Variational approach to Geometric Function Theorem, Nonlinear PDEs and Hyperelasticy

Syracuse University, Syracuse NY

Investigators

Abstract

Applied sciences are important in formulating and solving interesting mathematical problems. Conversely, researchers in science and engineering fields often seek improvements in both theory and practice via mathematical arguments to explain and confirm experimental results. This project involves partial differential equations, geometric function theory, calculus of variations and related problems that arise in real-world applications including: nonlinear elasticity, microstructure of materials, and crystals to name a few. The proposed physical interpretations of mathematical objects, like Sobolev homeomorphisms that are viewed as elastic deformations, have proven useful in understanding and solving challenging problems in nonlinear elasticity. The primary aim here is to develop new, and improve old, methods to meet these challenges. The new mathematical concepts such as free Lagrangians allow one to establish the existence of traction free energy-minimal deformations. Further advances in Hopf-Laplace differentials could lead to predictions of the formation of cracks in deformations, and in understanding the principle of interpenetration of matter. Theoretical prediction of failure of elastic bodies caused by cracks would have a broad impact to both mathematical analysts and researchers in the engineering fields. The research topics in this project, although in general mathematically challenging, also take on questions that are suitable for graduate students. The ultimate goal is to attract graduate students and young scholars, both women and men, to geometric function theory with a wide range of applications, to encourage them to participate in the interdisciplinary efforts, and to prepare and help them to develop meaningful interactions with physicists and engineers. The principal investigator is currently working with Jani Onninen on a joint monograph designed for researchers as well as graduate students in geometric function theory. The principal investigator has a history of strong efforts to gain from an interplay between pure and applied mathematics. It resulted in the solution of several mathematical problems such as the Nitsche conjecture in the theory of minimal surfaces, the Evans-Ball conjecture on approximation of Sobolev mappings with diffeomorphisms, the novel concept of free Lagrangians in the calculus of variations, formation of cracks along trajectories of Hopf differentials, rigorous description of the phenomenon of interpenetration of matter, and the partial answers in the dimension two to the legendary Morrey's conjecture on quasiconvexity of the rank-one convex functionals; in particular, sharp inequalities of Burkholder's stochastic integrals. The latter include the complex Beurling-Ahlfors singular integral transform. The eminence of this transform lies in the fact that it connects two homotopy classes of the first order elliptic systems in the complex plane: one whose solutions are orientation preserving mapping and the other with orientation reversing solutions. The notoriously difficult problem has been to identify the Lp-norm of the Beurling-Ahlfos transform. The status of the problem goes on as never before. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.