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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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| _version_ | 1866909328782393344 |
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| author | Kalayanamit, Panas Henao, Duvan |
| author_facet | Kalayanamit, Panas Henao, Duvan |
| 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 $u_j$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The positivity of the Jacobian in the weak limit of generalised axisymmetric maps Kalayanamit, Panas Henao, Duvan Analysis of PDEs 74B20 (Primary), 26B10, 46E35 (Secondary) 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 $u_j$. |
| title | The positivity of the Jacobian in the weak limit of generalised axisymmetric maps |
| topic | Analysis of PDEs 74B20 (Primary), 26B10, 46E35 (Secondary) |
| url | https://arxiv.org/abs/2409.19473 |