Jacobian of weak limits of Sobolev homeomorphisms

Abstract

Let Omega be a domain in R-n, where n = 2, 3. Suppose that a sequence of Sobolev homeomorphisms f(k) : Omega -> R-n with positive Jacobian determinants, J(x, f(k)) > 0, converges weakly in W-1,W- p(Omega, R-n), for some p >= 1, to a mapping f.

We show that J(x, f) >= 0 a.e. in Omega. Generalizations to higher dimensions are also given.

Keywords

• Sobolev homeomorphism
• weak limits
• Jacobian