Related papers: Moser-Trudinger inequality for the complex Monge-A…
We analyze solutions of the Toda system and establish an optimal Moser-Trudinger inequality
We study the asymptotics of complete Kaehler-Einstein metrics on strictly pseudoconvex domains in C^n and derive a convergence theorem for solutions to the corresponding Monge-Ampere equation. If only a portion of the boundary is analytic,…
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…
In this paper, we introduce the pluricomplex Green function of the Monge-Amp\`{e}re equation for $(n-1)$-plurisubharmonic functions by solving the Dirichlet problem for the form type Monge-Amp\`{e}re and Hessian equations on a punctured…
In this paper we study an obstacle problem for Monge-Amp\`ere type functionals, whose Euler-Lagrange equations are a class of fourth order equations, including the affine maximal surface equations and Abreu's equation.
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 present a numerical method for solving the Monge-Ampere equation based on the characterization of the solution of the Dirichlet problem as the minimizer of a convex functional of the gradient and under convexity and nonlinear…
In this paper, using blow-up analysis, we prove a singular Hardy-Morser-Trudinger inequality, and find its extremal functions. Our results extend those of Wang-Ye (Adv. Math. 2012), Yang-Zhu ( Ann. Glob. Anal. Geom. 2016), Csat\'{o}- Roy…
We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Amp\`ere operator on a bounded strongly pseudoconvex domain in $\C^n$. We show that the eigenfunction is…
We compare various notions of weak subsolutions to degenerate complex Monge-Amp\`ere equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge-Amp\`ere inequalities due to Kolodziej and Dinew.
We study the geometric properties of complex manifolds possessing a pair of plurisubharmonic functions satisfying Monge-Amp\`ere type of condition. The results are applied to characterize complex manifolds biholomorphic to $\C^{N}$ viewed…
In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.
We prove the existence of unique smooth solutions to the quaternionic Monge-Amp\`{e}re equation for $(n-1)$-quaternionic plurisubharmonic functions on a hyperK\"{a}hler manifold and thus obtain solutions for the quaternionic form type…
In this note, we solve the complex Monge-Amp\`ere equation for measures with a pluripolar part in compact K\"ahler manifolds. This result generalizes the classical results obtained by Cegrell in bounded hyperconvex domains. We also discuss…
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 prove the existence of extremals for fractional Moser-Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler-Lagrange equation, which requires new sharp…
In this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality, and obtain $L^1$…
We study Moser-Trudinger type functionals in the presence of singular potentials. In particular we propose a proof of a singular Carleson-Chang type estimate by means of Onofri's inequality for the unit disk in $\mathbb{R}^2$. Moreover we…
Wiegerinck has shown that a separately subharmonic function need not be subharmonic. Improving previous results of Lelong, of Avanissian, of Arsove and of us, Armitage and Gardiner gave an almost sharp integrability condition which ensures…
Probability measures with either finite Monge-Amp\`ere energy or finite entropy have played a central role in recent developments in K\"ahler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose…