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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 -Laplacian system. We prove...
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.
We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in around a -point and the existence of a point of particular type of a Mumford-Shah minimal set in , which is very close to a . This will give a local description of minimal sets of dimension 3 in around a singular point and a property of Mumford-Shah minimal sets in .
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.
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...
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.
Let be a smooth connected complete manifold of dimension , and be a smooth nonholonomic distribution of rank on . 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 on . 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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