Asymptotic expansion at infinity of solutions to Monge-Amp\`{e}re equation with $C^\alpha$ right term

TL;DR

This paper introduces a non-local method to analyze the asymptotic behavior at infinity of solutions to the Monge-Ampère equation with a Hölder continuous right-hand side, extending previous results that required higher regularity.

Contribution

It develops a novel non-local approach to handle solutions with less regular right-hand side functions, specifically those that are only Hölder continuous outside a bounded region.

Findings

Established asymptotic expansion at infinity for solutions with $C^eta$ right term

Extended previous results to cases with less regularity in the right-hand side

Provided a new analytical framework for Monge-Ampère equations with Hölder continuous data

Abstract

We develop a non-local method to establish the asymptotic expansion at infinity of solutions to Monge-Amp\`{e}re equation $\det(D^2v)=f$ on $\rn$, where $f$ is a perturbation of $1$ and is only assumed to be H\"{o}lder continuous outside a bounded subset of $\rn$, compared to the previous work that $f$ is at least $C^2$.