a1 Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland e-mail: firstname.lastname@example.org
We study interior C1,1 regularity of generalised solutions of the Monge-Ampére equation det D2u = ψ, ψ ≥ 0, on a bounded convex domain Ω in n with u = on ∂Ω. We prove in particular that u C1,1(Ω) if either i) = 0 and ψ1/(n − 1) C1,1 (Ω) or ii) Ω is C1,1 strongly convex, C1,1 (), ψ1/(n − 1) C1,1() and ψ > 0 on U Ω, where U is a neighbourhood of ∂Ω. The main tool is an improvement of Pogorelov's well known C1,1 estimate so that it can be applied to the degenerate case.
(Received December 05 2002)