Jacobian of weak limits of Sobolev homeomorphisms

Abstract

Let Ω be a domain in Rn, where n=2,3. Suppose that a sequence of Sobolev homeomorphisms fk:Ω→Rn with positive Jacobian determinants, J(x,fk)>0, converges weakly in W1,p(Ω,Rn), for some p⩾1, to a mapping f. We show that J(x,f)⩾0 a.e. in Ω. Generalizations to higher dimensions are also given.