On the Dirichlet problem for general augmented Hessian equations

TL;DR

This paper extends boundary value problem techniques to a broad class of augmented Hessian equations, providing new derivative estimates and barrier methods for Dirichlet problems in Euclidean space.

Contribution

It generalizes previous results on Monge-Ampère and k-Hessian equations to a wider class of augmented Hessian equations using advanced derivative estimates and barrier constructions.

Findings

Established first and second derivative estimates for augmented Hessian equations.

Developed barrier functions suitable for Dirichlet boundary conditions.

Extended existing methods to new classes of equations in Euclidean space.

Abstract

In this paper we apply various first and second derivative estimates and barrier constructions from our treatment of oblique boundary value problems for augmented Hessian equations, to the case of Dirichlet boundary conditions. As a result we extend our previous results on the Monge-Ampere and k-Hessian cases to general classes of augmented Hessian equations in Euclidean space