Calculus of Variations and Partial Differential Equations

在应用的激励下，PI将研究在高振荡或随机介质中由偏微分方程描述的各种动力系统的均匀和定量可控性问题。目前可用于研究此类问题的方法是均匀的或缓慢变化的介质，但可能不适用于许多经常具有异质性、噪声和随机性的现实世界问题。因此，需要一套新的思想和技术来在已知的均匀情况理论和与振荡现象有关的理论之间取得微妙的平衡。广泛应用于显示设备的液晶动力学也将被研究。这涉及到一些迷人而困难的理论问题，这些问题涉及流体动力学方程和某些几何物体的流动的非线性耦合，可能会导致一个新的研究方向。该项目是研究院研究生和博士后培养计划的重要组成部分。所取得的成果将通过出版物以及为教育目的而设计的讲座、研讨会、会议和暑期学校来传播。PI打算研究三组从变分学和偏微分方程的问题。第一组问题是关于分析具有高振荡系数的偏微分方程。在（周期和随机）均匀化的一般理论下，关于这一主题已经积累了大量的知识。然而，有许多有趣而深刻的开放问题，如与定量（大尺度和小尺度）唯一延拓有关的问题，进化问题的均匀和精确可控性，以及在这种高振荡介质中相关逆问题的稳定性估计。第二组问题是PI早期对液晶流体动力学研究的延续，重点是对潜在非线性耦合结构的探索。对于Ericksen-Leslie系统，我们特别感兴趣的是在三维中合适弱解的全局存在性和在二维中有限时间爆炸的存在性。项目的最后一部分是研究椭圆特征值的一些极值问题及其相关的自由边界。这些问题是由研究的最佳设计，模式形成，和其他应用在凝聚态物理。这是一个为期三年的研究计划，应该产生新的想法，方法，并在主题上的重要成果，它建立在PI以前的研究成果。该奖项反映了美国国家科学基金会的法定使命，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。

Motivated by applications, the PI will study problems concerning the uniform and quantitative controllability for various dynamical systems described by partial differential equations in highly oscillatory or random media. The methods currently available to investigate such problems for the case of homogeneous or slowly varing medium may not be applicable to many real-world problems that often possess heterogeneity, noise, and randomness. Thus, a new set of ideas and techniques are needed to strike a delicate balance between the known theory for the homogeneous situation and the one related to the oscillating phenomena. The dynamics of liquid crystals that are widely used in display devices will also be studied. This involves fascinating and diffcult theoretical questions that deal with the nonlinear coupling of fluid dynamical equations and flows of certain geometric objects that may lead to a new direction of research. The project is an important and integral part of the PI's training program for graduate students and postdoctoral researchers. The results obtained will be disseminated through publications and through lectures, seminars, conferences, and summer schools that are designed for educational purposes.The PI intends to study three sets of problems from calculus of variations and partial differential equations. The first set of problems is concerned with analysis of partial differential equations with highly oscillatory coefficients. A tremendous amount of knowledge has been accumulated on this topic under the general theory of (periodic and stochastic) homogenization. However, there are many interesting and deep open problems such as those related to quantitative (for both large and small scales) unique continuations, uniform and exact controllability of evolution problems, and stability estimates on the related inverse problems in such highly oscillatory medium. The second set of problems are a continuation of the PI's earlier studies on the hydrodynamics of liquid crystals with an emphasis on the exploration of the underlying nonlinear coupling structure. Of particular interest are global existence of suitable weak solutions in 3D and the existence of finite time blow ups in 2D for the Ericksen-Leslie system. The last part of the project is to investigate some extremum problems of elliptic eigenvalues and the associated free boundaries. These problems are motivated by studies in optimal designs, pattern formation, and other applications in condense matter physics. This is a three year research program that should yield new ideas, methods, and important consequences on the subjects, and it builds upon the PI's previous research accomplishments.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.