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

* Corresponding author: Asaf Shachar

Received  May 2018 Revised  December 2018 Published  January 2019

Fund Project: This research was partially funded by the Israel Science Foundation (Grant No. 1035/17), and by a grant from the Ministry of Science, Technology and Space, Israel and the Russian Foundation for Basic Research, the Russian Federation

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.

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
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.  Google Scholar

[3]

L. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, 1998. doi: 10.1090/gsm/019.  Google Scholar

[4]

T. Iwaniec, Null lagrangians: Definitions, examples and applications, Warsaw Lectures, Part 2. Google Scholar

[5]

R. KupfermanC. Maor and A. Shachar, Reshetnyak Rigidity for Riemannian Manifolds, Arch. Rat. Mech. Anal., 231 (2019), 367-408.  doi: 10.1007/s00205-018-1282-9.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[3]

L. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, 1998. doi: 10.1090/gsm/019.  Google Scholar

[4]

T. Iwaniec, Null lagrangians: Definitions, examples and applications, Warsaw Lectures, Part 2. Google Scholar

[5]

R. KupfermanC. Maor and A. Shachar, Reshetnyak Rigidity for Riemannian Manifolds, Arch. Rat. Mech. Anal., 231 (2019), 367-408.  doi: 10.1007/s00205-018-1282-9.  Google Scholar

[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.  Google Scholar

Figure 1.  Illustration of the geometric setting of the Euclidean Piola identity
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