44 research outputs found
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 -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