EXTREMAL  MAPPINGS  OF  FINITE  DISTORTION

    (Tadeusz  Iwaniec,  Syracuse  University)



I will  present recent joint work with  K. Astala, G.J.  Martin and J.
Onninen. In this work we refine the  connections between the theory of
mappings   of finite distortion and   the  calculus of variation.  Our
primary aim is to extend the study of extremal quasiconformal mappings
by considering integral averages of the distortion function instead of
its  L-infinity  norm. We identify many  new  and unexpected phenomena
concerning   existence, uniqueness and  regularity   for  the extremal
problems. The principal  advantage of  minimizing  over the  family of
homeomorphisms in the above variational problems lies in the fact that
the   inverse  maps   are  also   extremal    for  their own    energy
integrals. Sometimes this associated  problem for the inverse  mapping
is easier to solve than the original one as  it may involve minimizing
a convex functional.  There are many natural reasons for studying such
problems. We eventually hope  to  lay down the analytical  foundations
for approaches to compactifying the moduli spaces, such as Teichmuller
spaces, where it is our expectation that a compactification will be by
mappings whose distortion function  lies in some natural integrability
class.