Regularity for partial differential equations

在这个项目中，主要研究人员从事的是非线性偏微分方程、变分法和奇异摄动理论的研究项目。研究课题来自三个应用背景：超导的Ginzburg-Landau模型、薄膜起泡的von Karman模型和微磁学的模型。在超导领域，P.I.将重点关注样品对大磁场的响应，特别关注从正常状态到超导状态的分叉。在薄膜起泡方面，P.I.将通过分析起泡生长和薄膜生长的各种动力学模型来研究起泡区域不稳定性的本质。最后，在微磁学领域，P.I.将分析性地探索一种模型，该模型被认为可以捕捉到一种与样品中的几何收缩相关的新型磁壁结构。这个项目涉及各种材料在受到外场作用或被迫呈现特定形状时的行为。这些系统的能量通常通过一个函数来描述，这个函数通常被称为‘序参数’，它的值表明在给定的一组环境(如几何、外加磁场等)下，材料呈现什么状态。通过这种类型的研究，人们希望了解对于给定的样本来说，什么形状是最佳的，以增强或减少各种物理效应。例如，在超导体的情况下，人们希望了解哪些形状最有利于产生没有电阻损耗的超导电流。相关的数学工具来自变分法和非线性偏微分方程组理论，以及前人应用的渐近分析方法

In this project, the principal investigator pursues a research program in nonlinear partial differential equations, the calculus of variations and singular perturbation theory. The researchtopics come from three applied settings: Ginzburg-Landau type models for superconductivity, von Karman type models for thin film blistering, and a model in micromagnetics. In the area of superconductivity, the P.I. will focus on the response of samples to large magnetic fields, with particular attention paid to the bifurcation from the normal state to a superconducting state. In the area of thin film blisters, the P.I. will investigate the nature of instabilities of the blistered region through the analysis of various dynamical models for blister growth and thin film growth. Finally, in the area of micromagnetics, the P.I. will analytically explore a model thought to capture a new kind of magnetic wall structure associated with a geometric constriction within the sample. This project concerns the behavior of various materials when subjected to outside fields or when forced to take on specific shapes. The energy of these systems is generally described through a function, often called an `order parameter,' whose values indicate what state is taken on by the material under a given set of circumstances (such as geometry, applied fields,etc.). Through this type of study, one hopes to gain an understanding of what shapes are optimal for a given sample in order to enhance or diminish various physical effects. For example, in the case of a superconductor, one hopes to learn which shapes are most conducive to producing a supercurrent that conducts without losses due to resistance. The relevant mathematical tools come from the calculus of variations and from the theory of nonlinear partial differential equations, as well as from methods of asymptotic analysis as applied to the previou