Boundary Estimates for the Monge-Amp\`ere Equation in the Polygons with Guillemin Boundary Conditions

TL;DR

This paper proves boundary regularity results for a singular Monge-Ampère equation on convex polygons with Guillemin boundary conditions, extending previous work to less regular right-hand sides using advanced Donaldson techniques.

Contribution

It extends Schauder boundary regularity results to cases with Hölder continuous right-hand sides for the Monge-Ampère equation on polygons, utilizing Donaldson's methods.

Findings

Established Schauder-type boundary regularity for Hölder continuous data.

Extended previous regularity results to less regular right-hand sides.

Utilized Donaldson's techniques for the Abreu equation in a new context.

Abstract

We establish a Schauder-type boundary regularity result for a two-dimensional singular Monge-Amp\'ere equation on convex polytopes with Guillemin boundary conditions. This extends the previous work of Rubin and Huang to the case where the right-hand side is less regular; specifically, H\"older continuous functions. Our method relies heavily on the sophisticated techniques developed by Donaldson in his series of papers on the Abreu equation.