Related papers: The Monge-Ampere system: convex integration with i…
This article is about the convex solution $u$ of the Monge--Amp\`ere equation on an at least 2-dimensional open bounded convex domain with Dirichlet boundary data and nonnegative bounded right-hand side. For convex functions with zero…
In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Amp\`{e}re equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate…
This paper proposes a regularization of the Monge-Amp\`ere equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is…
We prove a regularity result for Monge-Amp\`ere equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of $\beta$-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical…
We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data. Our results are affine invariant analogues…
The general solution to the Complex Monge-Amp\`ere equation in a two dimensional space is constructed.
Regularity properties of intrinsic objects for a large class of Stein Manifolds, namely of Monge-Amp\`ere exhaustions and Kobayashi distance, is interpreted in terms of modular data. The results lead to a construction of an infinite…
This is a continuation of our earlier work [14] on the Monge-Amp\`ere obstacle problem \[ \det D^2 v = v^q \chi_{\{v>0\}}, \quad v \geq 0 \text{ convex} \] with $q \in [0,n)$, where we studied the regularity of the strictly convex part of…
The purpose of this paper is to study convergence of Monge-Ampere measures associated to sequences of plurisubharmonic functions defined on a hyperconvex subset of ${\mathbb C^n}$.
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 review some recent results on existence and regularity of Monge-Amp\`ere exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of…
We establish a Schauder-type boundary regularity result for a two-dimensional singular Monge-Amp\'ere equation on convex polytopes with Guillemin boundary conditions. This extends the previous work of Rubin and Huang to the case where the…
In this paper, the author studies quaternionic Monge-Amp\`ere equations and obtain the existence of the solutions to the Dirichlet problem for such equations in strictly pesudoconvex domains in quaternionic space. The stability and…
We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[ \mathcal{A} \mu = \sigma, \] where $\mu \in \mathcal{M}(\Omega;V)$ and $\sigma\in \mathcal{M}(\Omega;W)$ are vector measures and the polar…
We generalize and strenghten Ko{\l}odziej's stability theorem. In particular we give sharp stability exponent and treat the case with more singular right hand side of the Monge-Amp\`ere equation.
Let $P$ be a convex body containing the origin in its interior. We study a real Monge-Amp\`ere equation with singularities along $\del P$ which is Legendre dual to a certain free boundary Monge-Amp\`ere equation. This is motivated by the…
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…
We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct…
In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Amp\'ere equation with H\"older-continuous first derivatives of exponent $\beta<1/5$. Our approach is based on combining the…
In this note, we obtain sharp bounds for the Green's function of the linearized Monge-Amp\`ere operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge-Amp\`ere measure satisfying…