On the Dirichlet Problems for Symmetric Function Equations of the Eigenvalues of the Complex Hessian

Also Γk is symmetric in λ = (λ1, · · · , λn), which means that if λ = (λ1, · · · , λn) ∈ Γk, then λ̃ = (λi1 , · · · , λin) ∈ Γk where (i1, i2, · · · , in) is any permutation of 1, 2, · · · , n. We say that u is plurisubharmonic in D if λ(H(u)(z)) ∈ Γn, for all z ∈ D; we say that u is subharmonic in D if λ(H(u)(z)) ∈ Γ1 for all z ∈ D. We will let Γ be a convex cone which is symmetric in λ ∈ Γ, with vertex 0 so that Γn ⊆ Γ ⊆ Γ1. Let M(n,Γ) be subset of all n × n hermitian matices H over C so that λ(H) ∈ Γ where λ(H) is a vector in IR being formed by all eigenvalues of H. We will consider more general symmetric function than the kth symmetric function σ on M(n,Γ). Let D be a bounded domain in C with smooth boundary ∂D. We say that a real-valued function u is Γ-subharmonic if λ(H(u)(z)) ∈ Γ for all z ∈ D. we will consider the Dirichlet problem for a functional equation: