A new transportation distance with bulk/interface interactions and flux penalization (Q829425)

In this paper the author studies a new optimal transport problem on a bounded domain \(\Omega\subseteq\mathbb{R}^d\) defined via a dynamical Benamou-Brenier formulation, which penalizes the transfer of mass between interior and boundary. 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 classical Benamou-Brenier formula allows to rewrite the static problem as a dynamical fluid-mechanics problem, namely the minimization of the kinetic energy \[ \mathcal{W}_\Omega^2 (\rho_0,\rho_1) := \min_{\rho, w} \bigg\{ \frac{1}{2} \int_{[0,1]\times \bar{\Omega}} \rho |w|^2: \rho_t + \nabla\cdot(\rho w)=0 \bigg\} \] subject to the no flux condition \(\rho w\cdot n=0\) on \(\partial\Omega\) and initial/terminal data \(\rho_0,\rho_1\). This paper introduces a new model, namely minimizing \[ \mathcal{W}_\kappa^2 (\rho_0,\rho_1) := \min_{\rho, w} \bigg\{ \int_{[0,1]\times\bar{\Omega}} \frac{| F|^2}{2\omega} + \int_{[0,1]\times \bar{\Omega}} \frac{| G|^2}{2\gamma} + \kappa^2 \int_{[0,1]\times \bar{\Omega}} \frac{| f|^2}{2\gamma} \bigg\} \] subject to \[ \begin{cases} \omega_t +\nabla\cdot F=0 & \text{in } \Omega,\\ F\cdot n =f & \text{on }\partial\Omega,\\ \gamma_t + \nabla\cdot G=f & \text{on }\partial\Omega, \end{cases} \] where the endpoints \(\rho_i = (\omega_i,\gamma_i)\), \(i=0,1\), are prescribed, and \(\rho_i = \omega_i+\gamma_i\), \(i=0,1\). Intuitively, such an energy penalizes three types of mass transfers: within \(\Omega\), within \(\partial\Omega\), and between \(\Omega\) and its boundary \(\partial\Omega\). One can think \(\Omega\) as a city, with its network of local roads and related transport cost, and \(\partial\Omega\) as a ring highway around the city with its own (relatively cheaper) transport cost. Such a model preserves the total mass \(\rho_i\), but not the individual \(\omega_i\), \(\gamma_i\). Furthermore, the decomposition \(\rho_i\) into \(\omega_i+\gamma_i\) is far from unique, adding an additional layer of difficulty. A crucial issue is being able to differentiate between the interior and its boundary, since these play such a different role in the energy. The author overcomes this issue by constructing a suitable distance function \(d_\kappa\), and then reformulates this minimization problem as a Hamilton-Jacobi system. The main contributions consist in proving qualitative properties of minimizers, computing the distance between two Dirac masses (one in the interior, one on the boundary), and comparing the model with different distances.