Related papers: Maximal subextensions of plurisubharmonic function…

In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Amp\`ere equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates…

In this paper, we study the Dirichlet problem for Monge-Amp\`ere type equations for $p$-plurisubharmonic functions on Riemannian manifolds. The $a$ $priori$ estimates up to the second order derivatives of solutions are established. The…

We give a sufficient condition on the Monge-Amp\`ere mass of a plurisubharmonic function $u$ for $\exp (- 2 u)$ to be locally integrable. This gives a pluripotential theoretic proof of a theorem by J-P. Demailly.

We investigate existence and uniqueness of maximal plurisubharmonic functions on bounded domains with boundary data that are not assumed to be continuous or bounded. The result is applied to approximate (possibly unbounded from above)…

We consider the complex Monge-Amp\'{e}re equation on complete K\"{a}hler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some…

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 discuss pluripotential aspects of the Monge-Amp\`ere equations on compact Hermitian manifolds and prove $L^{\infty}$ estimates for any metric, as well as the existence of weak solutions under an extra assumption.

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 establish plurisubharmonicity of the envelope of Lelong functional on almost complex manifolds of real dimension four, thereby we generalize the corresponding result for complex manifolds.

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…

Let $X$ be a compact K\"ahler manifold and $\theta$ a smooth closed $(1,1)$-real form representing a big cohomology class $\alpha \in H^{1,1}(X,\R)$. The purpose of this note is to show, using pluripotential and viscosity techniques, that…

We consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth…

We study the Dirichlet problem for the complex Monge-Amp\`ere operator on a B-regular domain $\Omega$, allowing boundary data that is singular or unbounded. We introduce the concept of pluri-quasibounded functions on $\Omega$ and $\partial…

We study the Dirichlet problem for Monge-Amp\`ere equation in bounded convex polytopes. We give sharp conditions for the existence of global $C^2$ and $C^{2,\alpha}$ convex solutions provided that a global $C^2$, convex subsolution exists.

Uniform bounds are obtained using the auxiliary Monge-Amp\`ere equation method for solutions of very general classes of fully non-linear partial differential equations, assuming the existence of a ${C}$-subsolution in the sense of G.…

Let $u$ and $v$ be two plurisubharmonic functions in the domain of definition of the Monge-Amp\`ere operator on a domain $\Omega\subset {\bf C}^n$. We prove that if $u=v$ on a plurifinely open set $U\subset \Omega$ that is Borel measurable,…

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $\Omega$ and continuous on $\bar\Omega$, with functions that are plurisubharmonic on (shrinking) neighborhoods…

We construct examples of plurisubharmonic functions with isolated singularities at $0\in \mathbb{C}^n$, whose residual Monge-Amp\`{e}re masses at the origin can not be approximated by masses of analytic approximations obtained via…

In this paper, we extend the concept of finite entropy measures in K\"ahler geometry. We define the finite $p$-entropy related to $\omega$-plurisubharmonic functions and demonstrate their inclusion in an appropriate energy class. Our study…

Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$. Suppose that we have a function $\phi$ in this neighborhood with $1-\epsilon \le…