SMOOTH APPROXIMATIONS OF THE ALEKSANDROV SOLUTION OF THE MONGE-AMPERE EQUATION

We prove the existence of piecewise polynomial, strictly convex, smooth functions which converge uniformly on compact subsets to the Aleksandrov solution of the Monge-Ampere 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.