Related papers: Generalized Monge-Amp\`ere functionals and related…
We analyze the convergence of an iterative method for solving the nonlinear system resulting from a natural discretization of the Monge-Amp\`ere equation with $C^1$ conforming approximations. We make the assumption, supported by numerical…
A $C^2$ function on $\mathbb{R}^n$ is called strictly $(n-1)$-convex if the sum of any $n-1$ eigenvalues of its Hessian is positive. In this paper, we establish a global $C^2$ estimates to the Monge-Amp\`ere equation for strictly…
We study generalized complex Monge-Amp\`ere type equations on closed Hermitian manifolds. We derive {\em a priori} estimates and then prove the existence of admissible solutions. Moreover, the gradient estimate is improved.
Given a one-parameter family of $\mathbb{Q}$-Fano varieties such that the central fibre admits a unique K\"ahler-Einstein metric, we provide an analytic method to show that the neighboring fibre admits a unique K\"ahler-Einstein metric. Our…
We introduce a new family of intermediate operators between the fractional Laplacian and the Caffarelli-Silvestre nonlocal Monge-Amp\`ere that are given by infimums of integro-differential operators. Using rearrangement techniques, we…
In this paper, we study interior estimates for solutions to linearized Monge-Amp\`ere equations in divergence form with drift terms and the right-hand side containing the divergence of a bounded vector field. Equations of this type appear…
In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. As its application, we obtain a Liouville type theorem for the complex…
In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Amp\`ere equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.
We show that the pluripotential Cauchy-Dirichlet problem for the complex Monge-Amp\`ere flow is solvable for the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere measure of a bounded…
We describe a method to reduce partial differential equations of Monge-Amp\`ere type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the…
We generalize an inequality for mixed Monge-Amp\`ere measures. We also give an example that shows that our assumptions are sharp. The corresponding result in the setting of compact K\"ahler manifold is also discussed.
The main goal of this article is to find, following the approach given in [Ce1] and [Ce2], the largest possible sub-class of plurisubharmornic functions on a complex variety on which the complex Monge-Amp\`ere operator can be reasonably…
The discrete functional $L_p$ Minkowski problem is posed and solved. As a consequence, the general affine P\'{o}lya-Szeg\"{o} principle and the general affine Sobolev inequalities are established.
In this article we address the question whether the complex Monge-Amp\`{e}re equation is solvable for measures with large singular part. We prove that under some conditions there are no solution when the right-hand side is carried by a…
We characterize the class of probability measures on a compact Kahler manifold such that the associated Monge-Amp\`ere equation has a solution of finite pluricomplex energy. Our results are also valid in the big cohomology class setting.
We study a Monge-Amp\`ere type equation that interpolates the classical {\sigma_2} -Yamabe equation in conformal geometry and the 2-Hessian equation in dimension 4.
In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampere type equations under sharp hypotheses. In particular we assume that the matrix function in the…
We propose an extension to our monotone and convergent method for the Monge-Amp\`{e}re equation in dimension $d \geq2$, that incorporates the idea of filtered schemes. The method combines our original monotone operator with a more accurate…
We prove the long time existence and uniqueness of solution to a parabolic quaternionic Monge-Amp\`{e}re type equation on a compact hyperK\"{a}hler manifold. We also show that after normalization, the solution converges smoothly to the…
We prove existence and regularity of entire solutions to Monge-Ampere equations invariant under an irreducible action of a compact Lie group.