Existence of elastic deformations with prescribed principal strains and triply orthogonal systems
type: Item
instance of: scholarly article
title: Existence of elastic deformations with prescribed principal strains and triply orthogonal systems
zbMATH Open document ID: 0544.53012
DOI: 10.1215/S0012-7094-84-05114-7
author: Dennis DeTurck
author: Deane Yang
published in: Duke Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 58C15
Mathematics Subject Classification ID: 58D25
zbMATH DE Number: 3866103
zbMATH Keywords: elastic deformation
zbMATH Keywords: principal strains
zbMATH Keywords: triply orthogonal system
zbMATH Keywords: diffeomorphism
zbMATH Keywords: moving frame
zbMATH Keywords: nonlinear hyperbolic systems
MaRDI profile type: Publication
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full work available at URL: s0012-7094-84-05114-7
OpenAlex ID: W1509957847
description: scientific article; zbMATH DE number 3866103