Deformations of finite conformal energy: existence and removability of singularities

This paper features a class of mappings h = ( h 1 , … , h m ) : → onto between bounded domains , ⊂ ℝ n , having finite n ‐harmonic energy, such that we have E [ h ] =∫ X‖ D h ( x ) ‖ n d x ,        ‖ D h ‖ 2 = Tr ( D * h   D h ) . The fundamental question is whether or not the domains , ⊂ ℝ n of the same topological type admit a homeomorphism h : X → onto Y in a given homotopy class having finite energy. The examples of non‐existence, somewhat testing our theory, arise when we remove from bounded smooth domains and thin subsets X ⊂ Xand Υ ⊂ Y , referred to as cracks or fractures. We are looking for homeomorphisms h : X \ X → onto Y \ ϒ of finite energy for which ϒ is the cluster set of h over X . In general, infinite energy is required in order to increase the dimension of a crack X ⊂ X that is, when X < dim ⁡ Υ ⩽ n − 1 . Suppose now that a bounded deformation h : X ∖ X → R nof finite energy is given. Does h extend continuously to and, if so, is the extension injective on ? We give affirmative answers to these questions.