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描述的。理解从实验中观察到的不同行为的挑战可以被视为理解相应的偏微分方程的解的行为。该项目涉及起源于物理学和材料科学活跃领域的非线性模型的数学分析。 重点是研究近晶A型液晶非线性模型的极限行为和薄膜方程层解的对称性，从而为材料科学的经典模型提供见解，并有助于解释实验中长期观察到的一些现象。 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。在第一部分中，研究者将研究近晶A相液晶的非线性模型。从一个非线性近似能量模型开始，调查员将解决与有界能量的序列的紧凑性，并考虑发送液晶渗透长度为零时，该模型的伽马极限。接下来，研究人员将这些结果扩展到完全非线性能量模型，并研究一些明确的例子，这是相关的欧拉-拉格朗日方程的解决方案的属性。第二部分解决二维薄膜模型的层的解决方案，强调他们的一维对称性。本研究与De Giorgi关于Allen-Cahn方程层解的一维对称性的猜想密切相关。薄膜模型的扩展是具有挑战性的，由于矢量和非本地的功能的现象。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。

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.