arXiv subjects

Quang Dieu Nguyen

Publications and source records attributed to Quang Dieu Nguyen.

2 recordsLinked to original sources

arXiv2025

Removable sets for pseudoconvexity for weakly smooth boundaries

We show that for bounded domains in $\mathbb C^n$ with $\mathcal C^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial Ω\setminus F$ is $\mathcal C^{2}$-smooth and locally pseudoconvex at every point, then $Ω$ is globally pseudoconvex. Unlike in the globally $\mathcal C^{2}$-smooth case, the condition ``$F$ of (relative) empty interior'' is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity.

math.CV↗

arXiv2021

On the viscosity approach to a class of fully nonlinear elliptic equations

In this paper, we study some properties of viscosity sub/super-solutions of a class of fully nonlinear elliptic equations relative to the eigenvalues of the complex Hessian. We show that every viscosity subsolution is approximated by a decreasing sequence of smooth subsolutions. When the equations satisfy some conditions on the limit at infinity, we verify that the comparison principle holds, and as a sequence, we obtain a result about the existence of solution of the Dirichlet problem. Using the comparison principle, we show that, under suitable conditions, a Perron-Bremermann envelope can be approximated by a decreasing sequence of viscosity solutions.

math.AP↗