arXiv subjects
Publications and source records attributed to Nguyen Van Phu.
In this paper, we study Hessian type equations for $m-ω$ subharmonic functions. Using the recent results in \cite{KN23a}, \cite{KN23b}, we are able to show the existence of bounded solutions for such equations on bounded domains in $\mathbb{C}^n$.
In this note, we deal with the existence of solutions of the weighted complex $m$-Hessian equation $-χ(u)H_{m}(u)=μ$ in the class $\mathcal{E}_{m,χ}(f,Ω)$ if there exists a subsolution in this class, where the given boundary value $f\in\mathcal{E}_m(Ω)\cap MSH_m(Ω).$ This is a generalization of the result in the paper \cite{PDtaiwan} where we proved that the subsolution theorem is true in the class $\mathcal{E}_{m,χ}(f,Ω)$ in the case when the given boundary value $f\in\mathcal{N}_m(Ω)\cap MSH_m(Ω).$
In this note, we establish several results concerning the continuity (or weak convergence) of the complex Monge-Ampère operator on compact Hermitian manifolds. At the end of this note, we find a weak solution of the complex Monge-Ampère equation on a compact Hermitian manifold under the assumption of the existence of a smooth subsolution.
In this paper, we establish the weak*-convergence of a sequence of the complex Hessian measures $H_m(u_j)$ to the complex Hessian measure $H_m(u)$ in the class $\mathcal{E}(X,ω,m)$ under hypotheses that $u_j$ is convergent to $u$ in $(ω,m)$-capacity.
In this paper, we investigate the continuity of solutions to the Dirichlet problem for complex Hessian-type equations associated with $(ω, m)-β$-subharmonic functions on a ball in $\mathbb{C}^n$, where $ β=d d^c\|z\|^2=\frac{i}{2} \sum_{j=1}^n d z_j \wedge d \bar{z}_j $ is denoted the flat metric on $\mathbb{C}^n$.
Let $Ω$ be a bounded strictly $m$-pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Hessian equations of the form $(ω+ dd^c φ)^m\wedgeβ^{n-m} = μ$ in the generalized Cegrell classes $\mathcal{K}_m(Ω,ω,ϕ)$, where $ϕ\in \mathcal{E}_m(Ω)$ is a $m$-maximal function, $ω$ is a smooth real $(1,1)$-form defined in a neighborhood of $\barΩ$ and $μ$ is a positive Radon measure which is dominated by a Hessian measure of $m$-subharnomic functions in Cegrell class.
We first study subextensions of m-subharmonic functions in weighted energy classes with given boundary values. The results are used to approximate an m-subharmonic function in weighted energy classes with given boundary values by an increasing sequence of m-subharmonic functions defined on larger domains.
In this paper, we study Hessian type equations for $(ø,m)-β$-subharmonic functions on a ball in $\mathbb{C}^n$, where $β=dd^c\|z\|^2=\frac{i}{2}\sum\limits_{j=1}^n dz_j\w d\bar{z}_j$ is the flat metric on $\cn$. Using the recent results in \cite{KN23b}, we are able to show the existence of bounded solutions for Hessian type equations.
In this paper, we study maximal subextension of $m$-subharmonic functions with given boundary values. We also prove stability on $m$-capacity of maximal subextension of $m$-subharmonic functions with given boundary values.
Let $φ$ be a negative plurisubharmonic function in a pseudoconvex domain $Ω$ in $\mathbb{C}^{n}$ and $f$ be a bounded holomorphic function belonging to $L^{2}(Ω, φ)$. For all negative plurisubharmonic functions $ψ$ satisfying that $ψ$ is close to $φ$ in the $L^1(Ω)$ norm. Using the recent results about $L^2-$estimate for $\overline{\partial}-$equation and strong openness conjecture, we prove that there exists a holomorphic function $g$ belonging to $L^{2}( Ω, ψ)$ such that $g$ is close to $f$ in the $L^2$ norm on some relatively compact open subset $Ω^{'}$ of $Ω.$
In this paper, we concern with the existence of solutions of the complex $m-$Hessian type equation $-χ(u)H_{m}(u)=μ$ in the class $\mathcal{E}_{m,χ}(f,Ω)$ if there exists subsolution in this class, where the given boundary value $f\in\mathcal{N}_m(Ω)\cap MSH_m(Ω).$
In this paper, we first study subextensions in the classes $\mathcal{F}_{m}(Ω)$ and $\mathcal{E}_{m,χ}(Ω)$. These results are then used to study approximation in the classes $\mathcal{F}_{m}(Ω)$ and $\mathcal{E}_{m,χ}(Ω)$.
In this paper, we first study the comparison principle for the operator $H_{χ,m}$. This result is used to solve certain weighted complex $m-$ Hessian equations.