×

Existence of elastic deformations with prescribed principal strains and triply orthogonal systems. (English) Zbl 0544.53012

Two geometric problems are studied. The first is prescribing the principal strains of an elastic deformation. An elastic deformation may be viewed as a diffeomorphism between two Riemannian manifolds; the metrics are usually the standard flat one. The principal strains of the deformation are the eigenvalues of the metric in the range manifold viewed as a symmetric matrix with respect to the metric in the domain. Given distinct positive smooth functions on the domain, we prove the local existence of a smooth elastic deformation with these functions as principal strains. We also prove the local existence of a triply orthogonal system on a Riemannian 3-manifold. This is equivalent to the existence of local coordinates with respect to which the metric is diagonal. Each problem has two distinct approaches to it, depending on whether one solves for an appropriately defined diffeomorphism or for an appropriate moving frame. One way works and the other one doesn’t, but for the two problems the situations are opposite. By solving for the diffeomorphism in the first and the moving frame in the second, the problems reduce to nonlinear hyperbolic systems, which are locally solvable. Both are striking examples of natural problems in Riemannian geometry which reduce to nonelliptic partial differential equations.

MSC:

53B20 Local Riemannian geometry
35L70 Second-order nonlinear hyperbolic equations
74B20 Nonlinear elasticity
58C15 Implicit function theorems; global Newton methods on manifolds
58D25 Equations in function spaces; evolution equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] R. Bryant, P. Griffiths, and D. Yang, Characteristics and existence of isometric embeddings , Duke Math. J. 50 (1983), no. 4, 893-994. · Zbl 0536.53022
[2] E. Cartan, Les systèmes différentiels extérieurs et leurs applications géométriques , Actualités Sci. Ind., no. 994, Hermann et Cie., Paris, 1945. · Zbl 0063.00734
[3] D. M. DeTurck, Existence of metrics with prescribed Ricci curvature: local theory , Invent. Math. 65 (1981/82), no. 1, 179-207. · Zbl 0489.53014
[4] D. M. DeTurck and J. L. Kazdan, Some regularity theorems in Riemannian geometry , Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 249-260. · Zbl 0486.53014
[5] H. Flanders, Differential forms with applications to the physical sciences , Academic Press, New York, 1963. · Zbl 0112.32003
[6] B. M. Fraeijs de Veubeke, A course in elasticity , Applied Mathematical Sciences, vol. 29, Springer-Verlag, New York, 1979. · Zbl 0419.73001
[7] K. O. Friedrichs, Symmetric hyperbolic linear differential equations , Comm. Pure Appl. Math. 7 (1954), 345-392. · Zbl 0059.08902
[8] J. Gasqui, Sur l’existence locale de certaines métriques riemanniennes plates , Duke Math. J. 46 (1979), no. 1, 109-118. · Zbl 0403.53020
[9] R. S. Hamilton, The inverse function theorem of Nash and Moser , Bulletin Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65-222. · Zbl 0499.58003
[10] S. Klainerman, Global existence for nonlinear wave equations , Comm. Pure Appl. Math. 33 (1980), no. 1, 43-101. · Zbl 0405.35056
[11] J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I , Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 265-315. · Zbl 0174.47801
[12] F. D. Murnaghan, Finite Deformation of an Elastic Solid , John Wiley & Sons Inc., New York, N. Y., 1951. · Zbl 0045.26504
[13] M. Spivak, A comprehensive introduction to differential geometry. Vol. III , Publish or Perish Inc., Wilmington, Del., 1979. · Zbl 0439.53003
[14] M. E. Taylor, Pseudodifferential operators , Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. · Zbl 0453.47026
[15] D. Yang, Involutive hyperbolic differential systems , Ph.D. thesis, Harvard U., 1983.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.