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Boundary behaviour of solutions of the non-parametric least area problem

Published online by Cambridge University Press:  17 April 2009

Leon Simon
Leon Simon
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, PO Box 4, Canberra, ACT 2600, Australia.
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Abstract

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Previous work concerning boundary regularity of solutions of the non-parametric least area problem leaves open the question of regularity of solutions at points where the mean curvature of the boundary of the domain vanishes. We here prove that the solutions may be discontinuous at such points, even when the given boundary data is smooth. We also give a sufficient condition which will ensure continuity at such points.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 26 , Issue 1 , August 1982 , pp. 17 - 27
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Giusti, Enrico, “Superfici cartesiane di area minima”, Rend. Sent. Mat. Fis. Milano 40 (1970), 135–153.CrossRefGoogle Scholar
[2]Giusti, Enrico, “Boundary behavior of non-parametric minimal surfaces”, Indiana Univ. Math. J. 22 (1972), 435–444.CrossRefGoogle Scholar
[3]Serrin, J., “The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables”, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413–496.Google Scholar
[4]Simon, Leon, “Boundary regularity for solutions of the non-parametric least area problem”, Ann of Math. (2) 103 (1976), 429–455.CrossRefGoogle Scholar
[5]Simon, Leon, “Global estimates of Hölder continuity for a class of divergence-form elliptic equations”, Arch. Rational Mech. Anal. 56 (1974), 253–272.CrossRefGoogle Scholar
[6]Trudinger, Neil S., “The boundary gradient estimate for quasilinear elliptic and parabolic differential equations”, Indiana Univ. Math. J. 21 (1972), 657–670.CrossRefGoogle Scholar