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manuscripta mathematica

, Volume 28, Issue 1–3, pp 1–11 | Cite as

Existence and non existence of capillary surfaces

  • Robert Finn
Article

Abstract

A general criterion for existence of solutions of the capillary equation, introduced by Concus and Finn [1] and by Giusti [7], is shown to be equivalent to the question of existence of certain vector fields. The result is applied to particular boundary configurations, and it is shown that in some cases the local “corner condition” of [1] is both necessary and sufficient in the global configuration. In other situations a different kind of unstable dependence on the boundary geometry appears, that could not have been predicted by previous results.

Keywords

Vector Field Number Theory Algebraic Geometry Topological Group General Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    CONCUS, P., and R. FINN: On capillary free surfaces in the absence of gravity. Acta Math.132, 177–198 (1974)Google Scholar
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    DeGIORGI, E., and G. STAMPACCHIA: Sulle singolarità eliminabili delle ipersuperficie minimali. Rend. Accad. Naz. Lincei 38,352–357 (1965)Google Scholar
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    FINN, R.: Isolated singularities of solutions of nonlinear partial differential equations. Trans. Amer. Math. Soc. 75, 385–404 (1953)Google Scholar
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    FINN, R.: Capillarity Phenomena. Uspechi Math. Nauk 29, 131–152 (1974)Google Scholar
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    FINN, R. and E. GIUSTI: Non existence and existence of capillary surfaces. Manuscripta Math. followingGoogle Scholar
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    GIUSTI, E.: Boundary value problems for nonparametric surfaces of prescribed mean curvature. Ann. Scuola Norm Sup. Pisa 3, 501–548 (1976)Google Scholar
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    GIUSTI, E.: On the equation of surfaces of prescribed mean curvature-existence and uniqueness without boundary condition, to appearGoogle Scholar
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    NITSCHE, J. C. C.: Über ein verallgemeinertes Dirichletsches Problem für die Minimalflächengleichung und hebbare Unstetigkeiten ihrer Lösungen. Math. Ann. 158, 203–214 (1965)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Robert Finn
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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