Oblique boundary value problems for augmented Hessian equations I

TL;DR

This paper develops a comprehensive theory for the regularity of solutions to oblique boundary value problems involving augmented Hessian equations, covering various applications in geometry and optimal transportation.

Contribution

It introduces new global a priori estimates for classical solutions to a broad class of augmented Hessian boundary value problems under convexity conditions.

Findings

Established global second-order derivative estimates.

Unified framework for multiple geometric and PDE applications.

Extended regularity results to degenerate equations.

Abstract

In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge-Amp`ere type operators in optimal transportation and geometric optics, the general theory here embraces prescribed mean curvature problems in conformal geometry as well as oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations.