Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911761803771904 |
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| author | Campbell, Daniel |
| author_facet | Campbell, Daniel |
| contents | We present three novel classifications of the weak sequential (and strong) limits in $W^{1,p}$ of planar diffeomorphisms. We introduce a concept called the QM condition which is a kind of separation property for pre-images of closed connected sets and show that $u$ satisfies this property exactly when it is the limit of Sobolev homeomorphisms. Further, we prove that $u\in W^{1,p}_{\operatorname{id}}((-1,1)^2,\mathbb{R}^2)$ is the limit of a sequence of homeomorphisms exactly when there are classically monotone mappings $g_δ:[-1,1]^2\to \mathbb{R}^2$ and very small open sets $U_δ$ such that $g_δ = u$ on $[-1,1]^2 \setminus U_δ$. Also, we introduce the so-called three curve condition, which is in some sense reminiscent of the NCL condition of \cite{CPR} but for $u^{-1}$ instead of for $u$, and prove that a map is the $W^{1,p}$ limit of planar Sobolev homeomorphisms exactly when it satisfies this property. This improves on results in \cite{DPP} answering the question from \cite{IO2}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10639 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity Campbell, Daniel Analysis of PDEs We present three novel classifications of the weak sequential (and strong) limits in $W^{1,p}$ of planar diffeomorphisms. We introduce a concept called the QM condition which is a kind of separation property for pre-images of closed connected sets and show that $u$ satisfies this property exactly when it is the limit of Sobolev homeomorphisms. Further, we prove that $u\in W^{1,p}_{\operatorname{id}}((-1,1)^2,\mathbb{R}^2)$ is the limit of a sequence of homeomorphisms exactly when there are classically monotone mappings $g_δ:[-1,1]^2\to \mathbb{R}^2$ and very small open sets $U_δ$ such that $g_δ = u$ on $[-1,1]^2 \setminus U_δ$. Also, we introduce the so-called three curve condition, which is in some sense reminiscent of the NCL condition of \cite{CPR} but for $u^{-1}$ instead of for $u$, and prove that a map is the $W^{1,p}$ limit of planar Sobolev homeomorphisms exactly when it satisfies this property. This improves on results in \cite{DPP} answering the question from \cite{IO2}. |
| title | Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.10639 |