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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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References

1Aleksandrov, A. D.. Die innere Geometrie der konvexen Flächen (Berlin: Akademie, 1955).Google Scholar
2Cheng, S.-Y. and Yau, S.-T.. On the regularity of the Monge-Ampere equation det (∂2u/∂x i ∂x j) = F(x, u). Comm. Pure Appl. Math. 30 (1977), 41–68.CrossRefGoogle Scholar
3Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order, 2nd edn (Berlin: Springer, 1983).Google Scholar
4Guisti, E.. On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions. Invent. Math. 46 (1978), 111–137.CrossRefGoogle Scholar
5Pogorelov, A. V.. The Dirichlet problem for the n-dimensional analogue of the Monge-Ampere equation. Dokl. Akad. Nauk SSSR 201 (1971), 790–793 (in Russian). English translation. Soviet Math. Dokl. 12 (1971), 1727-1731.Google Scholar
6Pogorelov, A. V.. The Minkowski multidimensional problem (New York: J. Wiley, 1978).Google Scholar
7Trudinger, N. S. and Urbas, J. I. E.. On second derivative estimates for equations of Monge-Ampère type. Bull. Austral. Math. Soc. 30 (1984), 321–334.CrossRefGoogle Scholar
8Urbas, J. I. E.. The equation of prescribed Gauss curvature without boundary conditions. J. Differential Geometry 20 (1984), 311–327.CrossRefGoogle Scholar
9Urbas, J. I. E.. The generalized Dirichlet problem for equations of Monge-Ampère type. Ann. Inst. H. Poincare–Analyse Non Linéaire 3 (1986), 209–228.CrossRefGoogle Scholar
10Urbas, J. I. E.. Global Holder estimates for equations of Monge-Ampere type. Invent. Math. 91 (1988), 1–29.CrossRefGoogle Scholar
11Urbas, J. I. E.. Regularity of generalized solutions of Monge-Ampere equations. Math. Zeit. 197 (1988), 365–393.CrossRefGoogle Scholar
12Urbas, J. I. E.. On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations. Indiana Univ. Math. J. 39 (1990), 355–382.CrossRefGoogle Scholar