Calculus of Variations and Partial Differential Equations

受应用的启发，PI将研究有关的问题，在高度振荡或随机介质中的偏微分方程描述的各种动力系统的统一和定量可控性。目前研究均匀介质或缓变介质问题的方法可能不适用于许多具有非均匀性、噪声和随机性的实际问题。因此，需要一套新的思想和技术，以在均匀情况的已知理论和与振荡现象有关的理论之间取得微妙的平衡。此外，还将研究在显示装置中广泛使用的液晶的动力学。这涉及到迷人的和困难的理论问题，处理非线性耦合的流体动力学方程和流动的某些几何对象，可能会导致一个新的研究方向。该项目是PI研究生和博士后研究人员培训计划的重要组成部分。研究结果将通过出版物、讲座、研讨会、会议和暑期学校等形式进行传播。PI将研究变分法和偏微分方程中的三组问题。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，关于这个主题已经积累了大量的知识。然而，也有许多有趣的和深开放的问题，如有关的定量（无论是大规模和小规模）的唯一连续性，一致和精确的可控性的发展问题，以及稳定性估计的相关反问题，在这种高度振荡介质。第二组问题是PI早期对液晶流体力学研究的延续，重点是探索潜在的非线性耦合结构。特别令人感兴趣的是整体存在合适的弱解在3D和存在有限时间爆破在2D的Ericksen-Leslie系统。最后一部分是研究椭圆型特征值的极值问题及相应的自由边界。这些问题的动机是在凝聚态物理学中的最佳设计，图案形成和其他应用的研究。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。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.