Related papers: The Monge-Amp\`{e}re Equation

Following the authors' recent work \cite{Zhang-Zhou2025}, we further explore the convexity properties of solutions to the Dirichlet problem for the complex Monge-Amp\`ere operator. In this paper, we establish the $\log$-concavity of…

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…

In this paper, we give several new approaches to study interior estimates for a class of fourth order equations of Monge-Amp\`ere type. First, we prove interior estimates for the homogeneous equation in dimension two by using the partial…

We prove a local regularity (and a corresponding a priori estmate) for plurisubharmonic solutions of the nondegenerate complex Monge-Amp\'ere equation assuming that their $W^{2,p}$-norm is under control for some $p>n(n-1)$. This condition…

In this paper, we introduce a family of real Monge-Amp\`ere functionals and study their variational properties. We prove a Sobolev type inequality for these functionals and use this to study the existence and uniqueness of some associated…

We construct a counterexample to $W^{2,1}$ regularity for convex solutions to $$\det D^2u \leq 1, \quad u|_{\partial \Omega} = 0$$ in two dimensions. We also prove a result on the propagation of singularities in two dimensions that are…

We study the regularity of solutions to complex Monge-Amp\`ere equations $(dd^c u)^n=f dV$, on bounded strongly pseudoconvex domains $ \Omega \subset \C^n$. We show, under a mild technical assumption, that the unique solution $u$ to such an…

In this article, we introduce and study three numerical methods for the Dirichlet Monge Amp\`ere equation in two dimensions. The approaches consist in considering new equivalent problems. The latter are discretized by a wide stencil finite…

We present an iterative method based on repeatedly inverting the Monge-Amp\`ere operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain $\Omega \subset \mathbb{R}^n$. We prove that the iterates…

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…

We solve a general class of free boundary Monge-Amp\`ere equations given by \[ \det D^2u = \lambda \dfrac{f(-u)}{g(u^\star)h(\nabla u)}\chi_{\{u<0\}} \; \text{ in } \mathbb{R}^n, \quad \nabla u (\mathbb{R}^n) = P \] where $P$ is a bounded…

In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomogeneous term is only assumed to be Holder continuous. As a consequence of our approach, we also…

In this paper we consider the generalised solutions to the Monge-Amp{\`{e}}re type equations with general source terms. We firstly prove the so-called comparison principle and then give some important propositions for the border of…

The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,\alpha} estimates away from a divisor are obtained, by combining…

We prove that solutions to the Monge-Ampere inequality $$\det D^2u \geq 1$$ in $\mathbb{R}^n$ are strictly convex away from a singular set of Hausdorff $n-1$ dimensional measure zero. Furthermore, we show this is optimal by constructing…

We consider the Monge-Amp\`ere equation $\det(D^2u)=f$ in $\mathbb{R}^n$, where $f$ is a positive bounded periodic function. We prove that $u$ must be the sum of a quadratic polynomial and a periodic function. For $f\equiv 1$, this is the…

This paper is devoted to study the following degenerate Monge-Amp\`ere equation: \begin{eqnarray}\label{ab1} \begin{cases} \det D^2 u=\Lambda_q (-u)^q \quad \text{in}\quad \Omega,\\ u=0 \quad\text{on}\quad \partial\Omega \end{cases}…

Let $P$ be a convex body containing the origin in its interior. We study a real Monge-Amp\`ere equation with singularities along $\del P$ which is Legendre dual to a certain free boundary Monge-Amp\`ere equation. This is motivated by the…

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 consider Monge-Amp\`ere equations with right hand side $f$ that degenerate to $\infty$ near the boundary of a convex domain $\Omega$, which are of the type $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;\Omega,\quad\quad f\sim…