University of Pittsburgh

I am going to describe the main results of my recent joint work with T. Iwaniec, J. Maly and J. Onninen, ``Weakly differentiable mappings between manifolds". (accepted in Memoirs Amer. Math. Soc.). The paper is available at:

Paper

**Abstract**: We study the Sobolev mappings between compact
manifolds. Particular emphasis is on the case of Orlicz-Sobolev class of
mappings that is slightly larger than the Sobolev space of mappings in the
critical case p=n. We prove density of smooth mappings and study degree theory
in this case. It turns out that the degree theory is related to the topological
structure of the target manifold, more precisely we have good theory if the
target manifold is not a rational homology sphere and we have a counterexample
for the case of the sphere. We also generalize the theorem of Coifman Lions
Meyer and Semmes about the Hardy space regularity of Jacobians to the case of
Sobolev mappings between compact manifolds in the limiting case $p=n$.