Monge-Kantorovich equation for degenerate Finsler metrics
In this paper, the author studies how a generalization of the Monge-Kantorovich equation gives rise to a necessary and sufficient optimality condition for the Kantorovich dual problem and minimal flow problem associated with a very degenerate Finsler metric without any assumption on coerciveness. The Monge-Kantorovich problem, in a domain \(\Omega\), consists of minimizing, given a transport cost \(c\), and two measures \(\mu_0\), \(\mu_1\), the quantity \[ \inf\bigg\{ \int_{\Omega\times \Omega}c(x,y)d\gamma(x,y): \gamma \text{ has marginals } \mu_0 \text{ and } \mu_1 \bigg\}. \] The Monge-Kantorovich equation, first studied by Evans and Gangbo, and then generalized by Bouchitté and Buttazzo to allow for more general data, is \[ \begin{cases} -\nabla \cdot (\mu D_\mu u) = f_1-f_0,& \text{in } \mathcal{D}'(\bar\Omega),\\ |D_\mu u|=1, & \mu\text{-a.e.},\\ |\nabla u|\le 1, &\text{a.e.}, \end{cases} \] The Finsler metric associated to a given function \(F\) is defined as \[ d_F(x,y):=\inf_{\xi \text{ Lipschitz}} \bigg\{ \int_0^1 F(\xi(s),\dot \xi(s))\mathrm{d}s: \xi(0)=x,\ \xi(1)=y \bigg\}, \] i.e. \(d_F\) is the ``minimal action'' distance between \(x,y\). By relying on a duality argument, and a smooth approximation result, the author shows the following equivalency result: \begin{itemize} \item[(1)] First, given \(u\) such that \(F^*(x,\nabla u(x))\le 1\), where \(F^*\) denotes the polar function of \(F\), and \(\Phi \) satisfying \(-\nabla \cdot \Phi = f_1-f_0\) in \(\mathcal{D}'(\bar\Omega)\), it holds \[ \frac{\Phi}{|\Phi|}(x) D_{|\Phi|} u(x) \le F\Big( x, \frac{\Phi}{|\Phi|}(x)\Big) \qquad |\Phi|\text{-a.e.} \] \item[(2)] Second, if \(u,\Phi\) are optimal in the Kantorovich dual problem \[ \max_{u\in W^{1,\infty}(\Omega)} \bigg\{ \int_{\bar\Omega} u\mathrm{d}(f_1-f_0) : F^*(x,\nabla u(x))\le 1 \text{ a.e.} \bigg\} \] and the minimal flow problem \[ \min_{ \Phi } \bigg\{ \int_{\bar\Omega} F\Big( x, \frac{\Phi}{|\Phi|}(x)\Big) \mathrm{d}|\Phi|(x): -\nabla \cdot \Phi = f_1-f_0 \text{ in } \mathcal{D}'(\bar\Omega)\bigg\}, \] then \(u,\Phi\) also solves the generalized Monge-Kantorovich equation for degenerate Finsler metric \[ \begin{cases} -\nabla \cdot \Phi = f_1-f_0,& \text{in } \mathcal{D}'(\bar\Omega),\\ \frac{\Phi}{|\Phi|}(x) \cdot D_{|\Phi|} u(x)=F\Big( x, \frac{\Phi}{|\Phi|}(x)\Big) , & |\Phi|\text{-a.e.},\\ F^*(x,\nabla u(x))\le 1, &\text{a.e.} \end{cases} \] \end{itemize}
scientific article; zbMATH DE number 7317430