Differential equations methods for the Monge-Kantorovich mass transfer problem.

*(English)*Zbl 0920.49004
Mem. Am. Math. Soc. 653, 66 p. (1999).

The Monge-Kantorovich problem is a mathematical model for moving a pile of soil to another location (with prescribed position) in a way that minimizes the work. The mathematical model starts with two nonnegative \(L^1(\mathbb R^n)\) functions \(f^+\) and \(f^-\) (which represent the initial and final soil distribution densities, respectively) such that
\[
\int f^+ dx = \int f^- dx
\]
and asks for a smooth function \(\mathbf s\) such that
\[
f^+(x)=f^-( \mathbf s(x))\det D\mathbf s(x) \tag{*}
\]
for all \(x\in \mathbb R^n\). In addition, \(\mathbf s\) should minimize the functional \(I\) defined by
\[
I[\mathbf r] = \int | x-\mathbf r(x)| f^+ dx
\]
over the set of all functions satisfying (*). According to Monge, if there is a solution \(\mathbf s\), then there is a potential function \(u\) such that
\[
\frac {\mathbf s(x)-x}{| \mathbf s(x)-x} =-Du(x)
\]
for all \(x\) in the support of \(f^+\). There are two basic ideas in this monograph. The first is that the potential \(u\) is the limit as \(p\to\infty\) of a suitable set of solutions \(u_p\) of the \(p\)-Laplacian equation \(- \text{div} (| Du_p| ^pDu_p) = f\), where \(f= f^+ - f^-\). The second idea is that the function \(\mathbf s\) can be identified with the solution of an initial value problem for an ordinary differential equation involving the weak* limit \(A\) of \(A_p=| Du_p| ^pDu_p\). The reason for such a lengthy work is that this simple description of the method assumes a regularity for \(u\) and \(\mathbf s\) which is not generally true. The introduction gives a brief description of historical approaches to this problem and some related work. The actual proof comprises the remaining nine sections of this book. Despite the wealth of technical details in this undertaking, the authors do an excellent job of keeping everything in order with their exposition.

Reviewer: G.M.Lieberman (Ames)