Structure of Local Minimizers in Superconductivity and Models for Phase Transitions

9705774斯特恩伯格在这项建议中，首席研究员描述了一个关于非线性偏微分方程组和变分法的研究计划。研究课题来自三个领域：超导的Ginzburg-Landau模型，相变的Cahn-Hilliard模型，以及体积约束最小面积问题的稳定解的结构。整个提案的一个共同目标是研究当相关参数改变时，连续介质力学中出现的问题的局部极小值的存在、消失和结构。在以超导电性为中心的工作中，P.I.将重点研究外加磁场的大小与超导样品中出现的涡旋行为之间的复杂关系。与相变有关的项目涉及在卡恩-希利亚德相变理论的背景下连接两个(或更多)相的过渡层的结构。这项工作与体积约束最小面积问题的研究密不可分。这些几何问题相当微妙，它们的分辨率也应该有助于揭示卡恩-希利亚德环境中相变的结构。这项建议涉及与超导体研究和流体或合金相变研究有关的研究项目。在超导体的生产中，一个主要的问题是出现涡旋，它是样品中的小“缺陷”，导致完美导电性的损失。在这项提议中，P.I.将使用所谓的Ginzburg-Landau理论作为超导体的模型，以便研究当样品受到磁场影响时，这些“缺陷”的出现、消失和放置。在致力于相变的研究中，P.I.将研究二元合金(Cahn-Hilliard理论)和双流体系统(van der Waals理论)的数学模型。目标是了解介质不同相之间的边界的可能形状。将特别注意容纳介质的容器的边界对这些所谓过渡层的结构的影响。这些研究与限制在容器中的气泡可能形状的问题有关。由于这些气泡倾向于拥抱封闭容器的壁，人们想要了解容器的特定形状将如何影响稳定气泡的结构。

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.