Variational Models for Self-Assembly and Pattern Formation: Analysis and Computation

自组装是一种过程，在这种过程中，无序的组件系统仅作为组件相互作用的结果形成有组织的、结构化的模式，在本质上是无处不在的，对许多设计师材料的合成也是重要的。许多表现出自组装和图案形成的系统可以被视为能量驱动的；因此，人们可以通过最小化定义在所有可能的构型上的一些通常假设的自由能来成功地捕捉和分析这些现象。由此产生的变分问题涉及到高度非平凡的能量景观，在数学和计算上都是非常具有挑战性的。在这个方案中，我们将从数学分析和计算的角度研究三个这样的变分模型(以及它们之间的联系)。一个共同的主题将是对非局部问题和全球技术的关注，即那些视野超越对临界点的局部研究和对其稳定性的分析的技术。

Self-assembly, a process whereby a disordered system of components forms an organized, structured pattern solely as a consequence of component interactions, is both ubiquitous in nature and important for the synthesis of many designer materials. Many systems which exhibit self-assembly and pattern formation may be viewed as energy-driven; therefore, one can successfully capture and analyze the phenomena via minimization of some, often postulated, free energy defined over all possible configurations. The resulting variational problems involve highly non-trivial energy landscapes and are mathematically and computationally very challenging. In this proposal we will study three such variational models (and connections between them) from the point of view of mathematical analysis and computation. A common theme will be the focus on nonlocal problems and global techniques, i.e. those whose vision extends beyond the local study of critical points and the analysis of their stability.