Convergence of a fourth order singular perturbation of the $n$-dimensional radially symmetric Monge-Amp\`ere equation

TL;DR

This paper analyzes the convergence of a fourth order singular perturbation of the radially symmetric Monge-Ampère equation, establishing uniform convergence to the convex solution and providing convergence rates with numerical validation.

Contribution

It provides a detailed convergence analysis of a fourth order regularization of the Monge-Ampère problem, including a priori estimates, convexity preservation, and convergence rates.

Findings

Uniform convergence of the perturbed solution to the Monge-Ampère solution.

Explicit convergence rates in the $H^k$-norm for $k=0,1,2$.

Numerical experiments confirming theoretical results.

Abstract

This paper concerns with the convergence analysis of a fourth order singular perturbation of the Dirichlet Monge-Amp\`ere problem in the $n$-dimensional radial symmetric case. A detailed study of the fourth order problem is presented. In particular, various {\em a priori} estimates with explicit dependence on the perturbation parameter $\vepsi$ are derived, and a crucial convexity property is also proved for the solution of the fourth order problem. Using these estimates and the convexity property, we prove that the solution of the perturbed problem converges uniformly and compactly to the unique convex viscosity solution of the Dirichlet Monge-Amp\`ere problem. Rates of convergence in the $H^k$-norm for $k=0,1,2$ are established, and illustrating numerical experiment results are also presented in the paper.