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

翻译后摘要：几何函数理论的分析基础;变分积分和非线性弹性Tadeusz Iwaniec最近在高维几何函数理论的主要进展之一是基于寻找新的微分方程类似于Cauchy-Riemann系统在复平面上。霍奇理论的微分形式已经发挥了核心作用，在这一相当现代的方法。雅可比行列式和楔形产品的exactdifferential形式进行了大量的调查，因为他们提供的手段，实现连续性，紧凑性，或正常的familytype结果。重要的是要认识到，雅可比行列式的高可积性只能在Orlicz-Sobolev类中的映射中观察到。最近的发展强调了拟共形映射与S.S. Antman和J. Ball，1976-77年。这种联系是该建议的一个重要方面。这就是为什么我们远离通常的拟共形理论相当远的映射（弹性体的变形）与无限失真。然而，某些控制，如BMO界限，失真张量将是必要的，以实现concreate结果。有限畸变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，自然地产生为相关变分积分（变形的储能）的欧拉-拉格朗日方程。全纯函数和有限畸变映射的解析方面之间的类比在偶数维中特别明显。在研究这些映射时，一个富有成效的想法是将它们看作关于某些可测度量或共形结构的共形映射。许多这些概念延伸到黎曼流形，因此，虽然我们没有充分发展这方面的所有机器，我们建立准备和愿意为这些推广。 读者感兴趣的发展沿着这些路线和acomensive帐户的几何函数理论是热烈欢迎即将出版的专着G。马丁和派

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.