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Smoothness of the solution of the trivial Monge-Ampère equation

Published online by Cambridge University Press:  17 April 2009

Vitaly Ushakov
Vitaly Ushakov
Affiliation:
Department of MathematicsThe University of MelbourneParkville Vic 3052Australia e-mail: [email protected]
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Abstract

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A solution of the equation must be C2 smooth, need not be C3 smooth, but at the same time the ratio zxy: zxx is C1 (provided zxx ≠ 0). An analytical proof of this fact is given; the underlying geometrical interpretation is discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 439 - 445
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Courant, R. and Hilbert, D., Methods of mathematical physics II, Partial differential equations (Interscience publishers, New York, 1962).Google Scholar
[2]Hadamard, J., Cours d'analyse I (Herman, Paris, 1927).Google Scholar
[3]Hartman, P. and Nirenberg, L., ‘On spherical image maps whose Jacobians do not change sign’, Amer. J. Math. 81 (1959), 901–920.CrossRefGoogle Scholar
[4]Hartman, P. and Wintner, A., ‘On the fundamental equations of differential geometry’, Amer. J. Math. 72 (1950), 757–774.CrossRefGoogle Scholar
[5]Hartman, P. and Wintner, A., ‘On the asymptotic curves of a surface’, Amer. J. Math. 73 (1951), 149–172.CrossRefGoogle Scholar
[6]Ushakov, V., Riemannian manifolds and surfaces of constant nullity (in Russian), Thesis (St. Petersburg University, 1993).Google Scholar
[7]Ushakov, V., ‘Parametrization of developable surfaces by asymptotic lines’, Bull. Austral. Math. Soc. 54 (1996), 411–421.CrossRefGoogle Scholar
[8]Ushakov, V., ‘The explicit general solution of trivial Monge-Ampère equation’, (preprint, University of Melbourne, 1996).Google Scholar