Related papers: The pluricomplex Poisson kernel for convex finite …

We consider a generalised complex Monge-Amp\`ere equation on a compact K\"ahler manifold and treat it using the method of continuity. For complex surfaces, we prove an easy existence result. We also prove that (for three-folds and a related…

We present an explicit construction of the solution to the Dirichlet boundary value problem for the radial Schr\"odinger equation in the unit ball, with a complex-valued potential $V$ satisfying the condition $\int_0^1r|V(r)|dr<\infty$. The…

We classify global solutions of the Monge-Amp\`ere equation $\det D^2 u=1 $ on the first quadrant in the plane with quadratic boundary data. As an application, we obtain global $C^{2,\alpha}$ estimates for the non-degenerate Monge-Amp\`ere…

In this paper we consider Monge-Amp\`ere equations on compact Hessian manifolds, or equivalently Monge-Amp\`ere equations on certain unbounded convex domains $\Omega\subseteq \mathbb{R}^n$, with a periodicity constraint given by the action…

We prove several results on homogeneous plurisubharmonic polynomials on $\mathbb{C}^n$, $n\in\mathbb{Z}_{\geq 2}$. Said results are relevant to the problem of constructing local bumpings at boundary points of pseudoconvex domains of finite…

Given a continuous function $\phi$ defined on a domain $\Omega\subset\mathbb{R}^m\times\mathbb{R}^n$, we show that if a Pr\'ekopa-type result holds for $\phi+\psi$ for any non-negative convex function $\psi$ on $\Omega$, then $\phi$ must be…

Let $S_\epsilon$ be a set of $N$ points in a bounded hyperconvex domain in $C^n$, all tending to 0 as$\epsilon$ tends to 0. To each set $S_\epsilon$ we associate its vanishing ideal $I_\epsilon$ and the pluricomplex Green function…

In this paper, we obtain estimates on the quantitative strata of the critical set of non-trivial harmonic functions $u$ which vanish continuously on $V \subset \partial \Omega$, a relatively open subset of the boundary of a convex domain…

We consider the Szeg\"o kernel for non-pseudoconvex domains in C^2 given by \Omega = {(z,w): Im w > b(Re z)} for b a non-convex even-degree polynomial with positive leading coefficient. This is an extension of results previously obtained by…

The simplest genuinely multidimensional monopolist's problem involves minimizing a linearly perturbed Dirichlet energy among nonnegative convex functions $u$ on an open domain $X \subset [0, \infty)^2$. The geometry of the region of strict…

We solve a randomized version of the following open question: is there a strictly convex, bounded curve \gamma in the plane such that the number of rational points on \gamma, with denominator $n$, approaches infinity with $n$? Although this…

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…

This paper is concerned with approximations and related discretization error estimates for the normal derivatives of solutions of linear elliptic partial differential equations. In order to illustrate the ideas, we consider the Poisson…

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 show that if $\Omega\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(\sigma)$, where $\sigma$ is the surface measure of $\Omega$, then the outer…

We study positive solutions of the Dirichlet problem $-\Delta u = u^p$ in a uniformly convex domain $\Omega \subset \mathbb S^2$, $u= 0$ on $\partial\Omega.$ For $p=1$, we assume that the right-hand side is replaced by $\lambda_1 u$, where…

We consider the Szeg\"o kernel for domains \Omega in C^2 given by \Omega = {(z,w): Im w > b(Re z)} where b is a non-convex quartic polynomial with positive leading coefficient. Such domains are not pseudoconvex. We describe the subset of…

Using the logarithmic capacity, we give quantitative estimates of the Green function, as well as lower bounds of the Bergman kernel for bounded pseudoconvex domains in $\mathbb C^n$ and the Bergman distance for bounded planar domains. In…

The affine maximal type hypersurface has been a core topic in Affine Geometry. When the hypersurface is presented as a regular graph of a convex function $u$, the statement that the graph is of affine maximal type is equivalent to the…

We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincar\'e inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all…