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 Hk-norm for k = 0, 1, 2 are also established.
AMS Subject Classifications:
Acknowledgements
The work of this author was partially supported by the NSF grant numbers DMS-1016173 and DMS-1238711.