Jacobian of weak limits of Sobolev homeomorphisms

Abstract

Abstract Let Ω be a domain in ℝ n {\mathbb{R}^{n}} , where n = 2 , 3 {n=2,3} . Suppose that a sequence of Sobolev homeomorphisms f k : Ω → ℝ n {f_{k}\colon\Omega\to\mathbb{R}^{n}} with positive Jacobian determinants, J ⁢ ( x , f k ) > 0 {J(x,f_{k})>0} , converges weakly in W 1 , p ⁢ ( Ω , ℝ n ) {W^{1,p}(\Omega,\mathbb{R}^{n})} , for some p ⩾ 1 {p\geqslant 1} , to a mapping f. We show that J ⁢ ( x , f ) ⩾ 0 {J(x,f)\geqslant 0} a.e. in Ω. Generalizations to higher dimensions are also given.