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Interior regularity of the degenerate Monge-Ampère equation

Published online by Cambridge University Press:  17 April 2009

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

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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 ℝn with u = ϕ on ∂Ω. We prove in particular that uC1,1(Ω) if either i) ϕ = 0 and ψ1/(n − 1) ∈ C1,1 (Ω) or ii) Ω is C1,1 strongly convex, ϕ ∈ 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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2003

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