4,897 research outputs found
Optimal Transportation Theory with Repulsive Costs
This paper intents to present the state of art and recent developments of the
optimal transportation theory with many marginals for a class of repulsive cost
functions. We introduce some aspects of the Density Functional Theory (DFT)
from a mathematical point of view, and revisit the theory of optimal transport
from its perspective. Moreover, in the last three sections, we describe some
recent and new theoretical and numerical results obtained for the Coulomb cost,
the repulsive harmonic cost and the determinant cost.Comment: Survey for the special volume for RICAM (Special Semester on New
Trends in Calculus of Variations
On the duality between p-Modulus and probability measures
Motivated by recent developments on calculus in metric measure spaces
, we prove a general duality principle between
Fuglede's notion of -modulus for families of finite Borel measures in
and probability measures with barycenter in , with dual exponent of . We apply this general duality
principle to study null sets for families of parametric and non-parametric
curves in . In the final part of the paper we provide a new proof,
independent of optimal transportation, of the equivalence of notions of weak
upper gradient based on -Modulus (Koskela-MacManus '98, Shanmugalingam '00)
and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare
'11)Comment: Minor corrections, typos fixe
Global Lipschitz extension preserving local constants
The intent of this short note is to extend real valued Lipschitz functions on metric spaces, while locally preserving the asymptotic Lipschitz constant. We then apply this results to give a simple and direct proof of the fact that Sobolev spaces on metric measure spaces defined with a relaxation approach à la Cheeger are invariant under isomorphism class of mm-structures
Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope
IIn this paper we make a survey of some recent developments of the theory of Sobolev spaces W-1,W-q (X, d, m), 1 < q < infinity, in metric measure spaces (X, d, m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Gamma-convergence; this result extends Cheeger's work because no Poincare inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of m. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems
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