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On optimal matching measures for matching problems related to the Euclidean distance

José Manuel Mazón, Julio Daniel Rossi, Julián Toledo (2014)

Mathematica Bohemica

We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a p -Laplacian system. We prove...

On Perelman’s functional with curvature corrections

Rami Ahmad El-Nabulsi (2012)

Annales UMCS, Mathematica

In recent ten years, there has been much concentration and increased research activities on Hamilton’s Ricci flow evolving on a Riemannian metric and Perelman’s functional. In this paper, we extend Perelman’s functional approach to include logarithmic curvature corrections induced by quantum effects. Many interesting consequences are revealed.

On some properties of three-dimensional minimal sets in ℝ 4

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in ℝ 4 around a 𝕐 -point and the existence of a point of particular type of a Mumford-Shah minimal set in ℝ 4 , which is very close to a 𝕋 . This will give a local description of minimal sets of dimension 3 in ℝ 4 around a singular point and a property of Mumford-Shah minimal sets in ℝ 4 .

On the Curvature and Heat Flow on Hamiltonian Systems

Shin-ichi Ohta (2014)

Analysis and Geometry in Metric Spaces

We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation.We prove appropriate generalizations of the Bochner-Weitzenböck formula and Laplacian comparison theorem, and study the heat flow.

On the hessian of the optimal transport potential

Stefán Ingi Valdimarsson (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse...

On the Representation of Effective Energy Densities

Christopher J. Larsen (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the question raised in [1] of whether relaxed energy densities involving both bulk and surface energies can be written as a sum of two functions, one depending on the net gradient of admissible functions, and the other on net singular part. We show that, in general, they cannot. In particular, if the bulk density is quasiconvex but not convex, there exists a convex and homogeneous of degree 1 function of the jump such that there is no such representation.

On the stabilization problem for nonholonomic distributions

Ludovic Rifford, Emmanuel Trélat (2009)

Journal of the European Mathematical Society

Let M be a smooth connected complete manifold of dimension n , and Δ be a smooth nonholonomic distribution of rank m ≤ n on M . We prove that if there exists a smooth Riemannian metric on1for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of Δ on M . Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of...

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