Related papers: Complex Monge-Amp\`ere equations on quasi-projecti…
In this paper, we introduce a class of nonlinear optimisation problems. Under mild assumptions, we obtain the existence of potential functions and show that the potential function is a generalised solution of a Monge-Amp\`ere type equation.…
We prove a regularity result for Monge-Amp\`ere equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of $\beta$-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical…
We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel…
We obtain a quantitative expansion at infinity of solutions for a kind of Monge-Amp\`ere type equations that origin from mean curvature equations of Lagrangian graph $(x,Du(x))$ and refine the previous study on zero mean curvature equations…
In this paper, we give some precise characterizations of existence of solution to the complex Monge - Amp\`ere equation in the classes $\mathcal E_\chi(\Omega)$ and $\mathcal E_{\chi,loc}(\Omega)$.
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.
We give a new proof for the interior regularity of strictly convex solutions of the Monge-Amp\`ere equation. Our approach uses a doubling inequality for the Hessian in terms of the extrinsic distance function on the maximal Lagrangian…
The existence and regularity of the classical plurisubharmonic solution for complex Monge-Amp\`ere equations subject to the semilinear oblique boundary condition which is C^1 perturbation of the Neumann boundary condition, are proved in the…
We prove the long time existence and uniqueness of solution to a parabolic Monge-Amp\`ere type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth…
We obtain higher order estimates for a parabolic flow on a compact Hermitian manifold. As an application, we prove that a bounded $\hat{\omega}$-plurisubharmonic solution of an elliptic complex Monge-Amp\`{e}re equation is smooth under an…
We review recent advances in the numerical analysis of the Monge-Amp\`ere equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods…
This is an introduction to a particular class of auxiliary complex Monge-Amp\`ere equations which had been instrumental in $L^\infty$ estimates for fully non-linear equations and various questions in complex geometry. The essential…
For the Monge-Amp\`ere equation with a right-hand side bounded away from 0 and infinity, we show that the solution, subject to the natural boundary condition arising in optimal transport, is in $W^{2,1+\varepsilon}$ up to the boundary.
We study the continuity of solutions to complex Monge-Ampere equations with prescribed singularities. This generalizes the previous results of DiNezza-Lu and the author. As an application, we can run the Monge-Ampere flow starting at a…
We consider degenerate Monge-Amp\`ere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Amp\`ere…
We study asymptotic behaviors of solutions to the Monge-Amp\`ere equation in cones and use the expansion as a tool to study the regularity of solutions in polygonal domains.
In this paper, we establish local and global regularity estimates for linearized Monge-Amp\`ere equations in divergence form via critical Lorentz space estimates for the Green's function of the linearized Monge-Amp\`ere operator and its…
In this paper, we investigate the interior H\"older regularity of solutions to the linearized Monge-Amp\`ere equation. In particular, we focus on the cases with singular right-hand side, which arise from the study of the semigeostrophic…
A gradient estimate for complex Monge-Amp\`ere equations which improves in some respects on known estimates is proved using the ABP maximum principle.
We shall consider the regularity problem of solutions for complex Monge-Ampere equations. First we prove interior $C^2$ estimates of solutions in a bounded domain for complex Monge-Ampere equation with assumption of certain $L^p$ bound for…