Related papers: Weak solution of Parabolic complex Monge-Amp\'ere …
We study the complex Monge-Amp\`ere equation $(dd^c u)^n=\mu$ in a strictly pseudoconvex domain $\Omega$ with the boundary condition $u=\varphi$, where $\varphi\in C(\partial\Omega)$. We provide a non-trivial sufficient condition for…
We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})-Au+f(z,t)$ in domains of $\mathbb{C}^n$. This equation has a close connection with the K\"ahler-Ricci flow. In this paper, we consider the case where the boundary condition is…
Let $\mu$ be a non-negative measure defined on bounded $\mathcal F$-hyperconvex domain $\Omega$. We are interested in giving sufficient conditions on $\mu$ such that we can find a plurifinely plurisubharmonic function satisfying $NP (dd^c…
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,…
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.
The existence and regularity of the classical plurisubharmonic solution for complex Monge-Amp\`ere equations subject to the semilinear oblique boundary condition which is C^1 perturbation of the Neumann boundary condition, are proved in the…
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…
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.
Let $(u_j)$ be a deaceasing sequence of psh functions in the domain of definition $\cal D$ of the Monge-Amp\`ere operator on a domain $\Omega$ of $\mathbb{C}^n$ such that $u=\inf_j u_j$ is plurisubharmonic on $\Omega$. In this paper we are…
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…
We study the complex Monge-Amp\`ere operator in bounded hyperconvex domains of $\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Amp\`ere energy. They…
We consider the complex Monge-Amp\`ere equation on a compact K\"ahler manifold $(M, g)$ when the right hand side $F$ has rather weak regularity. In particular we prove that estimate of $\t\phi$ and the gradient estimate hold when $F$ is in…
We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})+f(t,z,u)$ in domains of $\mathbb C^n$. This equation has a close connection with the K\"ahler-Ricci flow. In this paper, we consider the case of the boundary conditions are…
We consider the complex Monge-Amp\'{e}re equation on complete K\"{a}hler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some…
We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.
In this paper, we study the Cauchy-Dirichlet problem for Parabolic complex Monge-Amp\`ere equations on strongly pseudoconvex domains using the viscosity method. We prove a comparison principle for Parabolic complex Monge-Amp\`ere equations…
We give a sufficient condition on a sequence of uniformly bounded $\omega$-plurisubharmonic functions, $\omega$ being a Hermitian metric, for which the sequence of associated Monge-Amp\`ere measures converges weakly. This criterion can be…
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 paper, we consider the following nonlinear eigenvalue problem for the Monge-Amp\'ere equation: find a non-negative weakly convex classical solution $f$ satisfying {equation*} {cases} \det D^2 f=f^p \quad &\text{in $\Omega$} f=\vp…
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{\`e}re equations, modelled on the K{\"a}hler-Ricci flow evolving on…