Mappings of Finite Distortion

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DMS 0400611PI: Jani OnninenUniversity of MichiganMappings of Finite DistortionAbstractGeometric function theory is largely concerned with generalizations of the theory of analytic functions to higher dimensions. It turns out that the category of maps with the same geometric and function theoretic properties of analytic functions are the mappings of bounded distortion, also called quasiregular mappings, or, if injective, quasiconformal mappings. Both kind of mappings solve uniformly elliptic partial differential equations in the plane. Moreover, these mappings preserve the natural Sobolev spaces which arise in the study of function theory and partial differential equations on subdomains of Euclidean n-space.In recent years there has been another well-known theory of mappings whose ideas have gotten to the core of geometry and analysis, non-linear elasticity theory. The mappings which naturally occur thereare not always quasiregular, but the governing partialdifferential equations are the same. This forces us to move from theclassical setting of uniformly elliptic partial differential equations to degenerate elliptic equations. Usually, however, some control of the ellipticity bounds will be necessary to achieve concrete results. These often take the form of integral estimates in some Lebesgue or Sobolev spaces. This is the theory of mappings of finite distortion.In this proposal we focus mainly on mappings of finite distortion between subsets of the Euclidean n-space. We also emphasize the fundamental role of the Jacobian determinant, which already has led toa very productive study of mappings of finite distortion. The PIstudies together with Haj\l asz, Iwaniec and Mal\'y, the Jacobiandeterminant (the pullback of the Riemannian volume forms) ofmappings between Riemannian n-manifolds. This study makes itpossible to discover new phenomena about such mappings. Also inthis proposal, we investigate the Hardy-Littlewood maximaloperators on Sobolev spaces, one of the most important tools inanalysis.Geometric function theory has been quite a successful theory, withmany diverse applications. The theory of non-linear elasticity forexample was based on practical problems from mathematics andphysics. It is necessary to study non-linear equations tounderstand certain physical phenomena such as bifurcation andphase transition. Our main motivation in the theory of mappings offinite distortion is to examine degenerate elliptic equationswhere important applications lie.