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A nonlinear Korn inequality in ℝn with an explicitly bounded constant.

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Title: A nonlinear Korn inequality in ℝn with an explicitly bounded constant.
Alternate Title: Une inégalité de Korn non linéaire dans ℝn avec une constante majorée explicitement.
Authors: Malin, Maria1 malinmaria@yahoo.com, Mardare, Cristinel2 cmardare@cityu.edu.hk
Source: Comptes Rendus. Mathématique. 2020, Vol. 358 Issue 5, p621-526. 6p.
Subjects: Tensor fields, Geodesic distance
Abstract (English): It is known that the W1,p-distance between an orientation-preserving mapping in W1,p (Ω; ℝn) and another orientation-preserving mapping Θ ∈ C¹ (...;ℝn), where Ω is a domain in ℝn, n ≥ 2, and p > 1 is a real number, is bounded above by the Lp-distance between the square roots of the metric tensor fields induced by these mappings, multiplied by a constant depending only on p, Ω, and Θ. 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 ℝn. [ABSTRACT FROM AUTHOR]
Abstract (French): Il est connu que la distance dans W1,p entre une application dans W1,p (Ω; ℝn) préservant l'orientation et une autre application Θ ∈ C¹ (...; ℝn) préservant l'orientation, où Ω est un domain de ℝn, n ≥ 2, et p > 1 est un nombre réel, est majorée par la distance dans Lp entre les racines carrées des champs de tenseurs métriques induits par ces applications, multipliée par une constante dépendant uniquement de p, Ω, et Θ. L'objet de cette Note est d'établir une meilleure inégalité de ce type, et de fournir en plus une borne supérieure explicitement calculable de la constante qui y apparaît. Un rôle essentiel est joué dans nos preuves par la notion de distance géodésique dans un ouvert de ℝn. [ABSTRACT FROM AUTHOR]
Copyright of Comptes Rendus. Mathématique is the property of Academie des Science, Centre Mersenne and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
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PubType: Academic Journal
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  Data: Une inégalité de Korn non linéaire dans ℝn avec une constante majorée explicitement.
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  Data: <searchLink fieldCode="AR" term="%22Malin%2C+Maria%22">Malin, Maria</searchLink><relatesTo>1</relatesTo><i> malinmaria@yahoo.com</i><br /><searchLink fieldCode="AR" term="%22Mardare%2C+Cristinel%22">Mardare, Cristinel</searchLink><relatesTo>2</relatesTo><i> cmardare@cityu.edu.hk</i>
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  Data: <searchLink fieldCode="JN" term="%22Comptes+Rendus%2E+Mathématique%22">Comptes Rendus. Mathématique</searchLink>. 2020, Vol. 358 Issue 5, p621-526. 6p.
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  Data: <searchLink fieldCode="DE" term="%22Tensor+fields%22">Tensor fields</searchLink><br /><searchLink fieldCode="DE" term="%22Geodesic+distance%22">Geodesic distance</searchLink>
– Name: Abstract
  Label: Abstract (English)
  Group: Ab
  Data: It is known that the W1,p-distance between an orientation-preserving mapping in W1,p (Ω; ℝn) and another orientation-preserving mapping Θ ∈ C¹ (...;ℝn), where Ω is a domain in ℝn, n ≥ 2, and p > 1 is a real number, is bounded above by the Lp-distance between the square roots of the metric tensor fields induced by these mappings, multiplied by a constant depending only on p, Ω, and Θ. 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 ℝn. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label: Abstract (French)
  Group: Ab
  Data: Il est connu que la distance dans W1,p entre une application dans W1,p (Ω; ℝn) préservant l'orientation et une autre application Θ ∈ C¹ (...; ℝn) préservant l'orientation, où Ω est un domain de ℝn, n ≥ 2, et p > 1 est un nombre réel, est majorée par la distance dans Lp entre les racines carrées des champs de tenseurs métriques induits par ces applications, multipliée par une constante dépendant uniquement de p, Ω, et Θ. L'objet de cette Note est d'établir une meilleure inégalité de ce type, et de fournir en plus une borne supérieure explicitement calculable de la constante qui y apparaît. Un rôle essentiel est joué dans nos preuves par la notion de distance géodésique dans un ouvert de ℝn. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
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  Data: <i>Copyright of Comptes Rendus. Mathématique is the property of Academie des Science, Centre Mersenne and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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      – SubjectFull: Tensor fields
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      – SubjectFull: Geodesic distance
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