Foundation of the Geometric Function Theory in R^n: The Governing differential Forms, Variational Integrals and Nonlinear Elasticity

文摘：几何函数理论的解析基础变分积分与非线性弹性最近高维几何函数理论的主要进展之一是基于在复平面上发现类似柯西-黎曼系统的新微分方程。霍奇的微分形式理论在这种相当现代的方法中发挥了核心作用。雅可比行列式和精确微分形式的楔形积受到大量的研究，因为它们提供了实现连续性，紧致性或正常家族型结果的手段。重要的是要认识到雅可比矩阵的高可积性只能在theOrlicz-Sobolev类的映射中观察到。最近的发展强调了拟共形映射和非线性弹性理论之间的联系，这些理论已经由S.S. Antman和J. Ball在1976-77年提出。这种联系是提案的一个重要方面。这就是为什么我们与通常的准共形理论相去甚远，而转向具有无界畸变的映射（弹性体的变形）。然而，为了获得具体的结果，对畸变张量的一些控制（如bmo边界）是必要的。有限畸变映射的控制方程是非线性一阶偏微分方程系统。也有相关的二阶系统，自然地作为相关变分积分（变形的存储能量）的欧拉-拉格朗日方程出现。全纯函数的解析方面与有限畸变的映射之间的类比在偶数维上特别明显。在研究这些映射时，一个富有成效的想法是将它们视为对某些可测量的度量或共形结构的共形。这些概念中的许多延伸到黎曼流形，因此，虽然我们没有完全发展这方面，但我们建立的所有机器都准备好并愿意进行这些推广。对这些方面的发展和几何函数理论的全面描述感兴趣的读者，热烈地参考即将出版的G. Martin和PI的专著。

Abstract :ANALYTICAL FOUNDATIONS OF THE GEOMETRIC FUNCTION THEORY;VARIATIONAL INTEGRALS AND NONLINEAR ELASTICITYTadeusz IwaniecOne of the major recent advances in the higher dimensional geometricfunction theory is based on finding new differential equations analogousto the Cauchy-Riemann system in the complex plane. Hodge theory ofdifferential forms has come to play a central role in this rather modernapproach. The Jacobian determinants and the wedge products of the exactdifferential forms are subjected to a great deal of investigation, as theyprovide the means of achieving continuity, compactness, or normal familytype results. It is important to realize that the higher integrabilityproperties of the Jacobians can only be observed for mapping in theOrlicz-Sobolev classes. More recent developments have emphasized theconnection between quasiconformal mappings and the theory of nonlinearelasticity already formulated by S.S. Antman and J. Ball in 1976-77. Thisconnection is an important aspect of the proposal. And that is why wedepart from the usual quasiconformal theory quite far towards mappings(deformations of elastic bodies) with unbounded distortion. However, somecontrol, such as BMO-bounds, of the distortion tensor will be necessary toachieve concreate results. The governing equations for mappings of finitedistortion are non-linear first order systems of PDEs. There are alsorelated second order systems which arrise naturally as the Euler- Lagrangeequations of the associated variational integrals (stored energy of thedeformation). An analogy between the analytic aspects of the holomorphicfunctions and mappings of finite distortion is particularly pronounced ineven dimensions. A fruitful idea when studying these mappings is to viewthem as conformal with respect to certain measurable metric or conformalstructures. Many of these notions extend to Riemannian manifolds, andaccordingly, while we do not develop this aspect in full all the machinerywe set up is ready and willing for these generalizations. The reader interested in developments along these lines and acomprehensive account of the geometric function theory is warmly referredto the forthcoming monograph of G. Martin and PI.