Free boundaries in optimal transport and Monge-Ampère obstacle problems. (English) Zbl 1196.35231

Caffarelli, Luis A.; McCann, Robert J.

Ann. Math. (2) 171, No. 2, 673-730 (2010).

Let the functions \(0 \leq f,g \in L^{1}({\mathbb R}_n)\) be compactly supported. The authors investigate the problem of transporting a fraction \(m \leq \min \{\|f\|_{L_1}, \|g\|_{L_1}\}\) of the mass of \(f\) onto \(g\) as cheaply as possible, where cost per unit mass transported is given by a cost function \(c\) which is quadratic, \(c\,({\mathbf x}, {\mathbf y}) = |{\mathbf x} - {\mathbf y}|^{2}/2\). It can be shown that this problem is equivalent to a double obstacle problem for the Monge-Ampère equation for which sufficient conditions are given to guarantee uniqueness of the solution, such as \(f\) vanishing on \(\text{supp} \, g\) in the quadratic sense. The Lagrange multiplier \(\lambda\) controlling the optimal mass parametrizes the distance between the upper and lower obstacles.
In a further section, the authors show the monotone dependence of the active domains \(U\) and \(V\) on the amount \(m\) of mass transferred. For the quadratic cost function, the semiconcavity of the free boundary is proved when \(f\) and \(g\) are compactly supported on opposite sides of a hyperplane. In the last two sections, interior and boundary regularity for the optimal mapping are shown under certain assumptions on \(f\) and \(g\). An appendix included makes the boundary regularity analysis self-contained. The bibliography contains 83 items.

Reviewer: Jürgen Socolowsky (Brandenburg an der Havel)

MSC:

35R35	Free boundary problems for PDEs
35B65	Smoothness and regularity of solutions to PDEs
35J87	Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
35J96	Monge-Ampère equations
49J20	Existence theories for optimal control problems involving partial differential equations

Keywords:

free boundaries; Monge-Ampère equation; obstacle problems; transport problem; regularity

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