Geometry and Analysis of Extremal Mappings of Finite Energy

建议的研究中心围绕极值问题的有限失真。这些问题的相关性来自于一个确认，即拟共形理论和非线性弹性力学共享令人信服的数学挑战。有限畸变映射理论的产生是因为需要将拟共形映射的经典理论的思想和应用推广到退化椭圆环境。在这样一个一般的退化环境中研究极值问题有许多自然的理由。我们最终希望为紧化模空间奠定分析基础。在这样的极值问题中，我们的映射不受边界的约束;对于紧致黎曼曲面，边界甚至不存在。环带是人们第一次观察到非平凡共形不变量的地方。具有最小平均偏差的映射的存在性与著名的Nitsche（1962）关于圆环上调和同胚的猜想有密切的关系。Nitsche猜想最初的动机是研究双连通极小图的不存在性。最近，PI与T. Iwaniec和L.科瓦廖夫对这个问题给予了肯定的回答。我们的方法开辟了几个未来的方向。关于极值映射的存在性（锤击现象）、唯一性、正则性和径向对称性的破坏，有许多新的和意想不到的现象;有些已经准备好了答案，而另一些仍然是长期的目标。数学分析的未来发展依赖于物理和几何直觉。近年来，这一趋势变得更加明显，导致理论数学家和应用数学家共同努力。PI在这里提出的几何函数理论和非线性弹性的交叉问题有助于这些努力。所提出的超弹性变形的存在性定理不仅可能引起数学家的兴趣，而且可能引起工程师和物理学家的兴趣。该项目还涉及与位于锡拉丘兹大学附近的小型学院合作，目的是吸引来自代表性不足群体的年轻研究人员从事数学研究，还涉及指导研究生。

The proposed research centers around extremal problems of finite distortion. The relevance of these problems comes out of a confirmation that quasiconformal theory and nonlinear elasticity share compelling mathematical challenges. The theory of mappings of finite distortion arose from the need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting. There are many natural reasons for studying extremal problems in such a general degenerated setting. We eventually hope to lay down the analytical foundations for compactifying the moduli spaces. In such an extremal problem our mappings are not constrained on the boundary; the boundary does not even exist for compact Riemann surfaces. Annuli are where one first observes nontrivial conformal invariants. There is a close relationship between the existence of mappings with smallest mean distortion and the well known conjecture of Nitsche (1962) on harmonic homeomorphisms between circular annuli. The Nitsche conjecture was originally motivated by the study of the non-existence of doubly connected minimal graphs. Very recently, the PI in collaboration with T. Iwaniec and L. V. Kovalev gave an affirmative answer to this question. Our approach opens several future directions. There are many new and unexpected phenomena concerning existence (hammering phenomenon), uniqueness, regularity and failure of radial symmetry of the extremal mappings; some already prepared for answers, while others remain long term goals. Mathematical analysis relies on physical and geometric intuition for its future development. In recent years, this trend has become more pronounced, leading to concerted efforts of pure and applied mathematicians to work together. Problems in the intersection of Geometric Function Theory and Non-linear Elasticity that the PI proposes here contribute to these efforts. The proposed existence theorems of hyperelastic deformations might interest not only mathematicians but also engineers and physicists. The project also involves working with small colleges located near to Syracuse University with the aim of attracting young researchers from underrepresented groups to mathematics and also involves mentoring graduate students.