论文标题
增强非对称$ k-$ hessian类型方程的可接受解决方案I
Admissible solutions to augmented nonsymmetric $k-$Hessian type equations I. The $d-$concavity of the $k-$Hessian type functions
论文作者
论文摘要
我们以$ 2 \ le k \ le n-1 $建立功能的严格凹度$ f_k（λ）= \ log（σ_k（λ））$在正锥$umγ_n= \ {λ_=（λ_{1}，λ_{2}，λ_{2}，\ cdots，λ_} n pastem的子集上λ_j> 0，j = 1，\ cdots，n \} $其中$σ_{k}（λ）$是度$ k的基本对称的多项式，$ 2 \ $ 2 \ leq k \ leqn。$，然后我们将结果应用结果来研究所谓的$ k- $ d- $ k- $ k- $ k- $ b-og $ k-$ $ hess类型$ f _} \左（s_ {k}（r）\右），$其中$ s_ {k}（r）=σ_{k}（λ（r）），λ（r）= \ left（λ_{1}，λ__{1}，λ__{2}，\ cdots，\ cdots，cdots，λ__{n} \ right） $ r \ in \ mathbb {r}^{n \ times n}的特征值 - 向量，$ $ r =ω+β，ω+β，ω^{t} =ω> 0，\quadβ^{tβ^{t} = - t} = - β。非对称$ k-$ hessian类型方程。
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.
