Asymptotic behavior of solutions to the Monge--Amp\`ere equations with slow convergence rate at infinity

TL;DR

This paper investigates the asymptotic behavior of solutions to the Monge--Ampère equations with slow convergence at infinity, extending previous results by analyzing the limits of the Hessian and gradient based on convergence rates.

Contribution

It introduces a revised level set method, spherical harmonic expansion, and iteration techniques to analyze solutions with slow convergence, providing new insights into their asymptotic limits.

Findings

Determined the limits of Hessian and gradient at infinity based on convergence rate.

Extended previous results to cases with slower convergence.

Developed new analytical methods for asymptotic analysis.

Abstract

We consider the asymptotic behavior of solutions to the Monge--Amp\`ere equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao--Li--Zhang [Calc. Var PDE. 52(2015). pp. 39-63]. Different from known results, we obtain the limit of Hessian and/or gradient of solution at infinity relying on the convergence rate. The basic idea is to use a revised level set method, the spherical harmonic expansion and the iteration method.