Related papers: H\"older continuous solutions to Monge-Amp\`ere eq…
This article is concerned with the parabolic Monge-Amp\`ere equation $-u_t\det D_x^2u=f$, where $f=f_1(x)f_2(t)$ and $f_1,f_2$ are positive periodic functions. We prove that any classical parabolically convex ancient solution $u$ must be of…
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 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…
In this paper, we establish a local regularity result for $W^{2,p}_{\mathrm{loc}}$ solutions to complex degenerate nonlinear elliptic equations $F(D^2_{\mathbb{C}} u)=f$ when they are comparable to the Monge-Amp\`ere equation. Notably, we…
In this paper we consider Monge-Amp\`ere equations on compact Hessian manifolds, or equivalently Monge-Amp\`ere equations on certain unbounded convex domains $\Omega\subseteq \mathbb{R}^n$, with a periodicity constraint given by the action…
We study the stability and H\"older continuity of solutions to degenerate complex Monge--Amp\`ere equations associated with a (non-closed) big form on compact Hermitian manifolds. We also show that the solution is globally continuous when…
In this paper, we establish boundary H\"older gradient estimates for solutions to the linearized Monge-Amp\`ere equations with $L^{p}$ ($n<p\leq\infty$) right hand side and $C^{1,\gamma}$ boundary values under natural assumptions on the…
In this paper, we study the eigenvalue problem for the Monge-Amp\`ere operator on general bounded convex domains. We prove the existence, uniqueness and variational characterization of the Monge-Amp\`ere eigenvalue. The convex…
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…
We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function…
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…
We establish a Schauder-type boundary regularity result for a two-dimensional singular Monge-Amp\'ere equation on convex polytopes with Guillemin boundary conditions. This extends the previous work of Rubin and Huang to the case where the…
We establish global $C^{1,\beta}$ and $W^{2, p}$ regularity for singular Monge-Amp\`ere equations of the form \[\det D^2 u \sim \text{dist}^{-\alpha}(\cdot,\partial\Omega),\quad \alpha\in (0, 1),\] under suitable conditions on the boundary…
We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Amp\`ere flow on a compact K\"ahler manifold, with the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere…
The convexity of solutions to boundary value problems for fully nonlinear elliptic partial differential equations (such as real or complex $k$-Hessian equations) is a challenging topic. In this paper, we establish the power convexity of…
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…
This paper studies the complex Monge-Amp\`ere equations for $\mathcal F$-plurisubharmonic functions in bounded $\mathcal F$-hyperconvex domains. We give sufficient conditions for this equation to solve for measures with a singular part.
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some…
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 sufficient conditions for the existence and uniqueness, in bounded uniformly convex domains $\Omega$, of solutions of degenerate elliptic equations depending also on the nonlinear gradient term $H$, in term of the size of $\Omega$,…