Differential inclusions in quasiconformal analysis

该方案主要研究具有一阶弱导数的弱可微映射。这类包括但不限于拟共形和拟正则映射。微分包含将导数的本质范围限制在特定的一组矩阵上。微分包含的主要来源是微分几何和非线性偏微分方程组理论。通常，矩阵空间中的凸性和连通性的各种概念推动了解的存在性和正则性的分析。基本问题之一是，哪些微分包含意味着局部可逆性。可逆性通常分两步建立：第一步证明映射是离散的和开的(即分支覆盖)；第二步证明分支集是空的。实施每个步骤都会遇到问题，这些问题继续挑战现有的分析和拓扑方法。该建议还将几何函数论的工具引入了常微分方程组领域。在欧氏空间上存在正则坐标的情况下，自治微分方程组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，拟共形分析中引入的技术在光滑(例如调和)映射的研究中也是有效的，这反过来又在极小曲面理论中得到应用最小曲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，最小曲面的两条边界曲线在曲面崩溃之前可以移动多远？主要研究人员将把几何函数论的技巧应用于这类极值问题。这种方法不仅适用于薄膜模型，也适用于固体材料的弹性变形研究。提案的另一部分涉及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组理论的标准结果不适用的情况下建立解的唯一性。PI与博士后和研究生一起工作，并组织锡拉丘兹分析研究小组。

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.