The positivity of the Jacobian in the weak limit of generalised axisymmetric maps

Abstract

Let $(u_j{)}_j$ be a sequence of maps in $W^{1,2}(\Omega ;{\mathbb{R}}^3)$, where $\Omega$ is a domain in ${\mathbb{R}}^3$. When can we conclude that its weak limit $u$ has non-negative Jacobian a.e.? Hencl and Onninen shows that it is sufficient that each $u_j$ is an orientation-preserving homeomorphism, using an ingenious analysis of a topological invariant called the linking number. Following their approach, we show that if each $u_j$ is a generalised axisymmetric map that has positive Jacobian a.e. and is one-to-one a.e., then $\det Du\ge 0$ a.e. Our proof is based on using the divergence identities to control the sign of the linking numbers of the images of links in $\Omega$ under $u_j$.