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。这样的物理系统一般演化为基态，如果所涉及的能量足够规则，则基态表现出高度的规律性和周期性。例如，基态通常由沿着规则晶格放置的基本单元或“晶体”的一致副本组成(例如，见[15])。这样的过程被称为“结晶”，这是最相关和最受调查的过程之一。有几个结果在2D中可用，例如[6，21，43]，而这样的结果很少在3D中可用。在这些能量中，最简单的是与距离有关的能量。这些能量被用于广泛的问题，例如数据分析、机器学习、矢量量化、图像处理等。这个计划的目的是研究一些距离相关能量的最小化的经典问题，这种最小化经常被观察到的图案的形成，以及它们的应用。短期目标包括研究3D基态的几何结构。由于发展起来的分析基态的技术非常普遍，作为一个长期目标，可以使用相同的技术来研究几个经典问题，例如最优泡沫问题和球填充。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，潜在地允许更快、更准确、更有效的模型。材料科学中的偏微分方程。在几个领域中，演化过程通常用偏微分方程(PDE)来模拟。其中一个领域是外延：近年来，作为生产高质量晶体的关键工业过程之一，外延受到了极大的关注，这对半导体和纳米技术至关重要。这种过程本质上是高度非线性的(见[7，27，38])。有太多的偏微分方程组模拟了几种不同的外延过程，但我们目前的知识仍然远远不完整。另一个备受关注的领域是液晶，因为它们的广泛应用和与生物学的相关性。一个常见的问题是，偏微分方程组是高度非线性的，这使得它们的处理(无论是理论上还是数值上)都相当困难。作为短期目标，我们计划研究材料科学中产生的偏微分方程组，因为由于强烈的非线性，许多偏微分方程组仍然很难被理解。作为长期目标，我们的目标是开发新的技术来研究“行为不佳”能量的梯度流动，因为其中许多能量来自材料科学的热门研究领域。严格证明这种偏微分方程的适定性可能会给出一种预测理论，这对制造过程至关重要，因为它允许准确预测材料在不同条件下的行为。此外，可能需要定量估计，这将有助于对这些偏微分方程的数值分析。

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.