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Regularity of almost extremal solutions of Monge–Ampère equations

Published online by Cambridge University Press:  14 November 2011

John I. E. Urbas
John I. E. Urbas
Affiliation:
Centre for Mathematical Analysis, Australian National University, GPO Box 4, Canberra ACT 2601, Australia
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Synopsis

We show that for a large class of Monge-Ampère equations, generalised solutions on a uniformly convex domain Ω⊂ℝn are classical solutions on any pre-assigned subdomain Ω′⋐Ω, provided the solution is almost extremal in a suitable sense. Alternatively, classical regularity holds on subdomains of Ω which are sufficiently distant from ∂Ω. We also show that classical regularity may fail to hold near ∂Ω in the nonextremal case. The main example of the class of equations considered is the equation of prescribed Gauss curvature.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 1-2 , 1991 , pp. 21 - 29
Copyright
Copyright © Royal Society of Edinburgh 1991

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