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

文摘：几何函数论的分析基础；变分积分与非线性弹性伊万尼高维几何函数论的一个重要进展是在复平面中找到类似于柯西-黎曼系统的新的微分方程式。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。精确微分形式的雅可比行列式和楔积受到了大量的研究，因为它们提供了实现连续性、紧性或正规族类型结果的手段。重要的是要认识到，只有在Orlicz-Sobolev类中的映射才能观察到雅可比的更高可积性。最近的发展强调了拟共形映射与S.S.Antman和J.Ball在1976-77年所提出的非线性相对论之间的联系。这种联系是该提案的一个重要方面。这就是为什么我们偏离了通常的准共形理论，转向了具有无限变形的映射(弹性体的变形)。然而，需要对变形张量进行某些控制，例如BMO界，才能达到理想的结果。有限维映射的控制方程是一阶非线性偏微分方程组。也有相关的二阶系统自然地上升为相关的变分积分的欧拉-拉格朗日方程(变形的储能)。全纯函数的解析方面与有限变形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相对于某些可测量的度量或构象结构的共形。许多这些概念扩展到黎曼流形，因此，尽管我们没有完全发展这一方面，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对几何函数理论的全面描述感兴趣的读者，请参阅即将出版的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.