Archive for Rational Mechanics and Analysis

, Volume 196, Issue 1, pp 143–189 | Cite as

Minimum Principle for Indefinite Mean-Field Free Energies

  • Eugene C. GartlandJr.
  • Epifanio G. Virga


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 \({\fancyscript{F}_0}\) indefinite. To justify this extension, so far largely based on heuristic arguments, we prove here a minimum principle that also applies when \({\fancyscript{F}_0}\) 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 \({\fancyscript{F}_0}\) in the order tensor associated with the repulsive component of the molecular interaction.


Free Energy Helmholtz Free Energy Minimum Principle Order Tensor Minimax Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKent State UniversityKentUSA
  2. 2.Dipartimento di Matematica and CNISMUniversità di PaviaPaviaItaly

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