arXiv subjects

Bang Tran Van

Publications and source records attributed to Bang Tran Van.

3 recordsLinked to original sources

arXiv2022

Admissible solutions to augmented nonsymmetric $k$-Hessian type equations II. A priori estimates and the Dirichlet problem

Using the established $d$-concavity of the $k$-Hessian type functions $F_k(R)=\log(S_k(R)),$ whose variables are nonsymmetric matrices, we prove $ C^{2, α}(\overlineΩ) $ estimates for strictly $(δ, \widetildeγ_k) $-admissible solutions to the Dirichlet problem without the well-known regularity condition. A necessary condition for the existence of strictly $δ$-admissible solutions to the equations is given. By the method of continuity, we provide some sufficient conditions for the unique solvability in the class of strictly $(δ,\widetildeγ_k)$-admissible solutions to the Dirichlet problem, provided that those skew-symmetric matrices in the equations are sufficiently small in some sense.

math.AP↗

arXiv2020

Admissible solutions to augmented nonsymmetric $k-$Hessian type equations I. The $d-$concavity of the $k-$Hessian type functions

We establish for $2 \le k \le n-1$ the strict concavity of the function $f_k(λ)=\log(σ_k(λ))$ on a subset of the positive cone $Γ_n=\{λ=(λ_{1}, λ_{2}, \cdots,λ_{n})\in \mathbb{R}^n; λ_j>0,j=1,\cdots, n\}$ where $σ_{k}(λ)$ is the basic symmetric polynomial of degree $k,$ $2 \leq k \leq n.$ Then we apply the result to study the so-called $d-$concavity of the $k-$Hessian type function $F_{k}(R)=\log \left(S_{k}(R)\right),$ where $S_{k}(R)=σ_{k}(λ(R)), λ(R)= \left(λ_{1}, λ_{2}, \cdots, λ_{n}\right) \in \mathbb{C}^{n}$ is eigenvalue-vector of $R \in \mathbb{R}^{n \times n},$ $R=ω+β, ω^{T}=ω, ω>0, \quad β^{T}=-β.$ The $d-$concavity will be used in our next paper to study the existence of admissible solutions to the Dirichlet problem for the augmented nonsymmetric $k-$Hessian type equations.

math.AP↗

arXiv2018

On the existence, uniqueness and stability of $β$-viscosity solutions to a class of Hamilton-Jacobi equations in Banach spaces

This paper is concerned with the qualitative properties of viscocity solutions to a class of Hamilton-Jacobi equations (HJEs) in Banach spaces. Specifically, based on the concept of $β$-derivative \cite{DGZ93b} we establish the existence, uniqueness and stability of $β$-viscosity solutions for a class of HJEs in the form $u+H(x,u,Du)=0.$ The obtained results in this paper extend ealier works in the literature, for example, \cite{CL85}, \cite{CL86} and \cite{DGZ93b}.

math.AP↗