Oblique boundary value problems for augmneted Hessian equations II

TL;DR

This paper advances the theory of oblique boundary value problems for augmented Hessian equations by establishing global second derivative estimates and applying these results to prove existence theorems, including standard Hessian equations.

Contribution

It introduces a new global barrier function applicable under regularity conditions and extends previous work to less restrictive cases, improving the understanding of boundary value problems for augmented Hessian equations.

Findings

Established global second derivative estimates

Constructed a universal barrier function

Proved existence theorems for augmented Hessian equations

Abstract

In this paper, we continue our investigations into the global theory of oblique boundary value problems for augmented Hessian equations. We construct a global barrier function in terms of an admissible function in a uniform way when the matrix function in the augmented Hessian is only assumed regular. This enables us to derive global second derivative estimates in terms of boundary estimates which are then obtained by strengthening the concavity or monotonicity conditions in our previous work on the strictly regular case. Finally we give some applications to existence theorems which embrace standard Hessian equations as special cases.