the container will affect the structure of the stable bubble.