Quasiconformal Mappings, Harmonic Analysis and Nonlinear Elasticity from the Prospective of PDEs

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Iwaniec ABSTRACT Iwaniec will pursue questions about quasiregular mappings which arise as generalizations of geometric aspects of analytic functions of one complex variable. The mappings in question solve important first order systems of PDEs analogous in many respects to the Cauchy-Riemann equations. The solutions of these systems can be viewed as "absolute" minimizers of certain energy functionals. It is striking how tight the connection is between quasiregular mappings and the recent development of the nonlinear elasticity theory whose mathematical principles were already formulated by S.S. Antman and J. Ball in 1976-77. Roughly speaking, the theory of elasticity studies mappings, referred to as deformations of elastic bodies, which minimize the so-called stored energy functionals. These functionals are not always convex and the deformations need not be quasiconformal but the governing PDEs are much the same. The Jacobian determinant, in particular, has been subjected to a great deal of investigation. Ths project accounts for this study together with some generalizations concerning wedge products of closed differential forms. Quasiconformal mappings generalize to higher dimensions the concept of analytic functions that is so fruitful for science and engineering in two space dimensions. Trying to extend this theory to three or more space dimensions proved quite difficult and the mathematics of such extension is being developed now. Iwaniec will work on how to extend familiar concepts in the plane, like degree, to similar concepts in space. The theory of non linear elasticity, among others, will benefit from this developments.