Some Global Estimates for the Jacobians of Quasiregular Mappings

Abstract

Abstract. Some global estimates for the Jacobians of quasiregular mappings f =( f 1 ;f 2 ; ;f n ) of the Sobolev class W 1 ;n Ω ;R n ) in L φ (  )-domains and John domainsare established. 1. IntroductionThroughout this paper, we always assume that Ω is an open subset of R n , n  2, and f = ( f 1 ;f 2 ; ;f n ) 2 W 1 ;n Ω ; R n ) be a mapping. The distribu-tional fftial Df =( @f i @x j ) 1 i;j n : Ω ! GL ( n ) of f is an integrable func-tion on Ω with values in the space GL ( n ) of all n n -matrices. Denote by jDf ( x ) j = max fjDf ( x ) hj : h 2 S n 1 g and J ( x;f ) = det Df ( x ) the norm of theJacobian matrix Df ( x ) and the Jacobian determinant of f , respectively.De nition 1.1. A mapping f 2 W 1 ;n Ω ; R n ) is said to be K -quasiregular,1  K < 1 , if jDf ( x ) j n  KJ ( x;f ) ; a.e. ΩWe say that f is orientation preserving (reversing), if its Jacobian determinant J ( x;f ) is nonnegative (nonpositive) almost everywhere in Ω From De nition 1.1we know that any quasiregular mappings are orientation preserving.This paper mainly deals with the Jacobians of quasiregular mappings. The