Existence and Regularity for Variational Problems

变分问题的存在性和正则性

• 批准号: 1609198
• 负责人: Christine Breiner
• 金额: $ 12.72万
• 依托单位: Fordham University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-15 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1609198&HistoricalAwards=false
• 关键词: Existence; Regularity; Variational; Problems

This project concerns optimal objects for their respective energy functionals, and as such existence and structural results are of interest in engineering, physics, and chemistry. The most classically studied of these are minimal surfaces, which locally minimize area subject to a fixed boundary. Of particular interest in this project are so-called constant mean curvature (CMC) and minimal surfaces as well as harmonic maps. CMC surfaces are also critical for the area functional, but with constraint now given by enclosed volume. Delaunay determined a family of CMC examples in 1841, but it was another 150 years before any new examples were known, at which time Kapouleas produced infinitely many new examples via gluing techniques. The variational solutions studied in this project have characterizations in many areas of mathematics and the proposed questions and desired results are of broad interest in mathematics and beyond.The PI will continue her study of classical questions in geometric analysis related to the existence, regularity, and compactness of solutions to variational problems. The project will use and refine the gluing techniques pioneered by Kapouleas to produce new examples of minimal and CMC surfaces. The understanding of singularity development for a sequence of complete, properly embedded minimal disks, developed by Colding-Minicozzi, was of critical importance for the resolution of the uniqueness of the helicoid. In contrast to the picture developed when the disks are complete and proper, the structure of the singular set for sequences of embedded minimal disks with boundary in a ball can be pathological. These pathological examples are helpful in the resolution of uniqueness and regularity results. Gluing techniques will be used to produce even wilder singularities in settings where problems are intractable via former techniques. For CMC gluing, the project aims to extend the generalized gluing techniques developed in Euclidean space to more general manifolds. In the setting of harmonic maps, the aim of the project is to establish the existence of conformal harmonic maps into metric spaces with upper curvature bounds. This work generalizes a classical result of Sacks and Uhlenbeck on the existence of minimal 2-spheres. Existence of the established maps could help answer the unresolved portions of Thurston's Hyperbolization Conjecture.

该项目涉及各自能量泛函的最佳物体，因此存在性和结构结果引起了工程、物理和化学的兴趣。其中最经典的研究是最小曲面，它局部最小化受固定边界影响的区域。该项目特别感兴趣的是所谓的恒定平均曲率（CMC）和最小曲面以及调和图。 CMC 表面对于面积功能也至关重要，但现在受到封闭体积的限制。 Delaunay 于 1841 年确定了一系列 CMC 样品，但又过了 150 年才出现任何新样品，此时卡普莱斯通过粘合技术生产了无数新样品。该项目研究的变分解在数学的许多领域都有特征，提出的问题和期望的结果在数学及其他领域引起了广泛的兴趣。PI将继续研究几何分析中与存在性、正则性和变分问题解的紧凑性。该项目将使用并改进 Kapouleas 首创的粘合技术，以生产最小和 CMC 表面的新示例。由 Colding-Minicozzi 开发的一系列完整的、正确嵌入的最小圆盘的奇点发展的理解对于解决螺旋面的唯一性至关重要。与圆盘完整且正确时所显示的图片相反，边界为球的嵌入式最小圆盘序列的奇异集结构可能是病态的。这些病理例子有助于解决结果的独特性和规律性。粘合技术将用于在以前的技术难以解决问题的环境中产生更疯狂的奇点。对于 CMC 粘合，该项目旨在将欧几里得空间中开发的广义粘合技术扩展到更通用的流形。在调和映射的设置中，该项目的目的是在具有上曲率界限的度量空间中建立共形调和映射的存在性。这项工作概括了 Sacks 和 Uhlenbeck 关于最小 2-球体存在性的经典结果。已建立的地图的存在可以帮助回答瑟斯顿双曲化猜想中未解决的部分。