Bulletin of the Australian Mathematical Society

Research Article

Interior regularity of the degenerate Monge-Ampère equation

Zbigniew Błockia1

a1 Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland e-mail: blocki@im.uj.edu.pl

Abstract

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 xs211Dn with u = xs03D5 on ∂Ω. We prove in particular that u xs2208 C1,1(Ω) if either i) xs03D5 = 0 and ψ1/(n − 1) xs2208 C1,1 (Ω) or ii) Ω is C1,1 strongly convex, xs03D5 xs2208 C1,1 (), ψ1/(n − 1) xs2208 C1,1() and ψ > 0 on U xs2229 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)