Nonlinear Theory for Smectics and Layered Solutions in Thin Films

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.