Variational Methods and Dynamics

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

This project focuses on problems in the calculus of variations and partial differential equations. A theory, now called direct methods of the calculus of variations, was developed by Charles Morrey in the middle of the last century wherein he introduced various fundamental concepts of convexity. This theory has wide application to many fields, including John Ball's groundbreaking work in elasticity theory. Recently, John Ball published a list of outstanding problems in the calculus of variations whose solutions will not only advance that field but will also have major impacts in elasticity theory. In recent work, the principal investigator has started to develop tools to handle specific concrete problems from that list. This will, he hopes, lead to general theories. The project will extend this study to include important classes of variational problems that have defied existing theory, such as the stored energy functional of Ogden material, which appears in elasticity. Another class of problems to be studied arises in mass transportation theory. It was first formulated by Monge in 1771 as a simple geometry problem and later extended by Kantorovich to more general cost functions. It consists in finding the optimal (minimal work) method for transporting a pile of earth at one location to an excavation site at another location. These classical results have been extended to more general cost functions by the principal investigator, his students, postdocs, collaborators and many others, with special attention to the Wasserstein (or Kantorovich) metric. Building on these results, the project will investigate so-called axi-symmetric flows, which preliminary computations indicate are Hamiltonian systems on the Wasserstein space. The principal investigator has established a link between the study of these flows and a poorly understood (challenging) class of parameter-dependent Monge-Ampere equations. He will study the existence of paths satisfying a certain basic stability criterion and connecting two prescribed symplectic forms. These problems, interesting in their own right, include searching for geodesics in the set of symplectic forms. Tproject will have a number of important scientific and educational components. First, the theoretical advances will directly aid other scientific research areas, including meteorology, elasticity theory, and other applied sciences, that use variational methods in their algorithms. Second, the principal investigator will continue to disseminate the research by giving courses and lectures in international conferences, workshops, and seminars. Finally, in addition to continuing to mentor undergraduates and Ph.D. students at his home institution, he will extend his active support of underrepresented groups in mathematics, in part by mentoring students and faculty at Spelman College, a local HBCU. Current projects involve Spelman's Professor Yewande Olubummo, who together with the principal investigator plans to develop tools for use in the study of shape recognition.