Related papers: Monge Problem on infinite dimensional Hilbert spac…
The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In…
In this paper we continue the analysis of the two-scale method for the Monge-Amp\`ere equation for dimension $d \geq 2$ introduced in [10]. We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete…
We prove the convergence of meshfree method for solving the elliptic Monge-Ampere equation with Dirichlet boundary on the bounded domain. L2 error is obtained based on the kernel-based trial spaces generated by the compactly supported…
We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp…
In this paper, Monge-Kantorovich problem is considered in the infinite dimension on an abstract Wiener space $(W, H,\mu)$, where $H$ is Cameron-Martin space and $\mu$ is the Gaussian measure. We study the regularity of optimal transport…
We consider the simultaneous optimal transportation of measures, where the target marginal is not necessarily fixed. For this problem, we prove the existence of a solution for completely regular spaces and investigate the structure of the…
A generalization of the Lebesgue number lemma is obtained. It is proved that, if each countably infinite locally finite open cover of a chainable metric space $X$ has a Lebesgue number, then $X$ is totally bounded. A property of metric…
An examples of Monge-Ampere equations connected with six-dimensional generalization of the Plebanski four-dimensional space are considered. Their particular solutions are constructed.
In this short note, we prove the existence of solutions to a Monge-Amp\`ere equation of entire type derived by a weighted version of the classical Minkowski problem.
We show the existence of a strictly convex function $u: B_1 \to \mathbb{R}$ with associated Monge-Amp\`ere measure represented by a function $f$ with $0 < f < 1$ a.e. whose Hessian has a singular part. This extends the work [13] and answers…
We study the problem of the existence and the holomorphicity of the Monge-Amp\`ere foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Amp\`ere equation even at points of arbitrary degeneracy. We obtain…
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian…
We discuss eternal inflation in context of classical probability spaces defined by a triplet: sample space, $\sigma$-algebra and probability measure. We show that the measure problem is caused by the countable additivity axiom applied to…
We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function…
In this paper we study Monge solutions to stationary Hamilton-Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous…
We investigate the problem of balanced embedding of a non-compact complex manifold into an infinite-dimensional projective space. In this paper we prove the existence of such an embedding in a model case. The strategy is by using a gradient…
Let u be a function of n independent variables x^1, ..., x^n, and U=(u_{ij}) the Hessian matrix of u. The symplectic Monge-Ampere equation is defined as a linear relation among all possible minors of U. Particular examples include the…
We propose a two-scale finite element method for the Monge-Amp\`ere equation with Dirichlet boundary condition in dimension $d\ge2$ and prove that it converges to the viscosity solution uniformly. The method is inspired by a finite…
The current work focuses on the Gaussian-Minkowski problem in dimension 2. In particular, we show that if the Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a…
We rephrase Monge's optimal transportation (OT) problem with quadratic cost--via a Monge-Amp\`ere equation--as an infinite-dimensional optimization problem, which is in fact a convex problem when the target is a log-concave measure with…