Regularity of minimizers and pattern formation in geometric minimization problems

1. 与距离相关的能量。能源驱动的演变无处不在。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，基态通常由沿着规则晶格放置的基本单元或“晶体”的全等副本组成（参见例如[15]）。这样的过程被称为“结晶”，它是最相关和研究最多的过程之一。有多个二维结果可用，例如[6,21,43]，但很少有这样的结果以 3D 形式提供。其中最简单的能量是与距离相关的能量。这些能量用于解决广泛的问题，例如数据分析、机器学习、矢量量化、图像处理等。该计划的目的是研究一些与距离相关的能量最小化器的经典问题、此类最小化器经常观察到的模式形成及其应用。短期目标包括研究 3D 基态的几何形状。由于用于分析基态的技术非常通用，作为一个长期目标，几个经典问题，例如可以使用相同的技术来研究最佳泡沫问题和球体堆积问题。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，有可能实现更快、更准确、更高效的模型。2．材料科学中的偏微分方程。在多个领域中，演化过程通常通过偏微分方程 (PDE) 进行建模。其中一个领域是外延：近年来，它作为生产高质量晶体的关键工业工艺之一而受到广泛关注，这对于半导体和纳米技术至关重要。这样的过程本质上是高度非线性的（参见[7,27,38]）。有大量的偏微分方程模拟了几种不同的外延过程，但我们目前的知识还远未完成。另一个备受关注的领域是液晶，因为其广泛的应用以及与生物学的相关性。一个常见的问题是偏微分方程是高度非线性的，这使得它们的处理（理论和数值）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，因为由于强非线性，许多偏微分方程仍然知之甚少。作为长期目标，我们的目标是开发新技术来研究“不良行为”能量的梯度流，因为许多能量都来自材料科学的热门研究领域。严格证明此类偏微分方程的适定性将有可能给出一种预测理论，这对制造过程至关重要，因为它可以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，这对于这些偏微分方程的数值分析很有用。

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.