Skip to main content
SpringerLink
Log in

Minimizers of Anisotropic Surface Tensions Under Gravity: Higher Dimensions via Symmetrization

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We consider a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques, we establish the existence, convexity and symmetry of minimizers for a class of surface tensions admissible to the symmetrization procedure. In the case of smooth surface tensions, we obtain the uniqueness of minimizers via an ODE characterization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alvino A., Ferone V., Trombetti G., Lions P.: Convex symmetrization and applications. Ann. Inst. Henri Poincare (C) Nonlinear Anal. 14(2), 275–293 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, Oxford University Press, New York (2000)

  3. Avron J.E., Taylor J.E., Zia R.K.P.: Equilibrium shapes of crystals in a gravitational field: crystals on a table. J. Stat. Phys. 33, 493–522 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  4. Barchiesi M., Cagnetti F., Fusco N.: Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15(4), 1245–1278 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caffarelli, L., Mellet, A.: Capillary drops on an inhmogeneous surface. Perspectives in Nonlinear Partial Differential Equations. Contemporary Mathematics, Vol. 446. American Mathematical Society, Providence, 175–201, 2007

  6. Caffarelli L., Mellet A.: Capillary drops: contact angle hysteresis and sticking drops. Calc. Var. Partial Differ. Equ. 29(1–3), 431–452 (2007)

    MathSciNet  Google Scholar 

  7. Cerf, R.: The Wulff Crystal in Ising and Percolation Models. Lecture Notes in Mathematics, Vol. 1878. Springer, Berlin, 2006

  8. Chlebik M., Cianchi A., Fusco N.: The perimeter inequality under Steiner symmetrization: cases of equality. Ann. Math. 162, 525–555 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. De Giorgi, E.: Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita. (Italian) Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fisc. Mat. Nat. Sez. I (8) 5, 33–44 (1958)

  10. Elcrat A., Treinen R.: Floating drops and functions of bounded variation. Complex Anal. Oper. Theory 5(1), 299–311 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Elcrat A., Kim T.-E.: Treinen R.: Annular capillary surfaces. Arch. Math. (Basel) 82(5), 449–467 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Figalli A., Maggi F.: On the shape of liquid drops and crystals in the small mass regime. Arch. Ration. Mech. Anal. 201(1), 143–207 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Finn R.: The sessile liquid drop. I. Symmetric case. Pac. J. Math. 88(2), 541–587 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Finn, R.: Equilibrium Capillary Surfaces. Grundlehren der mathematischen Wissenschaften, Vol. 284. Springer. New York, 1986

  15. Fonseca I.: The Wulff theorem revisited. Proc. Roy. Soc. Lond. Ser. A 432(1884), 125–145 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Fonseca I., Müller S.: A uniquness proof for the Wulff theorem. Proc. Roy. Soc. Edinb. Sect. A. 119, 125–136 (1991)

    Article  MATH  Google Scholar 

  17. Fusco N.: The classical isoperimetric theorem. Rend. Acad. Sci. Fis. Mat. Napoli 4(71), 63–107 (2004)

    MathSciNet  Google Scholar 

  18. Gonzalez E.: Sul problema della goccia appoggiata. Rend. Sem. Mat. Univ. Padova 55, 289–302 (1976)

    MathSciNet  Google Scholar 

  19. Gonzalez E.: Regularity of the problem of the supported drop. Rend. Sem. Mat. Univ. Padova 58, 25–33 (1977)

    MathSciNet  MATH  Google Scholar 

  20. Gonzalez, E., Tamanini, I.: Convexity of the supported drop. Rend. Sem. Mat. Univ. Padova 58, 35–43 (1977)

  21. Gonzalez, E., Massari, U., Tamanini, I.: Existence and regularity for the problem of a pendent liquid drop. Pac. J. Math. 88(2), 399–420 (1980)

  22. Koiso, M., Palmer, B.: Geometry and stability of bubbles with gravity. Indiana Univ. Math. J. 54(1), 65–98 (2005)

  23. Koiso M., Palmer B.: Anisotropic capillary surfaces with wetting energy. Calc. Var. Par. Differ. Equ. 29(3), 295–345 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Koiso M., Palmer B.: A uniqueness theorem for stable anisotropic capillary surfaces. SIAM J. Math. Anal. 39(3), 721–741 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Koiso M., Palmer B.: Equilibria for anisotropic surface energies with wetting and line tension. Calc. Var. Par. Differ. Equ. 43, 555–587 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Maggi F.: Some methods for studying stability in isoperimetric type problems. Bull. Am. Mat. Soc. 45(3), 367–408 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory. Cambridge Studies in Advanced Mathematics, Vol. 135. Cambridge University Press, Cambridge, 2012

  28. McCann R.: Equilibrium shapes for planar crystals in an external field. Commun. Math. Phys. 195, 699–723 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Mellet, A., Nolen, J.: Capillary drops on a rough surface. Preprint 2011

  30. Nickolov R.: Uniqueness of the singular solution to the capillary equation. Indiana Univ. Math. J. 51(1), 127–169 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Taylor J.: Existence and structure of solutions to a class of non-elliptic variational problems. Symp. Math. 14, 499–508 (1974)

    ADS  Google Scholar 

  32. Treinen R.: A general existence theorem for symmetric floating drops. Arch. Math. (Basel) 94(5), 477–488 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Van Schaftingen, J.: Anisotropic symmetrization. Ann. Inst. Henri Poincaré, 23, 539–565 (2006)

  34. Vol’pert, A.I.: Spaces BV and quasi-linear equations. Math. USSR Sb. 17, 225–267 (1967)

  35. Wente H.: The symmetry of sessile and pendent drops. Pac. J. Math. 88(2), 387–397 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wente H.: The stability of the axially symmetric pendent drop. Pac. J. Math 88(2), 421–470 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  37. Winterbottom, W.-L.: Equilibrium shape of a small particle in contact with a foreign substrate. Acta Metall. 15(2), 303–310 (1967)

  38. Wulff, G.: Zur Frage der Geschwindigkeit des Wachsturms und der Auflösung der Kristallflächen. Z. Kristallogr. 34, 449–530 (1901)

  39. Zia, R., Avron, J., Taylor, J.: The summertop construction: crystals in a corner. J. Stat. Phys. 50(3–4), 727–736 (1988)

Download references

Author information

Authors and Affiliations

  1. Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139-4381, USA

    Eric Baer

Authors
  1. Eric Baer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Baer.

Additional information

Communicated by D. Kinderlehrer

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baer, E. Minimizers of Anisotropic Surface Tensions Under Gravity: Higher Dimensions via Symmetrization. Arch Rational Mech Anal 215, 531–578 (2015). https://doi.org/10.1007/s00205-014-0788-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-014-0788-z

Keywords

Search

Navigation