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Linearization stability of nonlinear partial differential equations

Fischer, Arthur E. and Marsden, Jerrold E. (1975) Linearization stability of nonlinear partial differential equations. Proceedings of Symposia in Pure Mathematics , 27 (2). pp. 219-263. ISSN 0082-0717 . http://resolver.caltech.edu/CaltechAUTHORS:20100806-095929800

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Abstract

In this article we study solutions to systems of nonlinear partial differential equations that arise in riemannian geometry and in general relativity. The systems we shall be considering are the scalar curvature equations R(g) = P and the Einstein equations Ric (^(4)g) = 0 for an empty spacetime. Here g is a riemannian metric and R(g) is the scalar curvature of g , ^P is a given function, (4)g is a Lorentz metric on a 4-manifold and Ric(^(4)g) denotes the Ricci curvature tensor of ^(4)g . To study the nature of a solution to a given system of partial differential equations, it is common to linearize the equations about the given solution, solve the linearized equations, and assert that the solution to these linearized equations can be used to approximate solutionsto the nonlinear equations in the sense that there exists a curve of solutions to the full equations which is tangent to the linearized solution. This assertion, however, is not always valid. In our study of the above equations we give precise conditions on solutions guaranteeing that such an assertion is valid-at these solutions, the equations are called linearization stable. Although such solutions are exceptional, they still point up the need to exercise caution when such sweeping assumptions are made.


Item Type:Article
Additional Information:© 1975, American Mathematical Society. Partially supported by NSF grants GP-39060 and GP-1573S.
Funders:
Funding AgencyGrant Number
NSFGP-39060
NSFGP-1573S
Classification Code:AMS: Primary 53C25, 5300, 58B20, 83C05; Secondary 35B20, 35J60. 35J70. S8FIO, 83C10, 83C15
Record Number:CaltechAUTHORS:20100806-095929800
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100806-095929800
Official Citation:Proc. Symp. Pure Math., 27, (1975), 219-263
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19317
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:09 Aug 2010 21:09
Last Modified:26 Dec 2012 12:17

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