Structure of Local Minimizers in Superconductivity and Models for Phase Transitions

9705774 斯滕贝格 在这个建议中，主要研究者描述了一个非线性偏微分方程和变分法的研究计划。研究课题来自三个方面： 金兹堡-朗道超导模型 相变，和结构的稳定解决方案，体积约束的最小面积问题。一个共同的目标，在整个建议是研究的存在，消失，和结构的局部极小化所产生的问题，在连续介质力学时，相关参数被改变。在工作中， 超导性，P.I.将集中在有关问题的复杂关系的大小之间的一个应用磁场 以及超导体内部出现的涡流的行为 sample. 与阶段过渡有关的项目涉及 过渡层的结构桥接内部的两个（或更多个）相 Cahn-Hilliard相变理论这 工作是密不可分的研究体积约束最小 地区问题。这些几何问题是相当微妙的，他们的决议应该阐明的结构相变在卡恩-希利亚德设置以及。 这项建议涉及与超导体研究和流体相变研究有关的研究项目， 合金.在超导体的生产中，一个主要的问题是 涡流的出现是样品中的小“缺陷”，导致完美导电性的损失。在这份提案中，P.I.将使用所谓的Ginzburg-Landau理论作为超导体的模型，以研究当样品受到磁场时这些“缺陷”的出现，消失和位置。 在致力于相变的研究中，P.I.将研究二元合金（凯恩-希利亚德理论）和双流体系统（货车德瓦尔斯理论）的数学模型。目标是了解不同阶段之间的边界的可能形状， 介质将特别注意的影响， 在这些所谓的过渡层的结构上保持介质的容器的边界。这些研究涉及到的问题是关于局限于容器中的气泡的可能形状。由于这些气泡倾向于拥抱封闭容器的壁，人们希望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.