Related papers: Complex Monge-Amp\`ere equations on quasi-projecti…
In this note, we consider complex Monge-Ampere equation posed on a compact K\"ahler manifold. We show how to get $L^p$($p<\infty$) and $L^{\infty}$ estimate for the gradient of the solution in terms of the continuity of the right hand side.
We propose an extension to our monotone and convergent method for the Monge-Amp\`{e}re equation in dimension $d \geq2$, that incorporates the idea of filtered schemes. The method combines our original monotone operator with a more accurate…
In this paper, we prove the asymptotic expansion of the solutions to some singular complex Monge-Amp\`ere equation which arise naturally in the study of the conical K\"ahler-Einstein metric.
In this paper, the author studies quaternionic Monge-Amp\`ere equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper not only answers to…
We study the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudoconvex domain in Cn or a Hermitian manifold. Under the condition that the right-hand side lies in Lp function and the boundary data are H\"older…
Let $(X,\omega)$ be a compact K\"ahler manifold. We prove the existence and uniqueness of solutions to complex Monge-Amp\`ere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is…
We study MA-positivity, a notion of positivity relevant to a vector bundle version of the complex Monge--Amp\`ere equation introduced in an earlier work, and show that for rank-two holomorphic bundles over complex surfaces, MA-semi-positive…
Generalized Monge-Amp\`ere equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian-Yang Mills equations, and some Z-critical equations. Solvability of these…
We study continuity properties of generalized Monge-Amp\`ere operators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a…
In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact K\"ahler manifold. We prove that a precise bound on the complex Monge-Amp\`ere…
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 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 study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two…
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…
In this paper, we study existence, regularity, classification, and asymptotical behaviors of solutions of some Monge-Amp\`ere equations with isolated and line singularities. We classify all solutions of $\det \nabla^2 u=1$ in $\R^n$ with…
On a compact K\"ahler manifold $(X,\omega)$, we study the strong continuity of solutions with prescribed singularities of complex Monge-Amp\`ere equations with integrable Lebesgue densities. Moreover, we give sufficient conditions for the…
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 construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity…
We first obtain the interior $C^{1,1}$-regularity and solvability for the degenerate real Monge-Amp\`ere equation in a bounded, $C^3$-smooth and strictly convex domain in $\mathbb R^d$ ($d\ge2$), assuming that the boundary data is only…
Extending DiNezza-Lu's approach to the setting of big cohomology classes, we prove that solutions of degenerate complex Monge-Amp{\`e}re equations on compact K{\"a}hler manifolds are continuous on a Zariski open set. This allows us to show…