Deformations of finite conformal energy: Boundary behavior and limit theorems

We study homeomorphisms h Y between two bounded domains in ℝⁿ having finite conformal energy We consider the behavior of such mappings, including continuous extension to the closure of X and injectivity of h In general, passing to the weak W ^1,n-limit of a sequence of homeomorphisms h one loses injectivity. However, if the mappings in question have uniformly bounded ℒ¹-average of the inner distortion, then, for sufficiently regular domains X and Y, their limit map h: X Y is a homeomorphism. Moreover, the inverse map f=h^-1: Y X enjoys finite conformal energy and has integrable inner distortion as well.