The Sobolev closure of planar diffeomorphisms

Abstract: In studying variational problems in Non-linear Elasticity and in Geometric Function Theory a natural issue that appears is the understanding of the closure of planar diffeomoprhism in the weak or strong Sobolev topology. In this talk I will discuss the characterisation of the theses closures,  in particular I will show that both closures always  coincide. I will also provide some sufficient condition for a planar map to be approximate by diffeomorphism in terms of the connectedness of its counter image, in the spirit of Young’s characterisation of monotone functions.  This is a joint work with Aldo Pratelli.