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

Abstract

A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian-type equations on domains in ${ℂ}^n$. The arguments are modeled on the theory of viscosity solutions for real Hessian-type equations developed by Trudinger (1990). As a consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.