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

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Project Abstract: DMA 0301582T Iwaniec, Syracuse University Intellectual Merit. Prominent advances in analysis will continue toenhance both theory and applications. In this challenge, there is enormousopportunity for contributions from geometric function theory andnonlinear partial differential equations. Broadly speaking, this theory isabout weakly differentiable mappings which distort small circles in a controlled manner (finite distortion). It elevates allthe analytic and geometric spirit of holomorphic functions in the complexplane to higher dimensions; domains of the Euclidean n-space orRiemannian manifolds, Mappings of finite distortion owe much of theirimportance to recent developments inseveral fields of applied mathematics: continuum mechanics, nonlinearelasticity, material science, composites with microstructure, crystals,and so forth. However, in this proposal we focus on practical questionsonly through mathematical formulations, sometimes without proclaimingdefinite answers. We emphasize the fundamental role of the Jacobiandeterminant and other nonlinear differential expressions, called nullLagrangians, which have driven (over the past twelve years) a veryproductive study of mappings with unbounded distortion. The presentknowledge of null Lagrangians will tell us something about the regularityand topological behavior of those mappings. This also brings us closer tothe methods of harmonic analysis and nonlinear partial differential equations. However, we departfrom the earlier theories, which are governed byuniformly elliptic systems, and move into the realm of degenerate ellipticequations, where important new applications lie. Broader Impact.Hopefully this proposal gives some appreciation of thediversity of applications and directions in which current research of thePI is moving, as well as a glimpse of the substantial body of joint workwith young scholars. It is at this point where the proposed research meetsthe broader impact criteria, such as: effective training and educationalmaterials, enhancing partnerships, participation of under-representedgroups from non-PhD granting institutions in the USA and abroad, women andmen. The Principal Investigator plans to organize a conference at Syracuse to bring togetherresearchers in geometric function theory and in applied areas. Theproposed program reflects continued efforts of the PI to reach outgraduate students. In the final analysis, education is the path toopportunities for countless individuals, and the mathematical community asa whole is the beneficiary.