739 research outputs found
Continuity, curvature, and the general covariance of optimal transportation
Let M and \bar M be n-dimensional manifolds equipped with suitable Borel
probability measures \rho and \bar\rho. Ma, Trudinger & Wang gave sufficient
conditions on a transportation cost c \in C^4(M \times \bar M) to guarantee
smoothness of the optimal map pushing \rho forward to \bar\rho; the necessity
of these conditions was deduced by Loeper. The present manuscript shows the
form of these conditions to be largely dictated by the covariance of the
question; it expresses them via non-negativity of the sectional curvature of
certain null-planes in a novel but natural pseudo-Riemannian geometry which the
cost c induces on the product space M \times \bar M.
H\"older continuity of optimal maps was established for rougher mass
distributions by Loeper, still relying on a key result of Trudinger & Wang
which required certain structure on the domains and the cost. We go on to
develop this theory for mass distributions on differentiable manifolds --
recovering Loeper's Riemannian examples such as the round sphere as particular
cases -- give a direct proof of the key result mentioned above, and revise
Loeper's H\"older continuity argument to make it logically independent of all
earlier works, while extending it to less restricted geometries and cost
functions even for subdomains M and \bar M of R^n. We also give new examples of
geometries satisfying the hypotheses -- obtained using submersions and tensor
products -- and some connections to spacelike Lagrangian submanifolds in
symplectic geometry.Comment: 43 pages, 1 figur
A glimpse into the differential topology and geometry of optimal transport
This note exposes the differential topology and geometry underlying some of
the basic phenomena of optimal transportation. It surveys basic questions
concerning Monge maps and Kantorovich measures: existence and regularity of the
former, uniqueness of the latter, and estimates for the dimension of its
support, as well as the associated linear programming duality. It shows the
answers to these questions concern the differential geometry and topology of
the chosen transportation cost. It also establishes new connections --- some
heuristic and others rigorous --- based on the properties of the
cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page
On the second boundary value problem for Monge-Ampere type equations and geometric optics
In this paper, we prove the existence of classical solutions to second
boundary value prob- lems for generated prescribed Jacobian equations, as
recently developed by the second author, thereby obtaining extensions of
classical solvability of optimal transportation problems to problems arising in
near field geometric optics. Our results depend in particular on a priori
second derivative estimates recently established by the authors under weak
co-dimension one convexity hypotheses on the associated matrix functions with
respect to the gradient variables, (A3w). We also avoid domain deformations by
using the convexity theory of generating functions to construct unique initial
solutions for our homotopy family, thereby enabling application of the degree
theory for nonlinear oblique boundary value problems.Comment: Final version to appear in Archive for Rational Mechanics and
Analysi
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