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A free boundary problem with optimal transportation. (English) Zbl 1065.49030

Let \(Q\) be the \(n\)-cube of size \(l\) in \({\mathbb R}^n.\) Given \(x, y \in {\mathbb R}^n\) the torus distance is defined by \(| x - y| = \inf_{e \in {\mathbb Z}^n} | x - y - le| .\) Let \(u, v\) be probability densities in the cube \(Q\) extended periodically to \({\mathbb R}^n.\) The Wasserstein distance \(d_Q(u, v)\) is \[ d_Q(u, v)^2 = \inf_\gamma \;\bigg\{ \;{1 \over 2} \int_{{\mathbb R}^n \times {\mathbb R}^n} | x - y| ^2_Q d\gamma \;\bigg\} \, , \] the infimum taken over all positive Borel measures in \(Q \times Q\) with marginals \(u(x)\) and \(v(y).\) The author gives a characterization of the minimizers \(u\) of the functional \[ d_Q^2(u, u_0)^2 + {1 \over 2} \int_Q | \nabla u| ^2 dx \] \((u_0\) given) in terms of the potential of the transportation map from \(u\) to \(u_0.\) These results imply in particular regularity properties of the free boundary \(\partial\{u > 0\}.\) The author also provides an example where, although the probability distribution \(u_0\) is strictly positive, the set \(\{u = 0\}\) has positive measure.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
35R35 Free boundary problems for PDEs
35K55 Nonlinear parabolic equations
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