Deformations of Finite Conformal Energy: Boundary Behavior and Limit Theorems

We study homeomorphisms h : X onto −→ Y between two bounded domains in Rn having finite conformal energy E[h] = ∫ X ||Dh(x) || n dx < ∞ , h ∈ W (X,Y). We consider the behavior of such mappings, including continuous extension to the closure of X and injectivity of h : X → Y. In general, passing to the weak W 1,n-limit of a sequence of homeomorphisms hj : X → Y one loses injectivity. However, if the mappings in question have uniformly bounded L 1-average of the inner distortion, then, for sufficiently regular domains X and Y, their limit map h : X onto −→ Y is a homeomorphism. Moreover, the inverse map f = h−1 : Y onto −→ X enjoys finite conformal energy and has integrable inner distortion as well.