Stability theorem for spatial quasiconformal deformations

Abstract

Let G C R rL be a domain, n~ 2. A continuous function f:G --~ R m is called a quasiconformal deformation, abbreviated qc-deformation or qcd, if it has locally integrable first order distributional derivatives in G and llSfll = ess sup x~G llSf(x)ll ~o. Here S is Ahlfors' differential operator Sf(x) = (I/2)(Df(x)+Df(x)) (1/n)tr(Df(x))I where Df(x) is the Jacobian matrix of f, I is the unit matrix, A denotes the transpose and tr(A) the trace of an n~n-matrix A. The norm liA~i is the operator R n norm IIAIL = sup IAxl x E If ilSfll~ ~ k ~ 3, every trivial deformation f:G -~ R n is a restriction to G of a mapping which has the form