$L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative
For $n\ge 2$ and $1 we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\Omega \rightarrow \mathbb{R}^{n\,\times \,n}$ having $p$-integrable generalized $\underline{\operatorname{Curl}}\,$ and generalized vanishing tangential trace $P\,...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2021-09-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.216/ |
| _version_ | 1797651563802525696 |
|---|---|
| author | Lewintan, Peter Neff, Patrizio |
| author_facet | Lewintan, Peter Neff, Patrizio |
| author_sort | Lewintan, Peter |
| collection | DOAJ |
| description | For $n\ge 2$ and $1 we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\Omega \rightarrow \mathbb{R}^{n\,\times \,n}$ having $p$-integrable generalized $\underline{\operatorname{Curl}}\,$ and generalized vanishing tangential trace $P\,\tau _l=0$ on $\partial \Omega $, denoting by $\lbrace \tau _l\rbrace _{l=1,\,\ldots ,\,n-1}$ a moving tangent frame on $\partial \Omega $, more precisely we have:
\[ \left\Vert P \right\Vert _{L^p\left(\Omega ,\,\mathbb{R}^{n\,\times \,n}\right)}\le c\,\left(\left\Vert \operatorname{sym}P \right\Vert _{L^p\left(\Omega ,\,\mathbb{R}^{n \times n}\right)}+ \left\Vert \underline{\operatorname{Curl}}\,P \right\Vert _{L^p\left(\Omega ,\,\left(\mathfrak{so}(n)\right)^n\right)}\right), \]
where the generalized $\underline{\operatorname{Curl}}\,$ is given by $(\underline{\operatorname{Curl}}\,P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$ and $c=c(n,p,\Omega )>0$ |
| first_indexed | 2024-03-11T16:16:35Z |
| format | Article |
| id | doaj.art-c031b6369e4a46e1b6d5b91a86e92fc0 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:35Z |
| publishDate | 2021-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-c031b6369e4a46e1b6d5b91a86e92fc02023-10-24T14:19:23ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692021-09-01359674975510.5802/crmath.21610.5802/crmath.216$L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivativeLewintan, Peter0https://orcid.org/0000-0002-7188-4806Neff, Patrizio1https://orcid.org/0000-0002-1615-8879Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, GermanyFaculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, GermanyFor $n\ge 2$ and $1 we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\Omega \rightarrow \mathbb{R}^{n\,\times \,n}$ having $p$-integrable generalized $\underline{\operatorname{Curl}}\,$ and generalized vanishing tangential trace $P\,\tau _l=0$ on $\partial \Omega $, denoting by $\lbrace \tau _l\rbrace _{l=1,\,\ldots ,\,n-1}$ a moving tangent frame on $\partial \Omega $, more precisely we have: \[ \left\Vert P \right\Vert _{L^p\left(\Omega ,\,\mathbb{R}^{n\,\times \,n}\right)}\le c\,\left(\left\Vert \operatorname{sym}P \right\Vert _{L^p\left(\Omega ,\,\mathbb{R}^{n \times n}\right)}+ \left\Vert \underline{\operatorname{Curl}}\,P \right\Vert _{L^p\left(\Omega ,\,\left(\mathfrak{so}(n)\right)^n\right)}\right), \] where the generalized $\underline{\operatorname{Curl}}\,$ is given by $(\underline{\operatorname{Curl}}\,P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$ and $c=c(n,p,\Omega )>0$https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.216/ |
| spellingShingle | Lewintan, Peter Neff, Patrizio $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative Comptes Rendus. Mathématique |
| title | $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative |
| title_full | $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative |
| title_fullStr | $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative |
| title_full_unstemmed | $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative |
| title_short | $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative |
| title_sort | l p versions of generalized korn inequalities for incompatible tensor fields in arbitrary dimensions with p integrable exterior derivative |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.216/ |
| work_keys_str_mv | AT lewintanpeter lpversionsofgeneralizedkorninequalitiesforincompatibletensorfieldsinarbitrarydimensionswithpintegrableexteriorderivative AT neffpatrizio lpversionsofgeneralizedkorninequalitiesforincompatibletensorfieldsinarbitrarydimensionswithpintegrableexteriorderivative |