Minimum Principle for Indefinite Mean-Field Free Energies

The extension of Bogoliubov Jr.’s minimax principle to biaxial nematic liquid crystals has recently made it possible to derive the universal mean-field phase diagram for all quadrupolar interactions between nematogenic biaxial molecules, including those rendering the mean-field free energy F indefinite. To justify this extension, so far largely based on heuristic arguments, we prove here a minimum principle that also applies when F is indefinite. Bogoliubov Jr.’s minimax principle then emerges as a characterization of this minimum principle. The theory presented here is more general than its application to biaxial nematic liquid crystals could reveal. It essentially builds upon a concavity property of F in the order tensor associated with the repulsive component of the molecular interaction.