Differential inclusions in quasiconformal analysis

拟共形分析中的微分包含体

• 批准号: 0968756
• 负责人: Leonid Kovalev
• 金额: $ 10.02万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0968756&HistoricalAwards=false
• 关键词: Differential; inclusions; quasiconformal; analysis

The proposal focuses on weakly differentiable mappings with first-order weak derivatives. This class includes, but is not limited to, quasiconformal and quasiregular mappings. A differential inclusion restricts the essential range of the derivative to a certain set of matrices. Main sources of differential inclusions are differential geometry and the theory of nonlinear partial differential equations. Typically, various notions of convexity and connectedness in the matrix space drive the analysis of existence and regularity of solutions. One of basic questions is which differential inclusions imply local invertibility. Invertibility is often established in two steps: first it is shown that the mapping is discrete and open (that is, a branched cover); the second step is to prove that the branch set is empty. Implementation of each step encounters problems that continue to challenge the available methods of analysis and topology. The proposal also brings the tools of geometric function theory into the field of ordinary differential equations. In the presence of canonical coordinates on the Euclidean space the driving vector field in an autonomous system of differential equations can be identified with a mapping, and the geometry of this mapping turns out to be related to the uniqueness of solution. Thirdly, the techniques introduced in quasiconformal analysis are also effective in the studies of smooth (e.g., harmonic) mappings, which in turn find applications in the theory of minimal surfacesMinimal surfaces are mathematical models of thin films, for instance soap bubbles. Our understanding of their shapes develops through the solution of extremal problems. For example, how far apart can one move two boundary curves of a minimal surface before the surface breaks down? The principal investigator will apply the techniques of geometric function theory to such extremal problems. This approach is not limited to models of thin films and is also relevant in the studies of elastic deformation of solid materials. Another part of the proposal addresses uniqueness and stability of solutions of ordinary differential equations, which are commonplace in physics and engineering. They appear as equations of motion for one or several particles, with the number of particles affecting the dimensionality of the problem and the geometry of vector fields involved. Geometric function theory allows one to establish the uniqueness of a solution in situations where the standard results of the theory of ordinary differential equations do not apply. The PI works with post-docs and graduate students and organizes the Syracuse Analysis Study group.

本文主要研究具有一阶弱导数的弱可微映射。该类包括但不限于拟共形映射和拟正则映射。微分包含将导数的本质范围限制在一定的矩阵集合内。微分夹杂的主要来源是微分几何和非线性偏微分方程理论。通常，矩阵空间中各种凸性和连通性的概念驱动着解的存在性和正则性的分析。其中一个基本问题是，哪些微分包含意味着局部可逆性。可逆性通常分两步建立：首先证明映射是离散和开放的（即分支覆盖）；第二步是证明分支集是空的。每个步骤的实现都会遇到一些问题，这些问题继续挑战可用的分析和拓扑方法。该建议还将几何函数理论的工具引入了常微分方程领域。在欧氏空间上存在正则坐标的情况下，微分方程自治系统的驱动向量场可以用一个映射来识别，而这个映射的几何性质与解的唯一性有关。第三，在拟共形分析中引入的技术在光滑（如谐波）映射的研究中也很有效，这反过来又在最小表面理论中找到了应用。最小表面是薄膜的数学模型，例如肥皂泡。我们对它们形状的理解是通过解决极端问题而发展起来的。例如，在曲面破裂之前，一个最小曲面的两条边界曲线可以移动多远？主要研究者将运用几何函数理论的技术来解决这类极值问题。这种方法不仅适用于薄膜模型，也适用于固体材料弹性变形的研究。该提案的另一部分讨论了常微分方程解的唯一性和稳定性，这在物理和工程中是常见的。它们表现为一个或几个粒子的运动方程，粒子的数量影响问题的维度和所涉及的矢量场的几何形状。几何函数理论允许人们在常微分方程理论的标准结果不适用的情况下建立解的唯一性。PI与博士后和研究生合作，并组织雪城大学分析研究小组。