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A convexity principle for interacting gases. (English) Zbl 0901.49012
The author studies the minimization problem for an energy functional $$E(\rho)$$ which models a gas of interacting particles in $$\mathbb{R}^d$$ described by normalized density functions $$\rho\in L_1(\mathbb{R}^d)$$, $$\int\rho dx=1$$. The energy $$E= U+{1\over 2}G$$ consists of the internal energy $$U$$ due to compression and a potential energy $$G$$ due to the interaction where $$U(\rho)= \int A(\rho(x))dx$$ and $$G(\rho)= \iint d\rho(x)V(x- y)\rho(y)dy$$. The internal energy density $$A$$ is derived from an equation of state $$P= P(\rho)$$ relating pressure to density by means of $$A(\rho)= \int^\infty_1 P(\rho/v)dv$$; the following assumptions being made: $$P(\rho)\rho^{1-1/d}$$ is non-decreasing and $$P(\rho)/\rho^2$$ is not integrable at $$\rho=\infty$$; in particular, $$A$$ is convex. The potential $$V$$ which may be assumed to be even must be convex as well, i.e., the force between particles increases with distance. The main result of the paper (for strictly convex $$V$$) is the existence and uniqueness up to translations of a minimizer, provided that $$\inf E<+\infty$$. The key point of the paper is the introduction of a “displacement interpolation” $$\rho_t$$, $$0\leq t\leq 1$$, between two arbitrary densities $$\rho$$, $$\rho'$$: previous work of the author yields the existence of a convex function $$\psi$$ on $$\mathbb{R}^d$$ such that, viewing $$\rho$$ and $$\rho'$$ as measures, $$\rho'$$ is the pushforward of $$\rho$$ under $$\nabla\psi$$, $$\rho'= (\nabla\psi)_\sharp\rho$$. The $$\rho_t$$ is then defined as $$\rho_t= [(1- t)id+ t\nabla\psi]_\sharp\rho$$. The author studies various properties of this interpolation and establishes the convexity of $$U(\rho_t)$$ and $$G(\rho_t)$$ as functions of $$t$$, which is basic for the above existence and uniqueness theorem. As the author observes, the convexity of $$U(\rho_t)$$ with $$A(\rho)=- \rho^{(d-1)/d}$$ contains the classical Brunn-Minkowski inequality as a special case. The reviewer wishes to remark that this paper is extraordinarily well written; the presentation is very transparent and each step is very well motivated.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 76N15 Gas dynamics, general 26B25 Convexity of real functions of several variables, generalizations
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##### References:
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