Related papers: Boundary regularity for Monge-Amp\`ere equations w…
We prove that if a smoothly bounded strongly pseudoconvex domain $D \subset \mathbb C^n$, $n \geq 2$, admits at least one Monge-Amp\`ere exhaustion smooth up to the boundary (i.e. a plurisubharmonic exhaustion $\tau: \overline D \to [0,1]$,…
In this paper, we establish the global H\"older gradient estimate for solutions to the Dirichlet problem of the Monge-Amp\`ere equation $\det D^2u = f$ on strictly convex but not uniformly convex domain $\Omega$.
In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $\Omega$, $u=g$ in $\mathbb R^N\setminus\Omega$, in non-smooth domains $\Omega$. When $g$ is smooth enough, then it is easy to…
These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Amp\`ere and linearized Monge-Amp\`ere equations. The…
We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $\Omega$. By means of barriers, a nonlocal…
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 study the regularity of the solution for the obstacle problem associated with the linearized Monge-Amp\`ere operator: \begin{align*} \begin{cases} &u\geq\varphi \text{\quad in } \Omega &L_{ w}u=\tr( W D^{2}u)\leq 0…
We obtain pointwise $C^{2,\alpha}$ estimates at boundary points for solutions to the Monge-Ampere equation under appropriate local conditions on the right hand side and boundary data.
We prove the bounded subsolution theorem for the complex Monge-Amp\`ere type equation, with the right hand side being a positive Radon measure, on a compact Hermitian manifold with boundary.
We study the obstacle problem for a nonlocal, degenerate elliptic Monge--Amp\`ere equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary.
We consider smooth solutions to the Monge-Amp`ere equation subject to mixed boundary conditions on annular domains. We establish global $C^2$ estimates when the boundary of the domain consists of two smooth strictly convex closed…
In this paper, we establish the global $W^{2,p}$ estimate for the Monge-Amp\`ere obstacle problem: $(Du)_{\sharp}f\chi{_{\{u>\frac{1}{2}|x|^2\}}}=g$, where $f$ and $g$ are positive continuous functions supported in disjoint bounded $C^2$…
The regularity theory of the degenerate complex Monge-Amp\`{e}re equation is studied. The equation is considered on a closed compact K\"{a}hler manifold $(M,g)$ with nonnegative orthogonal bisectional curvature of dimension $m$. Given a…
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…
In this paper, we consider the global regularity for Monge-Amp\`ere type equations with the Neumann boundary conditions on Riemannian manifolds. It is known that the classical solvability of the Neumann boundary value problem is obtained…
We study the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudoconvex domain in Cn or a Hermitian manifold. Under the condition that the right-hand side lies in Lp function and the boundary data are H\"older…
Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Amp\`ere operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of…
We prove stability of solutions of the complex Monge-Amp\`ere equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be H\"older…
The Monge-Amp\`ere type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Amp\`ere type equations, which can be degenerate or singular near…
This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -\Delta u = f & \text{in} \Omega u= 0 & \text{on} \partial \Omega$$ where $f$ belongs to some $L^p(\Omega)$ and $\Omega$ is a…