An approximation result for solutions of Hessian equations

We show that W (2,p) weak solutions of the k-Hessian equation F (k) (D(2)u) = g(x) with k >= 2 can be approximated by smooth k-convex solutions v (j) of similar equations with the right hands sides controlled uniformly in C-0,C-1 norm, and so that the quantities f(Br)(Delta v(j))F-p-k+1(k-1)(D(2)v(j)) are bounded independently of j. This result simplifies the proof of previous interior regularity results for solutions of such equations. It also permits us to extend certain estimates for smooth solutions of degenerate two dimensional Monge-Ampere equations to W-2,W-p solutions.