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

Syracuse University, Syracuse NY

Investigators

Abstract

Abstract : ANALYTICAL FOUNDATIONS OF THE GEOMETRIC FUNCTION THEORY; VARIATIONAL INTEGRALS AND NONLINEAR ELASTICITY Tadeusz Iwaniec One of the major recent advances in the higher dimensional geometric function theory is based on finding new differential equations analogous to the Cauchy-Riemann system in the complex plane. Hodge theory of differential forms has come to play a central role in this rather modern approach. The Jacobian determinants and the wedge products of the exact differential forms are subjected to a great deal of investigation, as they provide the means of achieving continuity, compactness, or normal family type results. It is important to realize that the higher integrability properties of the Jacobians can only be observed for mapping in the Orlicz-Sobolev classes. More recent developments have emphasized the connection between quasiconformal mappings and the theory of nonlinear elasticity already formulated by S.S. Antman and J. Ball in 1976-77. This connection is an important aspect of the proposal. And that is why we depart from the usual quasiconformal theory quite far towards mappings (deformations of elastic bodies) with unbounded distortion. However, some control, such as BMO-bounds, of the distortion tensor will be necessary to achieve concreate results. The governing equations for mappings of finite distortion are non-linear first order systems of PDEs. There are also related second order systems which arrise naturally as the Euler- Lagrange equations of the associated variational integrals (stored energy of the deformation). An analogy between the analytic aspects of the holomorphic functions and mappings of finite distortion is particularly pronounced in even dimensions. A fruitful idea when studying these mappings is to view them as conformal with respect to certain measurable metric or conformal structures. Many of these notions extend to Riemannian manifolds, and accordingly, while we do not develop this aspect in full all the machinery we set up is ready and willing for these generalizations. The reader interested in developments along these lines and a comprehensive account of the geometric function theory is warmly referred to the forthcoming monograph of G. Martin and PI.