Deformations of non-linear partial differential equations

In this article we examine in what sense the linearization of a system of nonlinear partial differential equations approximates the full nonlinear system. These ideas are applied to study the deformations of the scalar curvature equation and Einstein's equations of general relativity, as well as the set of metrics wirth prescribed scalar curvature. We show that these systems are linearization stable under general hypotheses; in the exceptional cases of instability , we study the isolation of solutions