Necessary optimality conditions for geodesics in weighted Wasserstein spaces

Ambrosio, Luigi; Santambrogio, Filippo

doi:10.4171/rlm/479

Cited by 16 publications

(21 citation statements)

References 10 publications

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“…as in the Euclidean two-phase case (see e.g. [3]) we can use the Comparison Theorem with the convex set D to establish the existence of a minimizer of F 2 (·, E 0 , λ) and also that every minimizer E λ satisfies E λ ⊆ D.…”

Section: Existence Of Gmm For Bounded Partitionsmentioning

confidence: 99%

Minimizing Movements for Mean Curvature Flow of Partitions

Bellettini¹,

Kholmatov²

2018

SIAM J. Math. Anal.

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Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1 n+1 -Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1 2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

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Section: Existence Of Gmm For Bounded Partitionsmentioning

confidence: 99%

Minimizing Movements for Mean Curvature Flow of Partitions

Bellettini¹,

Kholmatov²

2018

SIAM J. Math. Anal.

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“…Notice that if one wants to keep the usual transport interpretation given by a "dynamic cost" to be minimized along the solution of the continuity equation, one can simply introduce the velocity vector fieldṽ t := ρ Therefore, in this model the usual p-energy R d ρ t |ṽ t | p dx of the moving masses ρ t with velocityṽ t results locally modified by a factor f (ρ t ) depending on the local density of the mass occupied at the time t. Different non-local models have been considered in [8,4].…”

Section: 4)mentioning

confidence: 99%

A new class of transport distances between measures

2008

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We introduce a new class of distances between nonnegative Radon measures in R d . They are modeled on the dynamical characterization of the Kantorovich-RubinsteinWasserstein distances proposed by BENAMOU-BRENIER [7] and provide a wide family interpolating between the Wasserstein and the homogeneous WFrom the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established KantorovichRubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.

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“…with g : X → [0, +∞] lower semicontinuous, which have been studied in detail in the papers [3] and [9].…”

Section: Lemmamentioning

confidence: 99%

“…Moreover, in the subsequent paper [3], there can be found some necessary optimality condition for a curve to be a minimizer, in the form of an Euler-Lagrange equation for the functional (1.1).…”

Section: Introductionmentioning

confidence: 98%

Curves of minimal action over metric spaces

Brasco

2009

Annali di Matematica

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Given a metric space $X$, we consider a class of action functionals, generalizing those considered in \cite{BBS} and \cite{AS}, which\ud measure the\ud cost of joining two given points $x_0$ and $x_1$, by means of an absolutely continuous curve. In the case $X$ is given by a space\ud of probability measures, we can think of these action functionals as giving the cost of some congested/concentrated mass transfer problem. We focus\ud on the possibility to split the mass in its {\it moving part} and its part that (in some sense) has\ud already reached its final destination: we consider new action functionals, taking into account only the contribution of the moving part

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Necessary optimality conditions for geodesics in weighted Wasserstein spaces

Cited by 16 publications

References 10 publications

Minimizing Movements for Mean Curvature Flow of Partitions

Minimizing Movements for Mean Curvature Flow of Partitions

A new class of transport distances between measures

Curves of minimal action over metric spaces

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