相关论文: The Monge-Amp\`ere equation in cones and polygonal…
We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct…
In this paper, we study the non-pluripolar complex Monge-Amp\`ere measure on bounded domains in \( \mathbb{C}^n \). We establish a general existence theorem for a non-pluripolar complex Monge-Amp\`ere type equation with prescribed…
In this paper, we establish several geometric properties of boundary sections of convex solutions to the Monge-Amp\`ere equations: the engulfing and separating properties and volume estimates. As applications, we prove a covering lemma of…
We generalize and strenghten Ko{\l}odziej's stability theorem. In particular we give sharp stability exponent and treat the case with more singular right hand side of the Monge-Amp\`ere equation.
In this paper, we prove the regularity of the free boundary in the Monge-Amp\`ere obstacle problem $\det D^2 v= f(y)\chi_{\{v>0\}}. $ By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of 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…
In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially…
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 study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in finite cones and establish optimal asymptotic expansions in terms of the corresponding solutions in infinite cones. The spherical domains over which cones are…
We study the stability and H\"older continuity of solutions to degenerate complex Monge--Amp\`ere equations associated with a (non-closed) big form on compact Hermitian manifolds. We also show that the solution is globally continuous when…
We study the problem of the existence and the holomorphicity of the Monge-Amp\`ere foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Amp\`ere equation even at points of arbitrary degeneracy. We obtain…
We construct convex functions on $\mathbb{R}^3$ and $\mathbb{R}^4$ that are smooth solutions to the Monge-Amp\`{e}re equation $\det D^2u = 1$ away from compact one-dimensional singular sets, which can be Y-shaped or form the edges of a…
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…
In this paper, we prove the H\"older continuity for solutions to the complex Monge-Amp\`ere equations on non-smooth pseudoconvex domains of plurisubharmonic type ${m}$.
In this paper, we study the eigenvalue problem for the Monge-Amp\`ere operator on general bounded convex domains. We prove the existence, uniqueness and variational characterization of the Monge-Amp\`ere eigenvalue. The convex…
In this paper, we study flexibility of weak solutions to the Monge-Amp\`ere system (MA) via convex integration. This new system of Pdes is an extension of the Monge-Amp\`ere equation in $d=2$ dimensions, naturally arising from the…
The main result asserts the existence of continuous solutions of the complex Monge-Amp\`ere equation with the right hand side in $L^p, p>1$, on compact Hermitian manifolds.
In this paper, the author studies quaternionic Monge-Amp\`ere equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper not only answers to…
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…