A Convexity Principle for Interacting Gases

A new set of inequalities is introduced, based on a novel but natural interpolation between Borel probability measures on R d . Using these estimates in lieu of convexity or rearrangement inequalities, the existence and uniqueness problems are solved for a family of attracting gas models. In these models, the gas interacts with itself through a force which increases with distance and is governed by an equation of state P=P(*) relating pressure to density. P(*) * (d&1)d is assumed non-decreasing for a d-dimensional gas. By showing that the internal and potential energies for the system are convex functions of the interpolation parameter, an energy minimizing stateunique up to translationis proven to exist. The concavity established for &\t & & pd q as a function of t # [0, 1] generalizes the BrunnMinkowski inequality from sets to measures. 1997 Academic Press

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