1. Introduction
This paper introduces some PDE and ODE methods for constructively solving one
version of the Monge-Kantorovich mass transfer problem.
The basic issue is this. Given two nonnegative, summable functions / ± on E n satisfying
the mass balance condition
/ f+dx= f f-dy, (1.1)
we consider the corresponding measures ^ + = f+dx, f.t~ = f~dy, and ask how we can
optimally rearrange /x+ onto \x~. If r : Rn —» Rn is a smooth, orientation-preserving, one-to-
one mapping, the requirement is that r transfer fi+ onto /i"; that is,
/+(x) = /"(r(x)) detDr{x) (x E n ). (1.2)
Denote by A the admissible class of smooth, one-to-one functions r satisfying (1.2). We then
seek a mass transfer plan s £ A which is optimal in the sense that
/[s] = min/[r], (1.3)
reA
where
I[r]= f \x-r{x)\f+(x) dx= [ \x - r(x)|d/x+ . (1.4)
This is a form of Monge's problem of the "deblais" and "remblais" (cf. Monge [M],
Dupin [D], Appell [A]), dating from the early 1780's. The physical interpretation is that
we are given a pile of soil or rubble (the "deblais"), with mass density / + , which we wish
to transport to an excavation or fill (the "remblais"), with mass density / " . For a given
transport scheme r, condition (1.2) is conservation of mass. Furthermore, as each particle
of soil moves a distance \x r(x)\, we can interpret I[r] as the total work involved. We
consequently are looking for a way to rearrange /i + = f+dx onto \T = f~dy, which requires
the least work.
This optimization problem, and its many, many variants and extensions (entailing for
example more general measures on more general spaces, different cost functionals, etc.)
has been intensively studied for over two hundred years. We review some of the principal
discoveries.
a. Monge. Monge himself contributed the essential insight that an optimal transfer plan
s should be in part determined by a potential u. More precisely, he deduced by heutistic,
°L.C.E. is supported in part by NSF Grant DMS-94-24342, and W.G. is supported in part by NSF
Grant DMS-96-22734.
Received by the editor May 1, 1996; and in revised form February 19, 1997.
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