An extension of Reshetnyak's theorem

Let F is an element of W-loc(1, n)(Omega; R-n) be a mapping with non negati ve Jacobian J(F)(x) = det DF(x) greater than or equal to 0 for a. e. x in a domain Omega subset of R-n. The dilatation of F is defined (almost everywh ere in Omega) by the formula K(x) = \DF(x)\(n)/J(F) (x). If K is bounded, the mapping F is said to be quasiregular. These are a gene ralization to higher dimensions of holomorphic functions. The theory of hig her dimensional quasiregular mappings began with Reshetnyak's theorem [R], stating that they are continuous, discrete and open, if they are nonconstan t. In some problems appearing in the nonlinear elasticity models suggested in [B1-2], the boundedness condition for K is too restrictive. Typically we on ly have that K-p is integrable for some p. In two dimensions, Iwaniec and S verak [IS] have shown that K is an element of L-loc(1) is enough to guarant ee the conclusion of Reshetnyak's theorem. In this paper we consider the hi gher dimensional case n greater than or equal to 3, and extend Reshetnyak's theorem to the case K is an element of L-loc(p), where p > n-1. This is kn own to be false for p < n-1 and is not known in the case p = n-1. We follow the footsteps of Reshetnyak's original proof, however our equatio ns are no longer strictly elliptic, We develop a method to deal with badly degenerate elliptic equations based on monotone functions estimates, that a llows us to establish a weak Harnack's inequality for log(1/\F\). A nontriv ial matter here, is the construction of appropriate test functions. We use a computer to exhibit an explicit smooth n-superharmonic "bump function" wh ich approximates log(1/\x\).