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

该项目致力于研究各种中心数学问题和来自物理、力学和计算机科学的问题。三个主要的主题是电磁波，索博列夫空间和相关问题，以及在弹性中的应用变分法。对于第一个主题，主要研究者打算研究麦克斯韦方程组的近似隐身问题和演化方程的广义阻抗边界条件问题。第一个方向的目标是了解是否可以在各种情况下使用变换光学（或变量方案的变化）近似地掩盖对象。关于第二个方向，主要研究者寻求获得高导电性障碍物的一般阻抗边界条件。项目的这一部分旨在提出新的问题，并回答由图像处理和Sobolev空间的最新方法引起的新问题。关于第三个主题，该项目研究了由薄弹性膜变形形成的图案。本文考虑了以下三个问题：(1)放置在开口圆柱体顶部的圆形弹性薄板在薄板中心受到向下的力作用时的变形；(2)漂浮的弹性薄膜的变形；(3)粘接在柔性基材上的薄膜的变形。这个项目希望找到的问题的解决方案的理解将推动许多技术应用（例如，隐形，图像处理，确定材料的性质）。所提出的方法利用了分析、变分演算和应用数学中的各种工具。这些似乎是新颖和强大的，它们可以广泛应用。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师、博士后学者和研究生之间产生重要的研究和教学互动。首席研究员计划在他的课程（微积分，偏微分方程和变分微积分）中使用该提案中开发的技术和结果。由于研究是多学科性质的，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.