Geometric Analysis of Deformations of Finite Distortiion via Nonlinear PDEs and Null Lagrangians

Syracuse University, Syracuse NY

Investigators

Abstract

Project Abstract: DMA 0301582 T Iwaniec, Syracuse University Intellectual Merit. Prominent advances in analysis will continue to enhance both theory and applications. In this challenge, there is enormous opportunity for contributions from geometric function theory and nonlinear partial differential equations. Broadly speaking, this theory is about weakly differentiable mappings which distort small circles in a controlled manner (finite distortion). It elevates all the analytic and geometric spirit of holomorphic functions in the complex plane to higher dimensions; domains of the Euclidean n-space or Riemannian manifolds, Mappings of finite distortion owe much of their importance to recent developments in several fields of applied mathematics: continuum mechanics, nonlinear elasticity, material science, composites with microstructure, crystals, and so forth. However, in this proposal we focus on practical questions only through mathematical formulations, sometimes without proclaiming definite answers. We emphasize the fundamental role of the Jacobian determinant and other nonlinear differential expressions, called null Lagrangians, which have driven (over the past twelve years) a very productive study of mappings with unbounded distortion. The present knowledge of null Lagrangians will tell us something about the regularity and topological behavior of those mappings. This also brings us closer to the methods of harmonic analysis and nonlinear partial differential equations. However, we depart from the earlier theories, which are governed by uniformly elliptic systems, and move into the realm of degenerate elliptic equations, where important new applications lie. Broader Impact. Hopefully this proposal gives some appreciation of the diversity of applications and directions in which current research of the PI is moving, as well as a glimpse of the substantial body of joint work with young scholars. It is at this point where the proposed research meets the broader impact criteria, such as: effective training and educational materials, enhancing partnerships, participation of under-represented groups from non-PhD granting institutions in the USA and abroad, women and men. The Principal Investigator plans to organize a conference at Syracuse to bring together researchers in geometric function theory and in applied areas. The proposed program reflects continued efforts of the PI to reach out graduate students. In the final analysis, education is the path to opportunities for countless individuals, and the mathematical community as a whole is the beneficiary.