Convergence of a fourth order singular perturbation of the $n$-dimensional radially symmetric Monge-Ampère equation

Abstract

This paper concerns with the convergence analysis of a fourth order singular perturbation of the Dirichlet Monge-Ampère 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-Ampère 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.

26 pages and 6 figures. arXiv admin note: substantial text overlap with arXiv:1109.1183

References in corpus (1)

• Smoothness of radial solutions to Monge-Ampere equations