Energy-Minimal Principles in Geometric Function Theory

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This project investigates the geometry and analysis of energy-minimizing deformations with applications to the study of nonlinear elasticity. The latter field, which studies the deformation of physical materials and bodies in response to stress and strain, is informed by developments in materials science and physics. A common challenge faced in the development of mathematical models of nonlinear elasticity is how to account for the physical impossibility of compressing a portion of an elastic body of positive volume into a space of zero volume. From a mathematical point of view, serious difficulties arise when trying to overcome the lack of injectivity as postulated by the physical principle of non-interpenetration of matter. While homeomorphic solutions would be ideal models, mathematical constraints linked to the necessity of complying with the aforementioned physical principle lead naturally to the conclusion that limits of homeomorphic solutions must be allowed as legitimate competitors. New analytic and geometric tools have been developed to accommodate these difficulties, and those tools will be developed further and in greater detail in this project. The project will also afford research opportunities for early career mathematicians, including graduate students.Weak limits of energy-minimizing sequences of Sobolev homeomorphisms are natural candidates of energy-minimizers. In two dimensions, weak and strong limits coincide and characterize the class of monotone Sobolev mappings. Non-injective energy-minimal solutions, being monotone, may squeeze two-dimensional plates or thin films, but may not fold them. Serious challenges arise in the investigation of energy-minimal solutions solely based on inner-variational equations. This project is largely concerned with questions similar in spirit to the Riemann conformal mapping problem. Such variational questions also lead to associated Sobolev mapping problems and, in the case of prescribed boundary values, require a preliminary investigation of Sobolev variants of the Jordan-Schoenflies theorem. A further goal of the project is to deepen the connections between geometric function theory and relevant areas within physics and engineering, by fostering the exchange of ideas among practitioners of these various subjects.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.