Boundary Estimates for solutions of the Monge-Ampere equation satisfying Dirichlet-Neumann type conditions in annular domains

TL;DR

This paper derives global second-derivative estimates for smooth solutions to the Monge-Ampère equation with mixed boundary conditions on annular domains bounded by convex hypersurfaces, advancing understanding of boundary behavior in nonlinear PDEs.

Contribution

It provides the first global $C^2$ estimates for solutions of the Monge-Ampère equation with Dirichlet-Neumann boundary conditions in annular domains.

Findings

Established global $C^2$ estimates for solutions

Extended boundary regularity results to annular domains

Enhanced understanding of boundary behavior in Monge-Ampère equations

Abstract

We consider smooth solutions to the Monge-Amp`ere equation subject to mixed boundary conditions on annular domains. We establish global $C^2$ estimates when the boundary of the domain consists of two smooth strictly convex closed hypersurfaces.