Linearization stability of nonlinear partial equations

Fischer, A. E. and J. E. Marsden

Proc. Symp. Pure Math., 27, (1975), 219-263

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) = $\rho$ 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 , $\rho$ 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.