Related papers: Estimates for complex singular Monge-Amp\`ere equa…
For the Monge-Amp\`{e}re equation $\det D^2 u=1$, we find new auxiliary curvature functions which attain respective maximum on the boundary. Moreover, we obtain the upper bounded estimates for the Gauss curvature and mean curvature of the…
We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Amp\`{e}re operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by…
In this note, we establish several results concerning the continuity (or weak convergence) of the complex Monge-Amp\`ere operator on compact Hermitian manifolds. At the end of this note, we find a weak solution of the complex Monge-Amp\`ere…
In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Amp\`ere equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.
The existence of a unique numerical solution of the semi-Lagrangian method for the simple Monge-Amp\`ere equation is known independently of the convexity of the domain or Dirichlet boundary data -- when the Monge-Amp\`ere equation is posed…
We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.
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…
A complex Monge-Amp\`ere equation for differential $(p,p)$-forms is introduced on compact K\"ahler manifolds. For any $1 \leq p < n$, we show the existence of smooth solutions unique up to adding constants. For $p=1$, this corresponds to…
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…
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…
In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.
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…
In this note, we derive a Liouville theorem for the complex Monge-Amp\`ere equation. Our result states that if the global solution $u$ of the complex Monge-Amp\`ere equation with constant right-hand side differs from a quadratic polynomial…
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…
We establish an analytic proof for the Krylov $C^{1,1}$ estimates for solutions of degenerate complex Monge-Amp\`ere equation. We also provide an analytic proof of the Bedford-Taylor interior $C^{1,1}$ estimate.
In this note, we give a proof of the uniform log-continuity of the solution to complex Monge-Amp\`ere equations on compact Hermitian manifolds, which is a generalization of the result of Guo-Phong-Tong-Wang in the K\"ahler case.
We provide a necessary and sufficient condition for the existence of H\"{o}lder continuous solutions to the complex Monge--Amp\`{e}re equation on bounded domains in $\mathbb{C}^n$. This condition is motivated by a paper by S.-Y. Li. We also…
We study the complex Monge-Amp\` ere operator on compact K\"ahler manifolds. We give a complete description of its range on the set of $\omega-$plurisubharmonic functions with $L^2$ gradient and finite self energy, generalizing to this…
We consider a generalised complex Monge-Amp\`ere equation on a compact K\"ahler manifold and treat it using the method of continuity. For complex surfaces, we prove an easy existence result. We also prove that (for three-folds and a related…
In this paper, we investigate regularity for solutions to the linearized Monge-Amp\`ere equations when the nonhomogeneous term has low integrability. We establish global $W^{1,p}$ estimates for all $p<\frac{nq}{n-q}$ for solutions to the…