Calculus of Variations and Partial Differential Equations

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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.