相关论文: Interior $C^{1,\alpha}$ regularity on the lineariz…

We review recent advances in the numerical analysis of the Monge-Amp\`ere equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods…

Let $(X,\omega)$ be a compact K\"ahler manifold. We obtain uniform H\"older regularity for solutions to the complex Monge-Amp\`ere equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample…

Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Amp\`ere equation ${\rm det} (D^2 z)=\varphi(x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that…

We obtain sharp local $C^{1,\alpha}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ \Delta_p u=\gamma(u-\varphi)^{\gamma-1}\,\text{ in }\,\{u>\varphi\}, $$ for $0<\gamma<1$ and…

This article is concerned with ``up to $C^{2, \alpha}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an…

In this paper, we show that $G$-invariant Calabi-Yau structures on the complexification $G^{\mathbb C}/K^{\mathbb C}$ of a symmetric space $G/K$ of compact type are constructed from solutions of a Monge-Amp$\grave{\rm e}$re type equation.…

We show here a "weak" H\"older-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Amp\`{e}re equation with data in the $L^p$ space and the boundary of the domain satisfying an $f$-property. The…

We present an explicit pluripotential and viscosity solution to the complex Monge-Amp\`ere equation with constant right-hand side on $\mathbb D\times\mathbb C^{n-1}\,(n\geq 2)$, which lies merely in $W^{1,2}_{loc}\cap W^{2,1}_{loc}$ and is…

We present an iterative approach to approximate the solution to the Dirichlet complex Monge-Amp\`ere eigenvalue problem on a bounded strictly pseudoconvex domain in $\C^n$. This approach is inspired by a similar approach initiated by F.…

In this paper, we study possibly non-closed big (1, 1)-forms on a compact Hermitian manifold satisfying the bounded mass property. We propose several criteria for the existence of rooftop envelopes. As applications, we establish the…

These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Amp\`ere and linearized Monge-Amp\`ere equations. The…

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$, equipped with a Hermitian metric $\omega$. Let $\beta$ be a possibly non-closed smooth $(1,1)$-form on $X$ such that $\int_X\beta^n>0$. Assume that there is a…

Given an $n$-dimensional compact K\"ahler manifold, we continue our study of $m$-positivity in two ways. We first propose generalisations of the notions of pseudo-effective and big Bott-Chern cohomology classes of bidegree $(1,\,1)$ by…

In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior H\"older estimates are obtained when the order of the…

This paper analyzes a regularization scheme of the Monge--Amp\`ere equation by uniformly elliptic Hamilton--Jacobi--Bellman equations. The main tools are stability estimates in the $L^\infty$ norm from the theory of viscosity solutions…

We study the eigenvalue problem for the complex Monge-Amp\`ere operator in bounded hyperconvex domains in $\C^n$, where the right-hand side is a non-pluripolar positive Borel measure. We establish the uniqueness of eigenfunctions in the…

We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})+f(t,z,u)$ in domains of $\mathbb C^n$. This equation has a close connection with the K\"ahler-Ricci flow. In this paper, we consider the case of the boundary conditions are…

In this paper we develop a new a posteriori error analysis for the Monge-Amp\`ere equation approximated by conforming finite element method on isotropic meshes in 2D. The approach utilizes a slight variant of the mixed discretization…

Let $\mu$ be a non-negative measure defined on bounded $\mathcal F$-hyperconvex domain $\Omega$. We are interested in giving sufficient conditions on $\mu$ such that we can find a plurifinely plurisubharmonic function satisfying $NP (dd^c…

On $(X,\omega)$ compact K\"ahler manifold, given a model type envelope $\psi\in PSH(X,\omega)$ (i.e. a singularity type) we prove that the Monge-Amp\`ere operator is an homeomorphism between the set of $\psi$-relative finite energy…