Communications in Mathematical Sciences

Volume 13 (2015)

Number 2

Smooth approximations of the Aleksandrov solution of the Monge-Ampère equation

Pages: 427 – 441

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n2.a8

Author

Gerard Awanou (Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Il., U.S.A.)

Abstract

We prove the existence of piecewise polynomial, strictly convex, smooth functions which converge uniformly on compact subsets to the Aleksandrov solution of the Monge-Ampère equation. We extend the Aleksandrov theory to the case of a right hand side which is only locally integrable and to convex bounded domains which are not necessarily strictly convex. The result suggests that for the numerical resolution of the equation, it is enough to assume that the solution is convex and piecewise smooth.

Keywords

Aleksandrov solution, Monge-Ampère, weak convergence of measures, convexity, finite elements

2010 Mathematics Subject Classification

35J96, 65N30

Published 3 December 2014