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 a priori estimates with explicit dependence on the perturbation parameter ε 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 also established.

Keywords:

• Monge–Ampère equation
• singular perturbation
• vanishing moment method
• convergence and rate of convergence

AMS Subject Classifications:

• Primary 35J96
• Secondary 35J40

Acknowledgements

The work of this author was partially supported by the NSF grant numbers DMS-1016173 and DMS-1238711.