A Remark on the Anisotropic Outer Minkowski content

We study an anisotropic version of the outer Minkowski content of a closed set in Rn. In particular, we show that it exists on the same class of sets for which the classical outer Minkowski content coincides with the Hausdorff measure, and we give its explicit form.Comment: We corrected an error in the orignal manuscript, on p. 14 (the boundaries of the regularized sets are not necessarily C^{1,1}

Gradient flows for non-smooth interaction potentials

We deal with a nonlocal interaction equation describing the evolution of a particle density under the effect of a general symmetric pairwise interaction potential, not necessarily in convolution form. We describe the case of a convex (or \lambda-convex) potential, possibly not smooth at several points, generalizing the results of [CDFLS]. We also identify the cases in which the dynamic is still governed by the continuity equation with well-characterized nonlocal velocity field. Reference: [CDFLS] J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, D. Slepcev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J. 156 (2011), 229--271.Comment: 35 pages, 1 figur

Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: equivalence and Gamma-convergence

This paper is devoted to the study of multi-agent deterministic optimal control problems. We initially provide a thorough analysis of the Lagrangian, Eulerian and Kantorovich formulations of the problems, as well as of their relaxations. Then we exhibit some equivalence results among the various representations and compare the respective value functions. To do it, we combine techniques and ideas from optimal transportation, control theory, Young measures and evolution equations in Banach spaces. We further exploit the connections among Lagrangian and Eulerian descriptions to derive consistency results as the number of particles/agents tends to infinity. To that purpose we prove an empirical version of the Superposition Principle and obtain suitable Gamma-convergence results for the controlled systems

Mean-field optimal control as Gamma-limit of finite agent controls

This paper focuses on the role of a government of a large population of interacting agents as a mean field optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a PDE of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a $\Gamma$-convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle

Nonlinear mobility continuity equations and generalized displacement convexity

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a non-rigorous argument indicating that they are not displacement semiconvex.Comment: 33 pages, 1 figur

Cahn-Hilliard and Thin Film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn-Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution - non-negativity, conservation of the total mass and dissipation of the energy - are automatically guaranteed by the construction from minimizing movements in the energy landscape