Calculus of Variations and Partial Differential Equations

在应用的推动下，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一致和定量能控性问题。目前可用于研究均匀或缓慢变化介质情况下这类问题的方法可能不适用于许多经常具有异质性、噪声和随机性的真实世界问题。因此，需要一套新的思想和技术来在已知的关于均匀情况的理论和与振荡现象有关的理论之间取得微妙的平衡。还将研究在显示设备中广泛使用的液晶的动力学。这涉及到一些有趣和困难的理论问题，这些问题涉及流体动力学方程和某些几何物体流动的非线性耦合，这可能会导致一个新的研究方向。该项目是国际和平研究所研究生和博士后研究人员培训计划的重要组成部分。这些成果将通过出版物、讲座、研讨会、会议和专为教育目的而设计的暑期班来传播。PI打算研究变分和偏微分方程的三组问题。第一组问题涉及具有高振荡系数的偏微分方程组的分析。在(周期和随机)齐化的一般理论下，关于这个主题已经积累了大量的知识。然而，在这种高振荡的介质中，仍然有许多有趣和深入的开放问题，如与定量(大尺度和小尺度)唯一延拓、发展问题的一致精确可控性以及相关反问题的稳定性估计有关的问题。第二组问题是PI早期对液晶流体动力学研究的继续，重点是对潜在的非线性耦合结构的探索。特别感兴趣的是Ericksen-Leslie系统在三维中适当弱解的整体存在性和在二维中有限时间爆破的存在性。项目的最后一部分是研究椭圆型本征值的极值问题和与之相关的自由边界。这些问题的动机是在优化设计、图案形成和凝聚态物理中的其他应用方面的研究。这是一个三年的研究计划，应该在主题上产生新的想法、方法和重要的结果，它建立在PI以前的研究成果的基础上。这个奖项反映了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。

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.