Geometric 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 there are not always quasiregular, but the governing partial differential equations are the same. This forces us to move from the classical 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 to a very productive study of mappings of finite distortion. The PI studies together with Hajl asz, Iwaniec and Mal'y, the Jacobian determinant (the pullback of the Riemannian volume forms) of mappings between Riemannian n-manifolds. This study makes it possible to discover new phenomena about such mappings. Also in this proposal, we investigate the Hardy-Littlewood maximal operators on Sobolev spaces, one of the most important tools in analysis.
Geometric function theory has been quite a successful theory, with many diverse applications. The theory of non-linear elasticity for example was based on practical problems from mathematics and physics. It is necessary to study non-linear equations to understand certain physical phenomena such as bifurcation and phase transition. Our main motivation in the theory of mappings of finite distortion is to examine degenerate elliptic equations where important applications lie.
