Related papers: An analytical solution of Monge differential equat…
It is shown that the Monge equation is equivalent to the ordinary differential equation $\ddot X=0$ of free motion. Equations of Monge type (with their general solutions) are connected with each ordinary differential equation of second…
Solution of Monge equation of arbitrary degree (non linear differential equation n-orden) is connected with solution of functional equation for 4 functions with 4 different arguments. Some number solutions of this equation is represented in…
A numerical method for the solution of the elliptic Monge-Ampere Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT) problem is presented. A local representation of the OT boundary…
We study second-order PDEs in 4D for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge-Ampere…
Monge-Amp\`ere equation $\det(D^2u)=f$ in two dimensional spaces is different in nature from their counterparts in higher dimensional spaces. In this article we employ new ideas to establish two main results for the Monge-Amp\`ere equation…
We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent…
In this paper, we prove the regularity of the free boundary in the Monge-Amp\`ere obstacle problem $\det D^2 v= f(y)\chi_{\{v>0\}}. $ By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of the…
We discuss the solution of the Mott transition problem in a fully frustrated lattice with a semicircular density of states in the limit of infinite dimensions from the point of view of a Landau free energy functional. This approach provides…
Consider the multidimensional SDE $\mathrm d X(t) = a(X(t))\mathrm d t + b(X(t))\mathrm d W(t).$ We study the asymptotic behavior of its solution $X(t)$ as $t \to \infty$, namely, we study sufficient conditions of transience of its solution…
The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image…
We consider the Monge-Amp\`ere equation $\det(D^2u)=f$ where $f$ is a positive function in $\mathbb R^n$ and $f=1+O(|x|^{-\beta})$ for some $\beta>2$ at infinity. If the equation is globally defined on $\mathbb R^n$ we classify the…
We study a fully nonlinear equation of complex Monge-Ampere type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.
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…
Given a function f(x, t), its fourth (symmetric) differential is a quartic form in dx, dt. It is well-known that any quartic form in two variables can be represented as a sum of three 4th powers of linear forms. The particular case of two…
In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the…
We introduce the so-called $d$-concavity, $d \geq 0,$ and prove that the nonsymmetric Monge-Amp\`{e}re type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for…
In this paper we solve the Monge problem on infinite dimensional Hilbert space endowed with a suitable Gaussian measure, that satisfies the Lebesgue differentiation theorem.
In this paper, we consider the following nonlinear eigenvalue problem for the Monge-Amp\'ere equation: find a non-negative weakly convex classical solution $f$ satisfying {equation*} {cases} \det D^2 f=f^p \quad &\text{in $\Omega$} f=\vp…
This paper solves the two-dimensional Dirichlet problem for the Monge-Amp\`ere equation by a strong meshless collocation technique that uses a polynomial trial space and collocation in the domain and on the boundary. Convergence rates may…
We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge-Amp\`ere equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb C^2$. Among these, the entire solutions defined on…