Mathematical analysis and its applications to electromagnetic waves, image processing, and materials science

这个项目致力于研究各种核心的数学问题，以及由物理、力学和计算机科学强烈驱动的问题。三个主要主题是电磁波、索波列夫空间和相关问题，以及变分法及其在弹性力学中的应用。对于第一个主题，主要研究者打算研究麦克斯韦方程的近似伪装问题和发展方程的广义阻抗边界条件问题。第一个方向的目标是了解在各种情况下是否可以使用变换光学(或可变方案的改变)近似地隐身对象。对于第二个方向，首席调查者寻求获得高导电障碍物的一般阻抗边界条件。该项目的这一部分旨在提出新的问题和回答新的问题，这些问题是由图像处理和最近对索博列夫空间的方法引起的。关于第三个主题，该项目研究了由弹性薄膜变形形成的图案。研究了以下三个问题：(1)放置在开口圆柱体上的弹性薄板在薄板中心受到向下压力的变形，(2)浮动弹性薄膜的变形，以及(3)粘结在柔顺衬底上的薄膜的变形。了解这些问题的解决方案将促进许多技术上的应用(例如，隐身、图像处理、材料性质的确定)。所提出的方法利用了分析、变分和应用数学中的各种工具。它们看起来既新颖又坚固，可能会得到广泛应用。该项目的主要目标之一将是在明尼苏达大学数学学院的教职员工、博士后学者和研究生之间产生重要的研究和教学互动。首席研究人员计划在他的课程(微积分、偏微分方程式和变分)中使用本提案中开发的技术和结果作为研究主题。由于这项研究本质上是多学科的，他还计划参加多学科和跨学科的会议和研讨会。

This project is devoted to the study of various central mathematical questions and problems strongly motivated from physics, mechanics, and computer sciences. The three main themes are electromagnetic waves, Sobolev spaces and related problems, and the calculus of variations with applications to elasticity. For the first theme, the principal investigator intends to study questions on approximate cloaking for Maxwell's equations and generalized impedance boundary conditions for evolution equations. The goal of the first direction is to understand whether or not one can approximately cloak an object using the transformation optics (or the change of variable scheme) under various circumstances. Regarding the second direction, the principal investigator seeks to obtain general impedance boundary conditions for highly conductive obstacles. This part of the project is intended to pose new problems and answer new questions motivated by image processing and recent approaches to Sobolev spaces. As to the third theme, the project investigates patterns formed by deformations of thin elastic membranes. The following three problems are considered: (1) the deformation of a thin circular elastic sheet placed on top of an open cylinder and subject to a downward force at the center of the sheet, (2) the deformation of a floating thin elastic membrane, and (3) the deformation of a thin film bonded to a compliant substrate.The understanding of the problems whose solutions this project hopes to find will advance many applications in technology (e.g., cloaking, image processing, determining the property of materials). The proposed methods make use of various tools in analysis, the calculus of variations, and applied mathematics. These appear to be novel and robust, and they could be widely applicable. One of the main goals of the project will be to generate significant research and teaching interactions among faculty, postdoctoral scholars, and graduate students associated with the University of Minnesota's School of Mathematics. The principal investigator is planning to use the techniques and results developed in this proposal for research topics in his classes (calculus, partial differential equations, and the calculus of variations). Since the research is multidisciplinary in nature, he is also planning to participate in multi- and interdisciplinary conferences and workshops.