$(INV)$ condition and regularity of the inverse
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909862774964224 |
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| author | Doležalová, Anna Hencl, Stanislav Onninen, Jani |
| author_facet | Doležalová, Anna Hencl, Stanislav Onninen, Jani |
| contents | Let $f \colon Ω\to Ω' $ be a Sobolev mapping of finite distortion between planar domains $Ω$ and $Ω'$, satisfying the $(INV)$ condition and coinciding with a homeomorphism near $\partialΩ$. We show that $f$ admits a generalized inverse mapping $h \colon Ω' \to Ω$, which is also a Sobolev mapping of finite distortion and satisfies the $(INV)$ condition.
We also establish a higher-dimensional analogue of this result: if a mapping $f \colon Ω\to Ω' $ of finite distortion is in the Sobolev class $W^{1,p}(Ω, \mathbb{R}^n)$ with $p > n-1$ and satisfies the $(INV)$ condition, then $f$ has an inverse in $W^{1,1}(Ω', \mathbb{R}^n)$ that is also of finite distortion.
Furthermore, we characterize Sobolev mappings satisfying $(INV)$ whose generalized inverses have finite $n$-harmonic energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18976 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $(INV)$ condition and regularity of the inverse Doležalová, Anna Hencl, Stanislav Onninen, Jani Functional Analysis Analysis of PDEs Let $f \colon Ω\to Ω' $ be a Sobolev mapping of finite distortion between planar domains $Ω$ and $Ω'$, satisfying the $(INV)$ condition and coinciding with a homeomorphism near $\partialΩ$. We show that $f$ admits a generalized inverse mapping $h \colon Ω' \to Ω$, which is also a Sobolev mapping of finite distortion and satisfies the $(INV)$ condition. We also establish a higher-dimensional analogue of this result: if a mapping $f \colon Ω\to Ω' $ of finite distortion is in the Sobolev class $W^{1,p}(Ω, \mathbb{R}^n)$ with $p > n-1$ and satisfies the $(INV)$ condition, then $f$ has an inverse in $W^{1,1}(Ω', \mathbb{R}^n)$ that is also of finite distortion. Furthermore, we characterize Sobolev mappings satisfying $(INV)$ whose generalized inverses have finite $n$-harmonic energy. |
| title | $(INV)$ condition and regularity of the inverse |
| topic | Functional Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2412.18976 |