Related papers: Remarks on the Green's function of the linearized …

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…

Let $1\leq m\leq n$ be two fixed integers. Let $\Omega \Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset \Omega \times ]0,+ \infty[$ a finite set of weighted poles. We define and study properties of the…

We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfying the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a…

We introduce a monotone (degenerate elliptic) discretization of the Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is consistent provided the solution hessian condition number is uniformly bounded. Our approach…

We study the solvability of the second boundary value problem for a class of highly singular fourth order equations of Monge-Amp\`ere type. They arise in the approximation of convex functionals subject to a convexity constraint using Abreu…

It is shown that the Green's function on a finite lattice in arbitrary space dimension can be obtained from that of an infinite lattice by means of translation operator. Explicit examples are given for one- and two-dimensional lattices.

We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Amp\`ere equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating…

This paper introduces a fast and robust iterative scheme for the elliptic Monge-Amp\`ere equation with Dirichlet boundary conditions. The Monge-Amp\`ere equation is a nonlinear and degenerate equation, with applications in optimal…

We prove stability of solutions of the complex Monge-Amp\`ere equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be H\"older…

We consider the complex Monge-Amp\`ere equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on compact Hermitian manifolds.

We prove a relative $L^\infty$ estimate for a class of complex Monge-Amp\`ere type equations on K\"ahler manifolds. It provides a unified approach to Tundinger type estimate and uniform estimate. It also improves the previous results about…

We construct an explicit Green's function for the conjugated Laplacian $e^{-\omega \cdot x/h}\Delta e^{-\omega \cdot x/h}$, which let us control our solutions on roughly half of the boundary. We apply the Green's function to solve a partial…

We investigate the Monge-Amp\`ere equation subject to zero boundary value and with a positive right-hand side unction assumed to be continuous or essentially bounded. Interior estimates of the solution's first and second derivatives are…

We study uniqueness of $p$-harmonic Green functions in domains $\Omega$ in a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality, with $1<p<\infty$. For bounded domains in unweighted $\mathbf{R}^n$,…

We consider mesh functions which are discrete convex in the sense that their central second order directional derivatives are positive. Analogous to the case of a uniformly bounded sequence of convex functions, we prove that the uniform…

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…

In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the…

We prove the long-time existence and convergence of solutions to a general class of parabolic equations, not necessarily concave in the Hessian of the unknown function, on a compact Hermitian manifold. The limiting function is identified as…

We consider mixed Monge-Amp\`ere products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one parameter limits of mixed Monge-Amp\`ere products of smooth functions,…

We calculate the second order derivatives of the Ronkin function in the case of an affine linear polynomial in three variables and give an expression of them in terms of complete elliptic integrals and hypergeometric functions. This gives a…