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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.19473 |
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Table of Contents:
- Let $(u_j)_j$ be a sequence of maps in $W^{1,2}(Ω;\mathbb R^3)$, where $Ω$ 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 $Ω$ under $u_j$.