For general non-linear elliptic PDEs, e.g. non-linear rubber
elasticity, linear stability analysis is false. This is because of the
possibility of point-instabilities. A point-instability is a non-linear
instability with zero amplitude threshold that occurs while linear
stability still holds. Examples include cavitation, fracture, and the formation
of a crease, a self-contacting fold in an otherwise free surface.
Each of which represents a kind of topological change. For any such