A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant
It is known that the $W^{1,p}$-distance between an orientation-preserving mapping in $W^{1,p}(\Omega ;\mathbb{R}^n)$ and another orientation-preserving mapping $\Theta \in C^1(\overline{\Omega };\mathbb{R}^n)$, where $\Omega $ is a domain in $\mathbb{R}^n$, $n\geqslant 2$, and $p>1$ is a real num...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2020-09-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.84/ |
| _version_ | 1797651557508972544 |
|---|---|
| author | Malin, Maria Mardare, Cristinel |
| author_facet | Malin, Maria Mardare, Cristinel |
| author_sort | Malin, Maria |
| collection | DOAJ |
| description | It is known that the $W^{1,p}$-distance between an orientation-preserving mapping in $W^{1,p}(\Omega ;\mathbb{R}^n)$ and another orientation-preserving mapping $\Theta \in C^1(\overline{\Omega };\mathbb{R}^n)$, where $\Omega $ is a domain in $\mathbb{R}^n$, $n\geqslant 2$, and $p>1$ is a real number, is bounded above by the $L^p$-distance between the square roots of the metric tensor fields induced by these mappings, multiplied by a constant depending only on $p$, $\Omega $, and $\Theta $.The object of this Note is to establish a better inequality of this type, and to provide in addition an explicitly computable upper bound on the constant appearing in it. An essential role is played in our proofs by the notion of geodesic distance inside an open subset of $\mathbb{R}^n$. |
| first_indexed | 2024-03-11T16:17:32Z |
| format | Article |
| id | doaj.art-a945f517e1764ba28978dd3a027c3e9e |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:17:32Z |
| publishDate | 2020-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-a945f517e1764ba28978dd3a027c3e9e2023-10-24T14:18:59ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692020-09-01358562162610.5802/crmath.8410.5802/crmath.84A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constantMalin, Maria0Mardare, Cristinel1Department of Mathematics, University of Craiova, Craiova, RomaniaDepartment of Mathematics, City University of Hong Kong, Kowloon, Hong KongIt is known that the $W^{1,p}$-distance between an orientation-preserving mapping in $W^{1,p}(\Omega ;\mathbb{R}^n)$ and another orientation-preserving mapping $\Theta \in C^1(\overline{\Omega };\mathbb{R}^n)$, where $\Omega $ is a domain in $\mathbb{R}^n$, $n\geqslant 2$, and $p>1$ is a real number, is bounded above by the $L^p$-distance between the square roots of the metric tensor fields induced by these mappings, multiplied by a constant depending only on $p$, $\Omega $, and $\Theta $.The object of this Note is to establish a better inequality of this type, and to provide in addition an explicitly computable upper bound on the constant appearing in it. An essential role is played in our proofs by the notion of geodesic distance inside an open subset of $\mathbb{R}^n$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.84/ |
| spellingShingle | Malin, Maria Mardare, Cristinel A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant Comptes Rendus. Mathématique |
| title | A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant |
| title_full | A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant |
| title_fullStr | A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant |
| title_full_unstemmed | A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant |
| title_short | A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant |
| title_sort | nonlinear korn inequality in protect mathbb r n with an explicitly bounded constant |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.84/ |
| work_keys_str_mv | AT malinmaria anonlinearkorninequalityinprotectmathbbrnwithanexplicitlyboundedconstant AT mardarecristinel anonlinearkorninequalityinprotectmathbbrnwithanexplicitlyboundedconstant AT malinmaria nonlinearkorninequalityinprotectmathbbrnwithanexplicitlyboundedconstant AT mardarecristinel nonlinearkorninequalityinprotectmathbbrnwithanexplicitlyboundedconstant |