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MINIMAL GRAPHS WITH DISCONTINUOUS BOUNDARY VALUES

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  01 February 2009

ROBERT HUFF  and
JOHN MCCUAN
ROBERT HUFF*
Affiliation:
Department of Mathematical Sciences, Indiana University, South Bend, IN 46634, USA (email: [email protected])
JOHN MCCUAN
Affiliation:
Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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We construct global solutions of the minimal surface equation over certain smooth annular domains and over the domain exterior to certain smooth simple closed curves. Each resulting minimal graph has an isolated jump discontinuity on the inner boundary component which, at least in some cases, is shown to have nonvanishing curvature.

Keywords

minimal surface equationWeierstrass representationextremal lengthboundary regularity

MSC classification

Secondary: 35J67: Boundary values of solutions to elliptic equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 1 , February 2009 , pp. 75 - 95
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Ahlfors, L., Conformal Invariants: Topics in Geometric Function Theory (McGraw-Hill, New York, 1973).Google Scholar
[2]Ekeland, I. and Témam, R., Convex Analysis and Variational Problems (SIAM, Philadelphia, PA, 1999).CrossRefGoogle Scholar
[3]Finn, R., ‘New estimates for equations of minimal surface type’, Arch. Ration. Mech. Anal. 14 (1963), 337–375.CrossRefGoogle Scholar
[4]Giusti, E., Minimal Surfaces and Functions of Bounded Variation (Birkhaüser, Boston, MA, 1984).CrossRefGoogle Scholar
[5]Hoffman, D. and Karcher, H., ‘Complete embedded minimal surfaces of finite total curvature’, in: Encyclopedia of Mathematics (ed. R. Osserman) (Springer, Berlin, 1997), pp. 5–93.Google Scholar
[6]Jenkins, H. and Serrin, J., ‘The Dirichlet problem for the minimal surface equation in higher dimensions’, J. Reine Angew. Math. 229 (1968), 170–187.Google Scholar
[7]Lancaster, K., ‘A proof of the Concus-Finn conjecture’, Preprint, 2006.Google Scholar
[8]Miranda, M., ‘Un principio di massimo forte per le frontiere minimali e una sua applicazione alla risoluzione del problema al contorno per l’equazione delle superfici di area minima’, Rend. Sem. Mat. Univ. Padova 45 (1971), 355–366.Google Scholar
[9]Simon, L., ‘Global estimates of Hölder continuity for a class of divergence-form elliptic equations’, Arch. Ration. Mech. Anal. 56 (1974), 253–272.CrossRefGoogle Scholar
[10]Simon, L., ‘Boundary behavior of solutions of the nonparametric least area problem’, Bull. Austral. Math. Soc. 26 (1982), 17–27.CrossRefGoogle Scholar
[11]Williams, G., ‘The Dirichlet problem for the minimal surface equation with Lipschitz continuous boundary data’, J. Reine Angew. Math. 354 (1984), 123–140.Google Scholar
[12]Williams, G., ‘Global regularity for solutions of the minimal surface equation with continuous boundary values’, Ann. Inst. H. Poincaré (C) Anal. Non Linéaire 3(6) (1986), 411–429.CrossRefGoogle Scholar
[13]Williams, G., ‘Solutions of the minimal surface equation-continuous and discontinuous at the boundary’, Comm. Partial Differential Equations 11 (1986), 1439–1457.CrossRefGoogle Scholar
[14]Williams, G., ‘The best modulus of continuity for solutions of the minimal surface equation’, Pacific J. Math. 129 (1987), 193–208.CrossRefGoogle Scholar