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Comparison principles in capillarity

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References

  1. T. Young: An essay on the cohesion of fluids. Philos. Trans. Roy. Soc. London 95 (1805), 65–87.

    Article  Google Scholar 

  2. P.S. Laplace: Traité de mécanique céleste; suppléments au Livre X, 1805 and 1806 resp. in Oeuvres Complete Vol. 4. Gauthier-Villars, Paris; see also the annotated English translation by N. Bowditch (1839); reprinted by Chelsea, New York, 1966.

    Google Scholar 

  3. C.F. Gauss: Principia Generalia Theoriae Figurae Fluidorum. Comment. Soc. Regiae Scient. Gottingensis Rec. 7 (1830). Reprinted as "Grundlagen einer Theorie der Gestalt von Flüssigkeiten im Zustand des Gleichgewichtes", in Ostwald’s Klassiker der exakten Wissenschaften, vol. 135. W. Engelmann, Leipzig, 1903.

    Google Scholar 

  4. R. Finn: Equilibrium Capillary Surfaces. Springer-Verlag, New York, 1986. (Grundlehren der Mathem. Wiss. # 284).

    Book  MATH  Google Scholar 

  5. K. Kenmotsu: Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann. 245 /1979) 89–99.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Bakker: Kapillarität und Oberflächenspannung. In: Handbuch der Experimentalphysik, Band 6. Akademische Verlagsgesellschaft, Leipzig, 1928.

    Google Scholar 

  7. B. Taylor: Concerning the ascent of water between two glass planes. Philos. Trans. Roy. Soc. London 27 (1712), 538.

    Article  Google Scholar 

  8. F. Hauksbee: Some further experiments, showing the ascent of water between two glass planes in an hyperbolic curve. Philos. Trans. Roy. Soc. London 28 (1713), 153.

    Article  Google Scholar 

  9. Petrus van Musschenbroek: Introductio ad Philosophiam Naturalem. Tom 1. S.J. et Luchtnams, Leiden, 1762, p. 376.

    Google Scholar 

  10. A. Ferguson and I. Vogel: On the "hyperbola" method for the measurement of surface tensions. Phys. Soc. Proc. 38 (1926) 193–203.

    Google Scholar 

  11. H.M. Princen: Capillary phenomena in assemblies of parallel cylinders. II. Capillary rise in systems with more than two cylinders. J. Colloid Interface Sci. 30 (1969), 359–371.

    Article  Google Scholar 

  12. H. Minkowski: Kapillarität. Encycl. Mathem. Wiss. VI, Teubner, Leibzig, 1903–1921, pp. 559–613.

    MATH  Google Scholar 

  13. C.V. Boys: Soap Bubbles, and the Forces Which Mould Them. Society for Promoting Christian Knowledge, London 1902; revised edition, 1916. Reprinted by Doubleday & Co., Garden City, 1959.

    Google Scholar 

  14. L.-F. Tam: Regularity of capillary surfaces over domains with corners. Pac. J. Math. 124 (1986), 469–482.

    Article  MathSciNet  MATH  Google Scholar 

  15. N.J. Korevaar: On the behavior of a capillary surface at a re-entrant corner. Pacific J. Math. 88 (1980), 379–385.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Spruck: On the existence of a capillary surface with a prescribed angle of contact. Comm. Pure Appl. Math. 28 (1975), 189–200.

    Article  MathSciNet  MATH  Google Scholar 

  17. N.J. Korevaar: The normal variations technique for studying the shape of capillary surfaces. Astérisque 118 (1984), 189–195.

    MathSciNet  MATH  Google Scholar 

  18. N.J. Korevaar: An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation. Proc. Symp. Pure Math. 45 (1986) Part 2, 81–89.

    Article  MathSciNet  MATH  Google Scholar 

  19. N.J. Korevaar: Maximum principle gradient estimates for the capillary problem. To appear.

    Google Scholar 

  20. G.M. Lieberman: Gradient estimates for capillary-type problems via the maximum principle. To appear.

    Google Scholar 

  21. G.M. Lieberman: Boundary behavior of capillary surfaces via the maximum principle. In: Proc. Conf. Var. Methods, ed. P. Concus & R. Finn, Vallombroso 1985; Springer-Verlag, 1986.

    Google Scholar 

  22. Jin-Tzu Chen and W.-S. Huang: Convexity of capillary surfaces in the outer space. Invent. Math. 67 (1982), 253–259.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.-T. Chen: Uniqueness of minimal point and its location of capillary free surfaces over convex domains. Astérisque 118 (1984), 137–143.

    Google Scholar 

  24. W.-H. Huang: Level curves and minimal points of capillary surfaces over convex domains. Bull. Inst. Math. Acad. Sin. 2 (1983), 390–399.

    MathSciNet  Google Scholar 

  25. N.J. Korevaar: Capillary surface convexity above convex domains. Indiana Univ. Math. J. 32 (1983), 73–81.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Finn: Existence criteria for capillary free surfaces without gravity. Indiana Univ. Math. J. 32 (1983), 439–460.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Siegel: Height estimates for capillary surfaces. Pacific J. Math. 88 (1980), 471–516.

    Article  MathSciNet  MATH  Google Scholar 

  28. A.D. Aleksandrov: Uniqueness theorems for surfaces in the large. V. Vestnik Leningrad Univ. 13 (1958) 5–8. Amer. Math. Soc. Translations (Series 2) 21, 412–416.

    MathSciNet  MATH  Google Scholar 

  29. J.B. Serrin: A symmetrty problem in potential theory. Arch. Rational Mech. Anal. 43 (1971), 304–318.

    Article  MathSciNet  MATH  Google Scholar 

  30. H.C. Wente: The symmetry of sessile and pendent drops. Pacific. J. Math. 88 (1980), 387–397.

    Article  MathSciNet  MATH  Google Scholar 

  31. T.I. Vogel: Weak conditions for capillary surfaces. Preprint, Texas A & M University; to appear.

    Google Scholar 

  32. W.N. Bond and D.A. Newton: Bubbles, drops, and Stokes’ Law (Paper 2). Phil. Mag. Ser. 7, no. 5 (1928), 794–800.

    Google Scholar 

  33. R. Finn: The sessile liquid drop I: Symmetric Case. Pacific J. Math. 88 (1980), 541–587.

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Finn: On the Laplace formula and the meniscus height for a capillary surface. Z. Angew. Math. Mech. 61 (1981), 165–173.

    Article  MathSciNet  MATH  Google Scholar 

  35. R. Finn: Addenda to my paper "on the Laplace formula and the meniscus height for a capillary surface". Z. Angew. Math. Mech 61 (1981), 175–177.

    Article  MathSciNet  MATH  Google Scholar 

  36. D. Siegel: Height estimates for the narrow capillary tube. Preprint, New Mexico Inst. of Mining and Technology.

    Google Scholar 

  37. D. Siegel: Uniqueness of the minimum point for capillary surfaces over convex domains. To appear.

    Google Scholar 

  38. D. Siegel: The behavior of a capillary surface for small Bond number. Proc. Conf. Var. Methods, Vallombrosa 1985, ed. P. Concus & R. Finn, Springer-Verlag 1986.

    Google Scholar 

  39. L.-F. Tam: The behavior of capillary surfaces as gravity tends to zero. Comm. P.D.E. 11 (1986), 851–901.

    MathSciNet  MATH  Google Scholar 

  40. R. Finn: Moon surfaces, and boundary behavior of capillary surfaces for perfect wetting and non wetting. Preprint, Univ. Bonn 1987, to appear.

    Google Scholar 

  41. P. Concus and R. Finn: On capillary free surfaces in a gravitational field. Acta Math. 132 (1974), 207–223.

    Article  MathSciNet  MATH  Google Scholar 

  42. D. Langbein and F. Rischbieter: Form, Schwingungen und Stabilität von Flüssigkeitsgrenzflächen. Forschungsbericht BMFT, Battelle Inst. Frankfurt/M., 1986.

    Google Scholar 

  43. M. Emmer: Esistenza, unicita e regolarita nelle superfici di equilibrio nei capillari. Ann. Univ. Ferrara Sez. VII 18 (1973), 79–94.

    MathSciNet  MATH  Google Scholar 

  44. N.N. Ural’tseva: Solution of the capillary problem (Russian). Vestnik Leningrad Univ. 19 (1973), 54–64.

    Google Scholar 

  45. C. Gerhardt: Existence and regularity of capillary surfaces. Boll. Un. Mat. Ital. 10 (1974), 317–335.

    MathSciNet  MATH  Google Scholar 

  46. R. Finn and C. Gerhardt: The internal sphere condition and the capillary problem. Ann. Mat. Pura App. 112 (1977), 13–31.

    Article  MathSciNet  MATH  Google Scholar 

  47. C. Gerhardt: Boundary value problems for surfaces of prescribed mean curvature. J. Math. Pures Appl. 58 (1979), 75–109.

    MathSciNet  MATH  Google Scholar 

  48. R. Finn: Global size and shape estimates for symmetric sessile drops. J. Reine Angew. Math. 335 (1982), 9–36.

    MathSciNet  MATH  Google Scholar 

  49. R. Finn: Some comparison properties and bounds for capillary surfaces. In Complex Analysis and its Applications (Russian). Moscow Math. Soc.: volume dedicated to I.N. Vekua, Scientific Press, Moscow, 1978.

    Google Scholar 

  50. M. Emmer: On the behavior of the surfaces of equilibrium in the capillary tubes when gravity goes to zero. Rend. Sem. Mat. Univ. Padova 65 (1981), 143–162.

    MathSciNet  MATH  Google Scholar 

  51. P. Concus and R. Finn: The shape of a pendent liquid drop. Philos. Trans. Roy. Soc. London Ser. A 292 (1979), 307–340.

    Article  MathSciNet  MATH  Google Scholar 

  52. P. Concus and R. Finn: A singular solution of the capillary equation. I. Existence. Invent. Math. 29 (1975), 143–148.

    Article  MathSciNet  MATH  Google Scholar 

  53. P. Concus and R. Finn: A singular solution of the capillary equation. II: Uniqueness. Invent. Math. 29 (1975), 149–160.

    Article  MathSciNet  MATH  Google Scholar 

  54. M.-F. Bidaut-Veron: Global existence and uniqueness results for singular solutions of the capillarity equation. Pacific. J. Math., 125 (1986) 317–334.

    Article  MathSciNet  MATH  Google Scholar 

  55. M.-F. Bidaut-Veron: New results concerning the singular solutions of the capillarity equation. Proc. Conf. Var. Methods, Vallombrosa, 1985, ed. P. Concus & R. Finn, Springer-Verlag 1986.

    Google Scholar 

  56. J.B. Serrin: The Dirichlet problem for surfaces of constant mean curvature. Proc. Lon. Math. Soc. (3) 21 (1970), 361–384.

    Article  MathSciNet  MATH  Google Scholar 

  57. O.A. Ladyzhenskaya, N.N. Ural’tseva: Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations. Comm. Pure Appl. Math. 23 (1970), 677–703.

    Article  MathSciNet  MATH  Google Scholar 

  58. E. Heinz: Interior gradient estimates for surfaces z=f(x,y) of prescribed mean curvature. J. Diff. Geom. 5 (1971), 149–157.

    MathSciNet  MATH  Google Scholar 

  59. R. Finn: On equations of minimal surface type. Annals of Math. 60 (1954), 397–416, esp. p. 397.

    Article  MathSciNet  MATH  Google Scholar 

  60. R. Finn and E. Giusti: Nonexistence and existence of capillary surfaces. Manuscripta Math. 28 (1979), 13–20.

    Article  MathSciNet  MATH  Google Scholar 

  61. Fei-tsen Liang: On nonparametric surfaces of constant mean curvature. Dissertation, Stanford University, 1986; to appear.

    Google Scholar 

  62. R. Finn: New estimates for equations of minimal surface type. Arch. Rat. Mech. Anal. 14 (1963), 337–375.

    Article  MathSciNet  MATH  Google Scholar 

  63. R.D. Gulliver, II: Regularity of minimizing surfaces of prescribed mean curvature. Annals of Math. 97 (1973), 275–305.

    Article  MathSciNet  MATH  Google Scholar 

  64. J. Spruck: Infinite boundary value problems for surfaces of constant mean curvature. Arch. Rat. Mech. Anal. 49 (1972/3), 1–31.

    Article  MathSciNet  MATH  Google Scholar 

  65. W.H. Fleming: On the oriented Plateau problem. Rend. Palermo 11 (1962), 69–90.

    Article  MathSciNet  MATH  Google Scholar 

  66. J.B. Serrin: A priori estimates for solutions of the minimal surface equation. Arch. Rat. Mech. Anal. 14 (1963), 376–383.

    Article  MathSciNet  MATH  Google Scholar 

  67. R. Finn: On partial differential equations whose solutions admit no isolated singularities. Scripta Math. 26 (1961), 107–115.

    MathSciNet  MATH  Google Scholar 

  68. L. Bers: Isolated singularities of minimal surfaces. Ann. of Math. 53 (1951), 364–386.

    Article  MathSciNet  MATH  Google Scholar 

  69. M.-F. Bidaut-Veron: A singular solution of a quasilinear partial differential equation. Technical report, Univ. Tours, 1986.

    Google Scholar 

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Stefan Hildebrandt Rolf Leis

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Finn, R. (1988). Comparison principles in capillarity. In: Hildebrandt, S., Leis, R. (eds) Partial Differential Equations and Calculus of Variations. Lecture Notes in Mathematics, vol 1357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082866

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