A viscosity approach to the Dirichlet problem for degenerate complex Hessian type equations

A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in ${\text{C}}^n$. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.