Singularity Behavior in Some Geometric Variational Problems

一些几何变分问题中的奇异性行为

• 批准号: 0604605
• 负责人: Robert Hardt
• 金额: $ 36.47万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604605&HistoricalAwards=false
• 关键词: Singularity; Behavior; Some; Geometric; Variational

This project lies in the area of geometric variational calculus, treating the behavior of singularities and energy concentration for various optimal or stationary functions, fields, measures, or geometric structures, possibly subject to constraints. The first specific class of projects involves continuing work with T. Riviere, on relations between the p energy of a map between Riemannian manifolds and its homotopy class. In case of nontorsion homotopy, our previous results on the geometric and topological structure of bubbles allows attack on open questions about the weak smooth approximability of Sobolev maps as well as refined questions about bubbling in p stationary maps and heat flows. We also are attacking higher order Sobolev spaces which seem more natural for certain homotopy classes, but for which basic approximation results and constructions have not been previously studied, A second class of projects involves continuing work with Thierry De Pauw on extending notions from geometric measure theory and solutions of Plateau-type problems in the context of chains in a metric space with coefficients in a general group. We consider a variety of mass-type functionals and the notion of a scan which generalizes the finite mass metric-space currents of Ambrosio-Kirchheim and the rectifiable and flat Euclidean-space G-chains of White. In the Euclidean space context we approximate the size functional of Almgren and prove optimal regularity for the minimizers of such approximate functionals. Solutions to many variational problems in both pure and applied mathematics often are forced to have singularities, that is, to involve regions where large oscillations occur. For example a nematic liquid crystal material in a spherical container whose optical axis is forced to point outward on the container necessarily will have singularities inside (observable through cross-polarizers or x-ray diffraction). In this example the optical axis has an energy density, which measures its local rate of change and whose integral tends to have a minimum value among all possible configurations. Our research proposes to understand the relationship between energies in such variational problems and the topological barriers imposed by the physics of these problems. We have derived new notions which allow the treatment and precise description of a wide variety of problems from soap films and their higher dimensional generalizations to optimal transport paths in various complex media. Geometric constraints which occur naturally in many physical problems have led to new mathematical and computational issues. In particular, two that we are studying are the constant cross-sectional area constraint exhibited in plant structure and gradient constraints in the microstructure formation in certain crystalline materials.

这个项目是在几何变分学领域，治疗各种最佳或固定功能，领域，措施，或几何结构，可能受到约束的奇异性和能量集中的行为。第一类特定项目涉及与T的持续合作。里维埃，关于黎曼流形与其同伦类之间映射的p能量的关系。在非挠同伦的情况下，我们以前的结果的几何和拓扑结构的泡沫允许攻击开放的问题弱光滑逼近的Sobolev映射，以及完善的问题起泡在p平稳映射和热流。 我们还攻击高阶索伯列夫空间似乎更自然的某些同伦类，但其中基本的近似结果和建设尚未事先研究，第二类项目涉及继续工作与蒂埃里德堡的概念，从几何测度理论和解决方案的高原型问题的范围内链的度量空间的系数在一般组。我们考虑了各种质量型泛函和扫描的概念，它推广了有限质量度量空间电流的安布罗休-基希海姆和可整流和平坦的欧几里得空间G-链的白色。 在欧氏空间的上下文中，我们近似的大小功能的Almgren和证明最佳的正则性，这种近似泛函的极小。在纯数学和应用数学中，许多变分问题的解经常被迫具有奇点，也就是说，涉及到发生大振荡的区域。 例如，在球形容器中的液晶材料（其光轴被迫在容器上向外指向）必然会在内部具有奇点（通过交叉偏振器或X射线衍射可观察到）。 在该示例中，光轴具有能量密度，该能量密度测量其局部变化率，并且其积分在所有可能的配置中趋于具有最小值。我们的研究旨在了解这些变分问题中的能量与这些问题的物理学所施加的拓扑障碍之间的关系。 我们已经得出了新的概念，允许治疗和精确描述各种各样的问题，从肥皂膜和他们的高维概括，以最佳的传输路径在各种复杂的介质。 几何约束在许多物理问题中自然存在，导致了新的数学和计算问题。特别是，我们正在研究的两个是恒定的横截面积的限制表现在植物结构和梯度限制在某些晶体材料的微观结构的形成。