Related papers: On regularity of complex Monge-Ampere equation
In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampere type equations under sharp hypotheses. In particular we assume that the matrix function in the…
Let $\Omega$ be a bounded strictly pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Monge-Amp\`ere equations of the form $(\omega + dd^c \varphi)^n = \mu$ in the generalized Cegrell classes $\mathcal{K}(\Omega,\omega,H)$,…
In this paper, we obtain gradient estimates and Laplacian estimates for the solution to the singular complex Monge-Amp\`ere equation by applying the integral method.
The existence, multiplicity and nonexistence of nontrivial radial convex solutions of a system of two weakly coupled Monge-Ampere equations are established with asymptotic assumptions for an appropriately chosen parameter. The proof of the…
In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Amp\`ere equation, with right hand side bounded away from zero and infinity, is $W_{\rm loc}^{2,1}$. This is obtained by showing higher integrability a-priori…
We define a general class of elliptic equations for 2-forms on 4-manifolds, of which the complex Monge-Ampere equation is a prototype. We obtain some regularity results and discuss various connections (some speculative) with modern…
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{\`e}re equations, modelled on the K{\"a}hler-Ricci flow evolving on…
Given a compact K\"ahler manifold, we survey the study of complex Monge-Amp\`ere type equations with prescribed singularity type, developed by the authors in a series of papers. In addition, we give a general answer to a question of…
It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class $C^{1,\alpha}, 0<\alpha<1$. An essential step is to establish that the rays can be extended as solutions of a…
In this paper, we study the regularity of the solution for the obstacle problem associated with the linearized Monge-Amp\`ere operator: \begin{align*} \begin{cases} &u\geq\varphi \text{\quad in } \Omega &L_{ w}u=\tr( W D^{2}u)\leq 0…
We prove several approximation theorems of the complex Monge-Ampere operator on a compact Kahler manifold. As an application we give a new proof of a recent result of Guedj and Zeriahi on a complete description of the range of the complex…
We introduce generalized Monge-Amp\`ere capacities and use these to study complex Monge-Amp\`ere equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is…
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 characterize the class of probability measures on a compact Kahler manifold such that the associated Monge-Amp\`ere equation has a solution of finite pluricomplex energy. Our results are also valid in the big cohomology class setting.
We study the Dirichlet problem for complex Monge-Ampere equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in the flat case. We…
This paper contains two results on the $L^p$ regularity problem on Lipschitz domains. For second order elliptic systems and $1<p<\infty$, we prove that the solvability of the $L^p$ regularity problem is equivalent to that of the…
We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Amp\`ere equations $$\det D^2u =f(x) \text{ in } \Omega , \quad D_{\beta}u = \phi(x) \text{ on } \partial \Omega.$$ In…
We prove that if the modulus of continuity of a plurisubharmonic subsolution satisfies a Dini type condition then the Dirichlet problem for the complex Monge-Amp\`ere equation has the continuous solution. The modulus of continuity of the…
In this paper, we obtain the Bedford-Taylor interior $C^{2}$ estimate and local Calabi $C^{3}$ estimate for the solutions to complex Monge-Amp\`ere equations on Hermitian manifolds.
We study the Dirichlet problem for the Monge-Amp\`ere equation on almost complex manifolds. We obtain the existence of the unique smooth solution of this problem in strictly pseudoconvex domains.