CAREER: Front Propagations and Viscosity Solutions

该项目涉及一些自然出现在物理学，经济学和工程学中的非线性偏微分方程（PDE），例如，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着深刻的联系，包括变分法、微分对策、动力系统、几何、均匀化理论和反问题。该项目的主要目标是发现新的基本原理和一般方法来理解所研究的偏微分方程解的性质。研究的一个重要对象是晶体生长模型，在该模型中，晶体在水平方向上通过吸附原子生长，在垂直方向上通过成核在过饱和介质中生长。为了实际应用该模型，深入了解晶体生长速度的定性和定量方面是非常重要的。该项目的一个组成部分是教育部分，包括通过各种活动在威斯康星大学麦迪逊分校培养PDE研究生。鼓励对PDE感兴趣的研究生参加主要研究者的PDE阅读研讨会，并与该领域的同行和博士后进行更多的互动。在本科生培训方面，主要研究者计划通过一些个人指导计划和两个本科生暑期学校来提高威斯康星大学麦迪逊分校本科数学专业学生对分析和PDE研究的兴趣。此外，主要研究者计划组织两次非线性偏微分方程及相关主题的会议，以供早期职业研究者使用。第一个是关于一个水平集平均曲率方程的驱动和源项，并应用到晶体生长模型，其中每个水平集的未知的演变，其平均曲率与单位恒定的力。第二个涉及Eikonal方程，和均匀化，其中的零级设置的未知移动在一个周期性的环境中具有高度振荡的正常速度。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一些开放问题提供了解决方案。新方法有望在该项目中得到进一步发展，从而为非线性偏微分方程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知识价值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.