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

We establish a Schauder-type boundary regularity result for a two-dimensional singular Monge--Ampère equation on convex polytopes subject to the Guillemin boundary condition.

Our result extends the work of Rubin and Huang to the case where the right-hand side is merely Hölder continuous.

In particular, we obtain Euclidean \(C^{1,\alpha/2}\) regularity up to the boundary, including along edges and at the vertices of the polytope, and then refine this to the sharp Euclidean \(C^{1,\alpha}\) regularity.

The analysis combines techniques introduced by Donaldson in his study of the Abreu equation with refined blow-up and localization arguments adapted to the degenerate boundary geometry.