Related papers: Viscosity solutions to quaternionic Monge-Amp\`{e}…
In this paper, we solve the Dirichlet problem for Monge-Amp\`ere type equations for $(n-1)$-plurisubharmonic functions on Hermitian manifolds.
We prove the long time existence and uniqueness of solution to a parabolic quaternionic Monge-Amp\`{e}re type equation on a compact hyperK\"{a}hler manifold. We also show that after normalization, the solution converges smoothly to the…
We prove the existence and uniqueness of weak solutions for the generalized Monge-Amp\`ere equation and the supercritical deformed Hermitian-Yang-Mills equation in cohomology classes lying on the boundary of the solvable region. Moreover,…
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 theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear Partial Differential Equations (PDEs) such as the elliptic Monge-Amp\`ere equation. The approximation theory of…
We study the Dirichlet problem for the complex Monge-Amp\`ere operator with bounded, discontinuous boundary data. If the set of discontinuities is b-pluripolar and the domain is B-regular, we are able to prove existence, uniqueness and some…
This paper introduces a fast and robust iterative scheme for the elliptic Monge-Amp\`ere equation with Dirichlet boundary conditions. The Monge-Amp\`ere equation is a nonlinear and degenerate equation, with applications in optimal…
We solve the Dirichlet problem for Monge-Amp\`ere equation for $(n-1)$-PSH functions possibly with degenerate right-hand side, through deriving a quantitative version of boundary estimate under the assumption of $(n-1)$-PSH subsolutions. In…
In this paper, we are interested in studying the Dirichlet problem for the complex Monge-Amp\`ere operator. We provide necessary and sufficient conditions for the problem to have H\"older continuous solutions.
In this paper, we study the interior C^{1,1} regularity of viscosity solutions for a degenerate Monge-Amp\`{e}re type equation \det[D^{2}u-A(x, u, Du)]=B(x, u, Du) when B \geq 0 and B^{\frac{1}{n-1}}\in…
The present paper provides two necessary and sufficient conditions for the existence of solutions to the exterior Dirichlet problem of the Monge-Amp\`ere equation with prescribed asymptotic behavior at infinity. By an adapted smooth…
Monge-Amp\`{e}re equation is a prototype second-order fully nonlinear partial differential equation. In this paper, we propose a new idea to design and analyze the $C^0$ interior penalty method to approximation the viscosity solution of the…
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 study a Dirichlet type problem for the non-pluripolar complex Monge - Amp\`ere equation with prescribed singularity on a bounded domain of $\mathbb{C}^n$. We provide a local version for an existence and uniqueness theorem…
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 consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth…
A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in $\C^n$. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by…
Given a psh function $\varphi\in\mathcal{E}(\Omega)$ and a smooth, bounded $\theta\geq 0$, it is known that one can solve the Monge-Amp\`{e}re equation $\mathrm{MA}(\varphi_\theta)=\theta^n\mathrm{MA}(\varphi)$, with some form of Dirichlet…
The proof of the comparison principlein [EGZ11] is not complete. We provide here an alternative proof, valid in the ample locus of any big cohomology class, and discuss the resulting modifications.
In this paper, we study weak solutions to complex Monge-Amp\`ere equations of the form $(\omega + dd^c \varphi)^n= F(\varphi,.)d\mu$ on a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, where $\omega$ is a smooth $(1,1)$-form,…