Nonlinear Theory for Smectics and Layered Solutions in Thin Films

连续介质力学和材料科学中的许多物理现象都是用偏微分方程来建模的。理解从实验中观察到的不同行为的挑战可以看作是理解相应偏微分方程解的行为。本项目涉及非线性模型的数学分析，这些模型起源于物理学和材料科学的活跃领域。重点是研究近晶A液晶非线性模型的极限行为和薄膜方程层解的对称性，从而增加对材料科学经典模型的见解，并帮助解释实验中一些长期观察到的现象。该项目将为研究生和本科生提供跨学科的培训机会。该项目分为两个主要部分。在第一部分，研究者将研究近晶A液晶的非线性模型。从非线性近似能量模型出发，研究人员将处理有界能量序列的紧性，并考虑将液晶穿透长度发送到零时模型的伽马极限。研究者接下来将把这些结果扩展到全非线性能量模型，并研究一些相关欧拉-拉格朗日方程解的显式例子的性质。第二部分讨论了二维薄膜模型的层解，重点讨论了它们的一维对称性。这一研究与De Giorgi关于Allen-Cahn方程层解的一维对称性的猜想密切相关。由于这种现象的矢量性和非局域性，对薄膜模型的扩展具有挑战性。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。

Many physical phenomena in continuum mechanics and materials science are modeled by partial differential equations. The challenge of understanding the different behaviors observed from experiments can be posed as understanding the behavior of the solutions of the corresponding partial differential equations. This project involves the mathematical analysis of nonlinear models that originated in active fields of physics and materials science. The emphasis is on the study of the limiting behavior of nonlinear models for smectic A liquid crystals and of the symmetry properties of layer solutions to thin film equations, so to add insights to classical models from materials science and help explain some long-observed phenomena in experiments. The project will offer cross-disciplinary training opportunities for graduate and undergraduate students.The project is divided in two main parts. In the first part, the investigator will study nonlinear models for smectic A liquid crystals. Starting from a nonlinear approximation energy model the investigator will address the compactness of sequences with bounded energy and consider the gamma limit of the model when sending the liquid crystal penetration length to zero. The investigator will next extend these results to the fully nonlinear energy model and study the properties of some explicit examples which are solutions of the associated Euler-Lagrange equation. The second part addresses layer solutions for two-dimensional thin film models, with an emphasis on their one-dimensional symmetry. This study is closely related to De Giorgi's conjecture on the one-dimensional symmetry of layer solutions to the Allen-Cahn equation. The extension to the thin film models is challenging due to the vectorial and nonlocal feature of the phenomena.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.