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

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.