CAREER: Front Propagations and Viscosity Solutions

这个项目涉及一些在物理、经济和工程中自然出现的非线性偏微分方程组(PDE)，这些方程在晶体生长、复合材料、燃烧、博弈论和最优控制理论的研究中出现。所研究的方程与许多其他数学领域有很深的联系，包括变分、微分对策、动力系统、几何学、齐次化理论和反问题。该项目的主要目标是发现新的基本原则和一般方法，以了解所调查的偏微分方程组的解的性质。研究的一个关键对象是晶体生长模型，在该模型中，晶体既通过吸附原子在水平方向生长，又通过在过饱和介质中成核在垂直方向生长。为了实际使用该模型，深入了解晶体生长速度的定性和定量方面是极其重要的。该项目的一个组成部分是教育部分，包括通过各种活动培养威斯康星大学麦迪逊分校PDE研究生的人数。对PDE感兴趣的即将入学的研究生被鼓励参加首席研究员的PDE阅读研讨会，并与该地区的同行和博士后进行更多的互动。在本科培养方面，首席调查员计划通过一些个人辅导计划和两个本科暑期班来提高威斯康星大学麦迪逊分校数学专业本科生学习分析和偏微分方程的兴趣。此外，首席研究员计划为早期职业研究人员组织两次关于非线性偏微分方程及相关主题的会议。第一部分是关于具有驱动项和源项的水平集平均曲率方程，以及它在晶体生长模型中的应用，在该模型中，未知的每个水平集通过其单位恒定力的平均曲率随时间演化。第二个涉及Eikonal方程和齐化，其中未知的零能级集在具有高度振荡的法向速度的周期环境中运动。首席研究员和他的合作者最近开发了一些新的方法，为这两个主题和相关领域的几个未决问题提供了解决方案。预计新方法将在该项目中得到进一步开发，从而为非线性PDE和粘度解决方案领域带来新的视角和见解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns some nonlinear partial differential equations (PDEs) that appear naturally in physics, economics, and engineering and that arise, for example, in the study of crystal growth, composite materials, combustion, game theory, and optimal control theory. The equations studied have deep connections with a host of other areas of mathematics, including the calculus of variations, differential games, dynamical systems, geometry, homogenization theory, and inverse problems. The main goal of the project is to discover new underlying principles and general methods to understand the properties of solutions of the PDEs under investigation. A key object of the research is a crystal growth model, in which the crystal grows in both horizontal direction by adatoms, and in vertical direction by nucleation in a supersaturated media. To make practical use of the model, it is extremely important to understand deeply the qualitative and quantitative aspects of the growth speed of the crystal. An integral part of the project is the educational component including bringing up the number of graduate PDE students at University of Wisconsin-Madison through various activities. The incoming graduate students interested in PDE are encouraged to participate in the principal investigator's PDE reading seminar, and to interact more with their peers and postdocs in the area. In term of undergraduate training, the principal investigator plans to increase the interest of University of Wisconsin-Madison undergraduate mathematics majors in the study of Analysis and PDE through some individual mentoring plans, and two undergraduate summer schools. Besides, the principal investigator plans to organize two conferences in nonlinear PDE and related topics for early career researchers.The proposed research involves two themes. The first is about a level-set mean curvature equation with driving and source terms, and applications to a crystal growth model, in which each level set of the unknown evolves in time by its mean curvature with unit constant force. The second involves Eikonal equations, and homogenization, where the zero-level set of the unknown moves in a periodic environment with highly oscillatory normal velocity. The principal investigator and his collaborators have recently developed some new approaches, which provided solutions to several open problems in these two themes and related areas. The new approaches are expected to be developed further in this project, thereby bringing fresh perspectives on and insights into the field of nonlinear PDE and viscosity solutions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.