Optimal asymptotic expansion of entire solutions to Monge-Amp\`{e}re equation with $C^\alpha$ perturbed periodic data

TL;DR

This paper investigates the asymptotic behavior of solutions to the Monge-Ampère equation with $C^eta$ perturbed periodic data, showing that solutions differ from a quadratic polynomial by a periodic function at infinity.

Contribution

It extends previous results by establishing the asymptotic expansion for solutions with only Hölder continuous perturbations of periodic data, using a nonlocal method.

Findings

Solutions differ from quadratic polynomials by a periodic function at infinity.

The asymptotic expansion holds under weaker regularity assumptions on $f$.

Nonlocal methods are effective in analyzing asymptotic behavior.

Abstract

We consider the asymptotic behavior at infinity of solution $u$ to Monge-Amp\`{e}re equation $\det(D^2u)=f$ in $\rn$, where $f$ is a perturbation of a periodic function and is only assumed to be H\"{o}lder continuous, compared to the previous work that $f$ is at least $C^{1,\az}$. The consequence established in this paper, by a nonlocal method, is that the difference between $u$ and a quadratic polynomial is asymptotically close to a periodic function.