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

A Pohožaev Identity and Critical Exponents of Some Complex Hessian Equations

“…[6]). For the recent study of complex Hessian equations, see [3], [10], [19] and the reference therein.…”

Weak solutions to the complex Hessian equation

“…The proof relies heavily on an a priori estimate for a special case of the complex Hessian equation which is presented in Section 4. We remark that a much more general estimate and a solution of the Dirichlet problem for non-degenerate equation was independently shown in [20] with essentially the same methods as below. Section 5 is devoted to the characterization of the class D m .…”

“…Of course many results could be generalized here to more general complex Hessian operators. The existence of strong solutions in domains in C n for such equations was recently proved in [20]. In particular, one could study the operator (dd c u) m ∧ ω n−m on manifolds, where ω is an arbitrary Kähler form.…”

A second order estimate for complex Hessian equations on a compact Kähler manifold

“…As we know, the Dirichlet problem of complex Hessian equation in C n was studied by S. Y. Li [10] and B locki [1]. The first named author [8] proved the uniqueness of the solution of (1.2); he [8] also proved the existence of a smooth admissible solution of (1.2), by assuming the nonnegativity of the orthogonal bisectional curvature of ω.…”