Related papers: Interior $C^2$ estimate for Monge-Amp\`ere equatio…
We study the Dirichlet problem for the Monge-Amp\`ere equation on almost complex manifolds. We obtain the existence of the unique smooth solution of this problem in strictly pseudoconvex domains.
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.
We prove a relative $L^\infty$ estimate for a class of complex Monge-Amp\`ere type equations on K\"ahler manifolds. It provides a unified approach to Tundinger type estimate and uniform estimate. It also improves the previous results about…
We study the solvability of the second boundary value problem for a class of highly singular fourth order equations of Monge-Amp\`ere type. They arise in the approximation of convex functionals subject to a convexity constraint using Abreu…
In this paper we develop a new a posteriori error analysis for the Monge-Amp\`ere equation approximated by conforming finite element method on isotropic meshes in 2D. The approach utilizes a slight variant of the mixed discretization…
We obtain the H\"older regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity. Our main tool is an interior H\"older…
In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Amp\`ere equation, with right hand side bounded away from zero and infinity, is $W_{\rm loc}^{2,1}$. This is obtained by showing higher integrability a-priori…
We solve the Dirichlet problem for the quaternionic Monge-Amp\`ere equation with a continuous boundary data and the right hand side in $L^p$ for $p>2$. This is the optimal bound on $p$. We prove also that the local integrability exponent of…
We present an iterative approach to approximate the solution to the Dirichlet complex Monge-Amp\`ere eigenvalue problem on a bounded strictly pseudoconvex domain in $\C^n$. This approach is inspired by a similar approach initiated by F.…
We improve our previous gradient estimate for the Monge-Amp\`ere equation on a compact Hermitian manifold and give a estimates for the non-mixed second order derivatives. These estimates are required to apply either the Evans-Krylov…
Uniform bounds are obtained using the auxiliary Monge-Amp\`ere equation method for solutions of very general classes of fully non-linear partial differential equations, assuming the existence of a ${C}$-subsolution in the sense of G.…
In this note we revisit previous Pogorelov type interior and global second derivative estimates of the author, F. Jiang and J. Liu for solutions of Monge-Amp`ere type partial differential equations. Taking account of recent strict convexity…
We use geometric methods to calculate a formula for the complex Monge-Amp\`ere measure $(dd^cV_K)^n$, for $K \Subset \RR^n \subset \CC^n$ a convex body and $V_K$ its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this…
We derive a priori $C^2$ estimates for a class of complex Monge-Ampere type equations on Hermitian manifolds. As an application we solve the Dirichlet problem for these equations under the assumption of existence of a subsolution; the…
We propose a new variational formulation of the elliptic Monge-Ampere equation and show how classical Lagrange elements can be used for the numerical resolution of classical solutions of the equation. Error estimates are given for Lagrange…
In this paper, we study the Neumann problem of Monge-Amp\`ere equations in Semi-space. For two dimensional case, we prove that its viscosity convex solutions must be a quadratic polynomial. When the space dimension $n\geq 3$, we show that…
In this paper, we study the interior $C^{2}$ estimates for Hessian quotient equations $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)}=1$ for $l=1, 2$, in arbitrary dimensions, under the natural ellipticity and semi-convexity conditions. We…
We study a Monge-Amp\`ere type equation in the class of $p$-plurisubharmonic functions and establish first and second order interior estimates. As an application of these we show that $p$-plurisubharmonic functions with constant operator…
In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our results…
We improve the result of Caffarelli-Li [CL03] on the asymptotic behavior at infinity of the exterior solution $u$ to Monge-Amp\`{e}re equation $det(D^2u)=1$ on $\mathbb{R}^n\backslash K$ for $n\geq 3$. We prove that the error term…