  In this paper we explain how to set up what is in fact the only possible
consistent construction scheme for a Landau theory of high pressure phase
transitions that systematically allows to take into account elastic
nonlinearities. We also show how to incorporate available information on the
pressure dependence of elastic constants taken from experiment or simulation.
We apply our new theory to the example of the high pressure cubic-tetragonal
phase transition in Strontium Titanate, a model perovskite that has played a
central role in the development of the theory of structural phase transitions.
Armed with pressure dependent elastic constants calculated by density
functional theory, we give a both qualitatively as well as quantitatively
satisfying description of recent high precision experimental data. Our
nonlinear theory also allows to predict a number of additional elastic
transition anomalies that are accessible to experiment.
