The complex Monge–Ampère equation on compact K?hler manifolds

We consider the complex Monge–Ampère equation on a compact K?hler manifold (M, g) when the right hand side F has rather weak regularity. In particular we prove that estimate of ${\triangle \phi}$ and the gradient estimate hold when F is in ${W^{1, p_0}}$ for any p ₀?>?2n. As an application, we show that there exists a classical solution in ${W^{3, p_0}}$ for the complex Monge–Ampère equation when F is in ${W^{1, p_0}}$ .     

The complex Monge–Ampère equation on weakly pseudoconvex domains

We show here a “weak” Hölder regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge–Ampère equation with data in the $L^p$ space and Ω satisfying an f-property. The f-property is a potential-theoretical condition that holds for all pseudoconvex domains of finite type and many examples of infinite-type ones.     

A local regularity of the complex Monge–Ampère equation

We prove a local regularity (and a corresponding a priori estimate) for plurisubharmonic solutions of the nondegenerate complex Monge–Ampère equation assuming that their W ^{2, p} -norm is under control for some p > n(n − 1). This condition is optimal. We use in particular some methods developed by Trudinger and an estimate for the complex Monge–Ampère equation due to Kołodziej.     

The obstacle problem for a fractional Monge–Ampère equation

Abstract

We study the obstacle problem for a nonlocal, degenerate elliptic Monge–Ampère equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary.     

Weak solutions to complex Monge–Ampère equations on compact Kähler manifolds

We show a general existence theorem of solutions to the complex Monge–Ampère type equation on compact Kähler manifolds.     

Monge–Ampère measures on pluripolar sets

In this article we solve the complex Monge–Ampère problem for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Kołodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge–Ampère measure, then it is a complex Monge–Ampère measure.     

A meshfree method for solving the Monge–Ampère equation

Numerical Algorithms - This paper solves the two-dimensional Dirichlet problem for the Monge-Ampère equation by a strong meshless collocation technique that uses a polynomial trial space and...     

Evans–Krylov Estimates for a nonconvex Monge–Ampère equation

Mathematische Annalen - We establish Evans–Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The...     

Optimal boundary regularity for a singular Monge–Ampère equation

In this paper we study the optimal global regularity for a singular Monge–Ampère type equation which arises from a few geometric problems. We find that the global regularity does not depend on the smoothness of domain, but it does depend on the convexity of the domain. We introduce $(a,\eta )$ type to describe the convexity. As a result, we show that the more convex is the domain, the better is the regularity of the solution. In particular, the regularity is the best near angular points.     

Monge–Ampère Type Equations on Almost Hermitian Manifolds

In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds. We derive C^∞ a priori estimates under the existence of an admissible C-subsolution. Finally,we obtain an existence result if there exists an admissible supersolution.     

The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities

Let (W,H,μ) be an abstract Wiener space, assume that $\text{d}\text{ν=L}\text{d}\text{μ}$ is a second probability measures on $\text{(W,}\text{B}\text{(W))}$ such that $\text{L=}\text{1}\text{c}\text{exp}\text{?f}$, with $\text{f∈}\text{D}{}_{\mathrm{2,1}}$ lower bounded and H-convex. Let $\text{T=I}{}_{\mathrm{W}}\text{+??,}\text{?∈}\text{D}{}_{\mathrm{2,1}}$, be the solution of the Monge problem transporting μ to ν and realizing the H-Wasserstein distance between μ and ν. We prove that $\text{?∈}\text{D}{}_{\mathrm{2,2}}$ hence the Gaussian Jacobian $\text{Λ=}\text{det}{}_2\text{(I+?}{}^2\text{?)}\text{exp}\text{{}\text{L}\text{??1/2|??|}{}_{\mathrm{H}}{}^2\text{}}$ is well-defined and T is the strong solution of the Monge–Ampère equation ΛL°T=1 a.s. on W. To cite this article: D. Feyel, A.S. Üstünel, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Monge–Ampère foliations for degenerate solutions

We study the problem of the existence and the holomorphicity of the Monge–Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge–Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a positive answer to a question of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge–Ampère equation and we obtain a generalization the work of Wong on the classification of complete weighted circular domains.     

W 2,1 regularity for solutions of the Monge–Ampère equation

In this paper we prove that a strictly convex Alexandrov solution u of the Monge–Ampère equation, with right-hand side bounded away from zero and infinity, is $W^{2,1}_{\mathrm{loc}}$ . This is obtained by showing higher integrability a priori estimates for D ² u, namely D ² u∈Llog ^k L for any k∈?.