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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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