Sobolev Mappings of Smallest Energy

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The main theme of this research program is to develop minimizing techniques in the geometric theory of functions and in the theory of elasticity. These two theories are related through challenging problems in the calculus of variations and nonlinear partial differential equations. They both rely on geometric intuition and an in-depth analysis of energy minimizing deformations of material bodies or, more mathematically, higher dimensional curved surfaces. In the search for mathematical models of hyperelasticity, one must accept and explore the limits of elastic deformations. This approach turns out to be particularly effective in the two-dimensional theory of flat plates and thin films. Theoretical prediction of failures of bodies caused by cracks and fractions is a good motivation that should appeal to both pure mathematicians and researchers in applied fields. Experimental answers to practical problems will lead to solutions or deeper insights into mathematical problems investigated in this project; for example, the existence and uniqueness of deformations of smallest average distortion. The research originated from the Riemann Mapping Theorem; conformal mappings being univalent solutions of the Cauchy-Riemann system. Moving to the second order variational equations and their homeomorphic solutions offers new challenges. The goal is to characterize energy-functionals whose minimizers exist and closely follow conformal maps. It is a common struggle in mathematical models of nonlinear elasticity to establish the existence of energy-minimal deformations which comply with the principle of no interpenetration of matter. To build a viable theory the PI will adopt monotone Sobolev mappings as legitimate deformations in 2-dimensional elasticity. Minimizing among Sobolev homeomorphisms the basic questions such as existence, uniqueness and regularity of energy-minimal mappings become challenging problems. Building new tools to solve such problems is the central part of this project.