Regularity of minimizers and pattern formation in geometric minimization problems

1. 距离相关能量。能源驱动的进化无处不在。这样的物理系统通常会演化到基态，如果所涉及的能量足够规则，那么基态就会表现出高度的规律性和周期性。例如，基态通常是由沿规则晶格放置的基本细胞或“晶体”的相同副本构成的（参见例如[15]）。这种过程被称为“结晶”，是最相关和研究最多的过程之一。一些结果在2D中可用，例如[6,21,43]，而很少有这样的结果在3D中可用。其中最简单的能量是与距离相关的能量。这些能量用于广泛的问题，例如数据分析，机器学习，矢量量化，图像处理等。本计划的目的是研究一些与距离相关的能量的最小化的经典问题，通常观察到的这种最小化的模式形成，以及它们的应用。短期目标包括研究三维基态的几何形状。由于用于分析基态的技术非常普遍，作为长期目标，可以使用相同的技术研究几个经典问题，例如最优泡沫问题和球体填充问题。对这些数学问题的更深入的理解（或完整的解决方案）将有利于它们的应用，可能允许更快、更准确、更有效的模型。

1. Distance related energies. Energy driven evolutions are ubiquitous. Such a physical system generally evolve to a ground state, and if the involved energy is sufficiently regular, then ground states exhibit a high degree of regularity and periodicity. For instance, ground states are often made of congruent copies of a basic cell, or “crystal”, placed along a regular lattice (see e.g. [15]). Such a process is known as “crystallization”, which is among the most relevant and investigated processes. Several results are available in 2D, e.g. [6, 21, 43], while few such results are available in 3D. Among the simplest such energies, are the distance related energies. These energies are used in a wide range of problems, e.g. data analysis, machine learning, vector quantization, image processing, etc.. The aim of this plan is to study classic problems about minimizers of some distance related energy, the pattern formation often observed for such minimizers, and their applications. Short term goals include investigating the geometry of the ground states in 3D. Since the techniques developed to analyze ground states are quite general, as a long term goal, several classic problems, e.g. the optimal foam problem, and sphere packing, can be studied using the same techniques. A deeper understanding (or a full solution) of these mathematical problems will be beneficial to their applications, potentially allowing for faster, more accurate, more efficient models. 2. PDEs in material sciences. Evolution processes, in several fields, are often modeled by Partial Differential Equations (PDEs). One such area is epitaxy: it has received a lot of attention in recent years since as one of the key industrial processes to produce high quality crystals, crucial for semiconductors and nanotechnology. Such a process is highly nonlinear in nature (see e.g. [7, 27, 38]). There is a plethora of PDEs modeling several different epitaxial processes, but our present knowledge is still far from complete. Another area receiving much attention is liquid crystals, due to their widespread applications, and relevance to biology. A common issue is that the PDEs are highly nonlinear, which makes their treatment (both theoretical and numerical) quite difficult. As short term goal, we plan to investigate PDEs arising from material sciences, since many are still poorly understood due to the strong nonlinearities. As long term goal, we aim to develop new techniques to study gradient flows of “badly behaved” energies, since so many arise from hotly studied areas of material sciences. Rigorously proving the well-posedness of such PDEs will potentially give a predictive theory, crucial to manufacturing processes since it allows to accurately predict the behavior of materials under different conditions. Moreover, quantitative estimates are likely to be required, which will be useful for the numerical analysis of these PDEs.