Linear and nonlinear Korn inequalities on curves in \(\mathbb R^3\).

*(English)*Zbl 1075.53002The classical Korn inequality is a tool in linear elasticity theory, providing the existence of solutions of linearised displacement-traction equations [cf. G. Duvaut and J.–L. Lions, “Inequalities in mechanics and physics” (Grundlehren 219, Springer, Berlin) (1976; Zbl 0331.35002); P. G. Ciarlet, “Mathematical elasticity. Vol. 3: Theory of shells” (Studies in Mathematics and its Applications 29, North-Holland, Amsterdam) (2000; Zbl 0953.74004)].

The results of the present paper extend to the case of curves the work of P. G. Ciarlet and S. Mardare [Math. Models Methods Appl. Sci. 11, No. 8, 1379–1391 (2001; Zbl 1036.74036)].

The idea is to estimate the distance (in the Lebesgue or Sobolev norm) between a parametrised submanifold (in \(\mathbb{R}^3\)) and another one, thought of as its deformation, in terms of the (linearised) change of the metric (curvature, torsion) along the displacement field. In the present paper, this is done for the case when the submanifolds are curves in \(\mathbb{R}^3\). The calculations proceed in an elementary way from the Frenet theory of curves.

It may be remarked that generalisations of the Korn inequality to Riemannian manifolds were provided recently by W. Chen and J. Jost [Calc. Var. Partial Differ. Equ. 14, No. 4, 517–530 (2002; Zbl 1006.74011)].

The results of the present paper extend to the case of curves the work of P. G. Ciarlet and S. Mardare [Math. Models Methods Appl. Sci. 11, No. 8, 1379–1391 (2001; Zbl 1036.74036)].

The idea is to estimate the distance (in the Lebesgue or Sobolev norm) between a parametrised submanifold (in \(\mathbb{R}^3\)) and another one, thought of as its deformation, in terms of the (linearised) change of the metric (curvature, torsion) along the displacement field. In the present paper, this is done for the case when the submanifolds are curves in \(\mathbb{R}^3\). The calculations proceed in an elementary way from the Frenet theory of curves.

It may be remarked that generalisations of the Korn inequality to Riemannian manifolds were provided recently by W. Chen and J. Jost [Calc. Var. Partial Differ. Equ. 14, No. 4, 517–530 (2002; Zbl 1006.74011)].

Reviewer: Aleksandar Perović (Berlin)

##### MSC:

53A04 | Curves in Euclidean and related spaces |

74B15 | Equations linearized about a deformed state (small deformations superposed on large) |

74K05 | Strings |

35Q72 | Other PDE from mechanics (MSC2000) |

##### Keywords:

Korn type inequalities; curves in Euclidean 3-space; linear elasticity theory; linearised displacement-traction equations; Frenet theory; Korn inequality
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\textit{S. Mardare} and \textit{M. Szopos}, Anal. Appl., Singap. 3, No. 3, 251--270 (2005; Zbl 1075.53002)

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##### References:

[1] | Blouza A., Quart. Appl. Math. 57 pp 317– |

[2] | DOI: 10.1007/s005260100113 · Zbl 1006.74011 |

[3] | Ciarlet P. G., Mathematical Elasticity, Volume I: Three Dimensional Elasticity (1988) |

[4] | Ciarlet P. G., Mathematical Elasticity, Volume III: Theory of Shells (2000) |

[5] | DOI: 10.1016/j.crma.2004.01.014 · Zbl 1183.74008 |

[6] | Ciarlet P. G., Math. Models Methods Appl. Sci. 8 pp 1379– |

[7] | DOI: 10.1002/mma.546 · Zbl 1174.74313 |

[8] | DOI: 10.1007/978-1-4612-9923-3 |

[9] | Lions J. L., Problèmes aux Limites Non Homogènes et Applications (1968) · Zbl 0235.65074 |

[10] | DOI: 10.1142/S0252959904000457 · Zbl 1080.53003 |

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