Related papers: Regularities for solutions to the $L_p$ dual Minko…

The central focus of this paper is the $L_p$ dual Minkowski problem for $C$-compatible sets, where $C$ is a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior. Such a problem deals with the characterization of the $(p,…

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…

We consider the Dirichlet problem for the complex Monge-Amp\`ere equation in a bounded strongly hyperconvex Lipschitz domain in $\C^n$. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is…

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…

We study H\"older continuity of solutions to the Dirichlet problem for measures having density in $L^p$, $p>1$, with respect to Hausdorff-Riesz measures of order $2n-2+\epsilon$ for $0<\epsilon \leq 2$, in a bounded strongly hyperconvex…

We study convex solutions to the Monge-Amp\`ere obstacle problem \[ \operatorname{det} D^2 v=g v^q\chi_{\{v>0\}}, \quad v \geq 0, \] where $q \in [0,n)$ is a constant and $g$ is a bounded positive function. This problem emerges from the…

In this paper, we consider the $L_p$ dual Minkowski problem for capillary hypersurfaces for $p>q$ and $q\leq 1$, which aims to find a capillary convex body with a prescribed capillary $(p,q)$-th dual curvature measure in the Euclidean…

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…

The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Amp\`ere equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and…

We show here a "weak" H\"older-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Amp\`{e}re equation with data in the $L^p$ space and the boundary of the domain satisfying an $f$-property. The…

The current state of art concerning the $L_p$ Minkowski problem as a Monge-Ampere equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the $p=0$ case are surveyed and connections to…

We study the Dirichlet problem for Monge-Amp\`ere equation in bounded convex polytopes. We give sharp conditions for the existence of global $C^2$ and $C^{2,\alpha}$ convex solutions provided that a global $C^2$, convex subsolution exists.

This paper studies the core problems in the $L_p$ dual Brunn-Minkowski theory, encompassing the $L_p$ Minkowski problem and $L_p$ Brunn-Minkowski inequality for dual quermassintegrals. For the case $0<p<q\leq n$, we establish $C^0$…

Let $(X,\omega)$ be a compact K\"ahler manifold. We obtain uniform H\"older regularity for solutions to the complex Monge-Amp\`ere equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample…

On a smooth domain $\Omega\subset\subset\mathbb C^n$, we consider the Dirichlet problem for the complex Monge-Amp\`ere equation $((dd^cu)^n=fdV,\,u|_{b\Omega}\equiv\phi)$. We state the H\"older regularity of the solution $u$ when the…

Let $\Omega$ be a bounded, pseudoconvex domain of $\mathbb C^n$ satisfying the "$f$-Property". The $f$-Property is a consequence of the geometric "type" of the boundary; it holds for all pseudoconvex domains of finite type but may also…

The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical…

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…

In this paper, we study the $L_p$ dual Minkowski problem for all $q, p \in \mathbb{R}$ from an algebraic perspective. We establish the existence of solutions for group-invariant convex bodies (not necessarily origin-symmetric), thereby…

In this paper we shall prove the existence, uniqueness and global H$\ddot{o}$lder continuity for the Dirichlet problem of a class of Monge-Amp\`ere type equations which may be degenerate and singular on the boundary of convex domains. We…