Abstract
A new set of inequalities is introduced, based on a novel but natural interpolation between Borel probability measures onRd. 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 stateP=P(ϱ) relating pressure to density.P(ϱ)/ϱ>(d−1)/dis assumed non-decreasing for ad-dimensional gas. By showing that the internal and potential energies for the system are convex functions of the interpolation parameter, an energy minimizing state—unique up to translation—is proven to exist. The concavity established for ¶ρt¶−p/dqas a function oft∈[0,1] generalizes the Brunn–Minkowski inequality from sets to measures.
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