On capillary free surfaces in the absence of gravity. (English) Zbl 0382.76003

Summary: The authors prove various existence and non-existence theorems for solutions of the differential equation for a surface film bounded by various shapes of cylinder. Some experiments on freely falling cylinders are quoted and it is claimed that the shapes of film seen are in accordance with theory.
Reviewer: H. N. V. Temperley


76A20 Thin fluid films
76D99 Incompressible viscous fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J67 Boundary values of solutions to elliptic equations and elliptic systems
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: DOI


[1] Adam, N. K.,The Physics and Chemistry of Surfaces. London, Oxford University Press, 1941.
[2] Bernstein, S., Sur les équations du calcul des variations,Ann. Sci. École Norm. Sup., 29 (1912), 431–486. · JFM 43.0460.01
[3] Concus, P. &Finn, R., On the behavior of a capillary surface in a wedge.Proc. Nat. Acad. Sci. U.S.A., 63 (1969), 292–299. · Zbl 0219.76104
[4] –, On a class of capillary surfaces.J. Analyse Math., 23 (1970), 65–70. · Zbl 0257.76007
[5] –, On capillary free surfaces in a gravitational field.Acta Math., 132 (1974), 207–223. · Zbl 0382.76004
[6] Delaunay, C. E., Sur la surface de révolution dont la courbure moyenne est constante.J. Math. Pures Appl., 6 (1841), 309–315, see also Sturm, M.,Note a l’occasion de l’article précédent, ibid, 315–320.
[7] Finn, R., Remarks relevant to minimal surfaces, and to surfaces of prescribed mean curvature.J. Analyse Math., 14 (1965), 139–160. · Zbl 0163.34604
[8] –, A note on capillary free surfaces.Acta Math., 132 (1974), 199–205. · Zbl 0382.76004
[9] Hopf, E., Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus.Sitzungsber. preuss. Akad. Wiss. Berlin, 19 (1927), 147–152. · JFM 53.0454.02
[10] Hsiung, C. C., Some integral formulas for closed hypersurfaces.Math. Scand., 2 (1954), 286–294. · Zbl 0057.14603
[11] Leray, J., Discussion d’un problème de Dirchlet.J. Math. Pures Appl., 18 (1939), 249–284. · Zbl 0023.04502
[12] Minkowski, H., Volumen und Oberfläche,Math., Ann. 57 (1903), 447–495;Gesammelte Abhandlungen, B. G. Teubner, Leipzig und Berlin, 1911, 230–276. · JFM 34.0649.01
[13] Spruck, J., Infinite boundary value problems for surfaces of constant mean curvature.Arch. Rational Mech. Anal., 49 (1972–3), 1–31. · Zbl 0263.53008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.