Sobolev Mappings and Energy-Integrals in Mathematical Models of Nonlinear Elasticity

该提案继续PI的长期研究计划，以推进几何函数论中的变分技术，并考虑到应用学科。它提供了一个深入的分析的概念之间的能量最小变形欧几里德域（主要是，牵引自由问题的二维）。目标是深入了解在变分法，最小曲面，非线性弹性，材料科学等领域遇到的许多新现象。历史上的例子是著名的数学结构-泰希米勒空间，其中人们寻求最小化失真函数的上确界范数。还可以举出其他例子。我们将最小化积分平均失真。具有最小平均变形的沿沿着边界滑动（无牵引）的映射的存在性和唯一性将挑战带到了一个全新的水平。 该项目继续PI的努力，使研究生和博士后学者几何分析与广泛的应用，并帮助他们成熟有意义的互动与物理学家和工程师已经在他们的博士后阶段。这也是激励学校或大学里有天赋的年轻学生进入数学领域的关键。该计划通过教育材料（正在编写的书籍），夏季/冬季迷你课程，初级-高级研讨会提供这样的环境。 PI有一个从事研究的气候和这些努力的强有力的国际互动的历史。他预计，提案中应用领域的存在将改善博士生和他的众多合作伙伴的就业机会。

The proposal continues a long term research program of the PI to advance variational techniques in Geometric Function Theory with applied disciplines in mind. It provides an in-depth analysis of the concept of energy-minimal deformations between Euclidean domains (predominantly, traction free problems in dimension two). The ambition is to gain insights into the many new phenomena encountered in the Calculus of Variations, Minimal Surfaces, Nonlinear Elasticity, Material Science, and so forth. Historical example would be the celebrated mathematical structure -Teichmüller spaces, where one seeks to minimize the supremum norm of the distortion function. Other examples could be cited. We shall minimize the integral mean distortion instead. The existence and uniqueness of the mappings slipping along the boundary (traction free) with smallest mean distortion takes the challenge to a whole new level. The project continues the efforts of the PI to bring graduate students and postdoctoral scholars to Geometric Analysis with a wide range of applications, and to help them to mature meaningful interaction with physicists and engineers already in their post-doc phase. This is also a key to motivate young gifted students at school or college to enter mathematics. The program provides such an environment through educational materials (book in progress), Summer/Winter mini-courses, junior-senior seminars. PI has a history of an engaging research climate and strong international interaction on these efforts. He anticipates that the presence of applied fields in the proposal will improve the job opportunities for the PhD students and his numerous co-advisees.