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:
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 $p=n$.