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Extended Monge-Kantorovich theory. (English) Zbl 1064.49036

Caffarelli, Luis A. (ed.) et al., Optimal transportation and applications. Lectures given at the C.I.M.E. summer school, Martina Franca, Italy, September 2–8, 2001. Berlin: Springer (ISBN 3-540-40192-X/pbk). Lect. Notes Math. 1813, 92-121 (2003).
The paper deals with a formulation of the Monge-Kantorovich mass transportation problem, given in an equivalent way in terms of generalized geodesics. Given two probability measures \(\rho_0\) and \(\rho_1\) on a bounded closed convex \(n\)-dimensional set \(D\), the Monge-Kantorovich cost to transport \(\rho_0\) on \(\rho_1\) is defined by \[ I_k(\rho_0, \rho_1)= \text{inf}\Biggl\{\int_{D\times D} k(x- y) d\mu(x,y): \pi^\#_1 \mu= \rho_0,\, \pi^\#_2 \mu= \rho_1\Biggr\} \] being \(\pi_1\) and \(\pi_2\) the projections on the first and second factors, and being \(k\) a convex function, typically \(k(x)= |x|^p/p\) with \(p> 1\).
In the paper the author considers pairs \((\rho, E)\) of measures on \([0, 1]\times D\) which satisfy the partial differential equation \[ \partial_t\rho+ \text{div\,}E= 0,\quad \rho(0,\cdot)= \rho_0,\quad \rho(1,\cdot)= \rho_1. \] The cost is then defined as \[ K(\rho, E)= \int k\Biggl({dE\over d\rho}\Biggr)d\rho, \] where \(k\) is as above and \(dE/d\rho\) denotes the Radon-Nikodým derivative of \(E\) with respect to \(\rho\). It is shown that the infimum of the functional \(K\) coincides with the quantity \(I_k(\rho_0,\rho_1)\) previously defined.
The paper contains several interesting remarks, comments and open problems about various possible choices of the functional \(K(\rho, E)\) together with some relations to relativistic heat equation, Moser’s lemma, generalized harmonic functions.
In Chapter 4 multiphasic Monge-Kantorovich transportation problems are considered; Chapter 5 deals with generalized extremal surfaces and their relations with electrodynamics; Maxwell equations are considered in Chapter 6.
For the entire collection see [Zbl 1013.00028].

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
35Q99 Partial differential equations of mathematical physics and other areas of application
82C70 Transport processes in time-dependent statistical mechanics
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