438 research outputs found
Structural results on convexity relative to cost functions
Mass transportation problems appear in various areas of mathematics, their
solutions involving cost convex potentials. Fenchel duality also represents an
important concept for a wide variety of optimization problems, both from the
theoretical and the computational viewpoints. We drew a parallel to the
classical theory of convex functions by investigating the cost convexity and
its connections with the usual convexity. We give a generalization of Jensen's
inequality for cost convex functions.Comment: 10 page
Global estimates for solutions to the linearized Monge--Amp\`ere equations
In this paper, we establish global estimates for solutions to the
linearized Monge-Amp\`ere equations under natural assumptions on the domain,
Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant
analogues of the global estimates of Winter for fully nonlinear,
uniformly elliptic equations, and also linearized counterparts of Savin's
global estimates for the Monge-Amp\`ere equations.Comment: v2: presentation improve
Upper bounds for the eigenvalues of Hessian equations
We prove some upper bounds for the Dirichlet eigenvalues of a class of fully
nonlinear elliptic equations, namely the Hessian equationsComment: 15 pages, 1 figur
Boundary definition of a multiverse measure
We propose to regulate the infinities of eternal inflation by relating a late
time cut-off in the bulk to a short distance cut-off on the future boundary.
The light-cone time of an event is defined in terms of the volume of its future
light-cone on the boundary. We seek an intrinsic definition of boundary volumes
that makes no reference to bulk structures. This requires taming the fractal
geometry of the future boundary, and lifting the ambiguity of the conformal
factor. We propose to work in the conformal frame in which the boundary Ricci
scalar is constant. We explore this proposal in the FRW approximation for
bubble universes. Remarkably, we find that the future boundary becomes a round
three-sphere, with smooth metric on all scales. Our cut-off yields the same
relative probabilities as a previous proposal that defined boundary volumes by
projection into the bulk along timelike geodesics. Moreover, it is equivalent
to an ensemble of causal patches defined without reference to bulk geodesics.
It thus yields a holographically motivated and phenomenologically successful
measure for eternal inflation.Comment: 39 pages, 4 figures; v2: minor correction
Local and global behaviour of nonlinear equations with natural growth terms
This paper concerns a study of the pointwise behaviour of positive solutions
to certain quasi-linear elliptic equations with natural growth terms, under
minimal regularity assumptions on the underlying coefficients. Our primary
results consist of optimal pointwise estimates for positive solutions of such
equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto
The Monge problem in Wiener Space
We address the Monge problem in the abstract Wiener space and we give an
existence result provided both marginal measures are absolutely continuous with
respect to the infinite dimensional Gaussian measure {\gamma}
Sobolev Regularity for Monge-Ampere Type Equations
In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly c-convex potentials arising in optimal transportation belong to W2,1+\u3baloc for some \u3ba>0. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge-Amp\`ere equation with right hand side bounded away from zero and infinity
The Momentum Constraints of General Relativity and Spatial Conformal Isometries
Transverse-tracefree (TT-) tensors on , with an
asymptotically flat metric of fast decay at infinity, are studied. When the
source tensor from which these TT tensors are constructed has fast fall-off at
infinity, TT tensors allow a multipole-type expansion. When has no
conformal Killing vectors (CKV's) it is proven that any finite but otherwise
arbitrary set of moments can be realized by a suitable TT tensor. When CKV's
exist there are obstructions -- certain (combinations of) moments have to
vanish -- which we study.Comment: 16 page
Monge's transport problem in the Heisenberg group
We prove the existence of solutions to Monge transport problem between two
compactly supported Borel probability measures in the Heisenberg group equipped
with its Carnot-Caratheodory distance assuming that the initial measure is
absolutely continuous with respect to the Haar measure of the group
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