On The Integrability For The Jacobian In Anisotropic Sobolev Spaces

This paper considers the integrabihty of the Jacobian of orientation-preserving mappings in anisotropic Sobolev space W^1,P-ε(Rⁿ, Rⁿ),P=(p1,p2,…,pn), with 1 < p1-ε,p2-ε,…,pn-ε<∞,1/p1 + 1/p2 +…+ 1/pn = 1 and 0 <ε< 1. The first result is a sufficient condition ensuring integrabihty of the Jacobian, and the second one is a result in Orlicz-Sobolev class. Both of the results can be regarded as refinements of the results of Iwaniec and Sbordone [IS].