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

TL;DR

This paper introduces a viscosity solution framework for complex Hessian equations, solving the Dirichlet problem and establishing regularity results, extending real Hessian theory to complex settings.

Contribution

It develops a viscosity approach for complex Hessian equations, enabling solutions to the Dirichlet problem and regularity analysis, which was not previously available.

Findings

Solved the Dirichlet problem for Hessian quotient and special Lagrangian equations.

Established basic regularity results for solutions.

Extended viscosity solution theory to complex Hessian 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 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.