Figure 1.
Illustration of the geometric setting of the Euclidean Piola identity
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March
2019, 11(1): 59-76.
doi: 10.3934/jgm.2019004
A geometric perspective on the Piola identity in Riemannian settings
Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel |
The Piola identity $ \operatorname{div}\; \operatorname{cof} \;\nabla f = 0 $ is a central result in the mathematical theory of elasticity. We prove a generalized version of the Piola identity for mappings between Riemannian manifolds, using two approaches, based on different interpretations of the cofactor of a linear map: one follows the lines of the classical Euclidean derivation and the other is based on a variational interpretation via Null-Lagrangians. In both cases, we first review the Euclidean case before proceeding to the general Riemannian setting.
Mathematics Subject Classification:
Primary: 53Zxx; Secondary: 74Bxx.
Citation:
Raz Kupferman, Asaf Shachar. A geometric perspective on the Piola identity in Riemannian settings. Journal of Geometric Mechanics,
2019, 11
(1)
: 59-76.
doi: 10.3934/jgm.2019004
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References:
| [1] |
P. Ciarlet, Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity, Elsevier, 1988. Google Scholar |
| [2] |
J. Eells and L. Lemaire, Selected Topics in Harmonic Maps, CBMS Regional Conference Series in Mathematics, 50. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983.
doi: 10.1090/cbms/050. |
| [3] |
L. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, 1998.
doi: 10.1090/gsm/019. |
| [4] |
T. Iwaniec, Null lagrangians: Definitions, examples and applications, Warsaw Lectures, Part 2. Google Scholar |
| [5] |
R. Kupferman, C. Maor and A. Shachar,
Reshetnyak Rigidity for Riemannian Manifolds, Arch. Rat. Mech. Anal., 231 (2019), 367-408.
doi: 10.1007/s00205-018-1282-9. |
| [6] |
J. Marsden and T. Hughes, Mathematical Foundations of Elasticity, Dover, 1983. Google Scholar |
| [7] |
D. Saunders, The Geometry of Jet Bundles, Cambridge University Press, 1989.
doi: 10.1017/CBO9780511526411.
Google Scholar
|
| [8] |
P. Steinmann, Geometrical Foundations of Continuum Mechanics, Springer, 2015.
doi: 10.1007/978-3-662-46460-1. |
show all references
References:
| [1] |
P. Ciarlet, Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity, Elsevier, 1988. Google Scholar |
| [2] |
J. Eells and L. Lemaire, Selected Topics in Harmonic Maps, CBMS Regional Conference Series in Mathematics, 50. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983.
doi: 10.1090/cbms/050. |
| [3] |
L. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, 1998.
doi: 10.1090/gsm/019. |
| [4] |
T. Iwaniec, Null lagrangians: Definitions, examples and applications, Warsaw Lectures, Part 2. Google Scholar |
| [5] |
R. Kupferman, C. Maor and A. Shachar,
Reshetnyak Rigidity for Riemannian Manifolds, Arch. Rat. Mech. Anal., 231 (2019), 367-408.
doi: 10.1007/s00205-018-1282-9. |
| [6] |
J. Marsden and T. Hughes, Mathematical Foundations of Elasticity, Dover, 1983. Google Scholar |
| [7] |
D. Saunders, The Geometry of Jet Bundles, Cambridge University Press, 1989.
doi: 10.1017/CBO9780511526411.
Google Scholar
|
| [8] |
P. Steinmann, Geometrical Foundations of Continuum Mechanics, Springer, 2015.
doi: 10.1007/978-3-662-46460-1. |
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