Skip to main content
SpringerLink
Log in

Conducting Flat Drops in a Confining Potential

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

Abstract

We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is confining and the Coulomb repulsion strength is below a critical value, we show existence and regularity estimates for volume-constrained minimizers. We also derive the Euler–Lagrange equation satisfied by regular critical points, expressing the first variation of the Coulombic energy in terms of the normal \(\frac{1}{2}\)-derivative of the capacitary potential.

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. Ambrosio, L., Caselles, V., Masnou, S., Morel, J.M.: Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc. 3(1), 39–92, 2001

    Article  MathSciNet  Google Scholar 

  2. Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  3. Barrero, A., Loscertales, I.G.: Micro- and nanoparticles via capillary flows. Annu. Rev. Fluid Mech. 39, 89–106, 2007

    Article  ADS  Google Scholar 

  4. Basaran, O.A., Scriven, L.E.: Axisymmetric shapes and stability of isolated charged drops. Phys. Fluids A 1, 795–798, 1989

    Article  ADS  Google Scholar 

  5. Burton, J.C., Taborek, P.: Simulations of Coulombic fission of charged inviscid drops. Phys. Rev. Lett. 106, 144501, 2011

    Article  ADS  Google Scholar 

  6. Castro-Hernandez, E., García-Sánchez, P., Tan, S.H., Gañán-Calvo, A.M., Baret, J.-C., Ramos, A.: Breakup length of AC electrified jets in a microfluidic flow focusing junction. Microfluid. Nanofluid. 19, 787–794, 2015

    Article  Google Scholar 

  7. Choksi, R., Muratov, C.B., Topaloglu, I.: An old problem resurfaces nonlocally: Gamow’s liquid drops inspire today’s research and applications. Not. Am. Math. Soc. 64, 1275–1283, 2017

    MathSciNet  MATH  Google Scholar 

  8. Dalibard, A.-L., Gérard-Varet, D.: On shape optimization problems involving the fractional Laplacian. ESAIM Control Optim. Calc. Var. 19, 976–1013, 2013

    Article  MathSciNet  Google Scholar 

  9. De Philippis, G., Hirsch, J., Vescovo, G.: Regularity of minimizers for a model of charged droplets. Preprint to appear on Commun. Math. Phys. (2019)

  10. De Silva, D., Savin, O.: Boundary Harnack estimates in slit domains and applications to thin free boundary problems. Rev. Mat. Iberoam. 32(3), 891–912, 2016

    Article  MathSciNet  Google Scholar 

  11. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573, 2012

    Article  MathSciNet  Google Scholar 

  12. Esposito, L., Fusco, N.: A remark on a free interface problem with volume constraint. Convex Anal. 18(2), 417–426, 2011

    MathSciNet  MATH  Google Scholar 

  13. Fernández de la Mora, J.: The fluid dynamics of Taylor cones. J. Ann. Rev. Fluid Mech. 39, 217–243, 2007

    Article  ADS  MathSciNet  Google Scholar 

  14. 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  Google Scholar 

  15. Fontelos, M.A., Friedman, A.: Symmetry-breaking bifurcations of charged drops. Arch. Ration. Mech. Anal. 172, 267–294, 2004

    Article  MathSciNet  Google Scholar 

  16. Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. 2(168), 941–980, 2008

    Article  MathSciNet  Google Scholar 

  17. Garzon, M., Gray, L.J., Sethian, J.A.: Numerical simulations of electrostatically driven jets from nonviscous droplets. Phys. Rev. E 89, 033011, 2014

    Article  ADS  Google Scholar 

  18. Gaskell, S.J.: Electrospray: principles and practice. J. Mass Spectrom. 32, 677–688, 1997

    Article  ADS  Google Scholar 

  19. Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer, Berlin, 1983

  20. Goldman, M., Novaga, M., Ruffini, B.: Existence and stability for a non-local isoperimetric model of charged liquid drops. Arch. Ration. Mech. Anal. 217, 1–36, 2015

    Article  MathSciNet  Google Scholar 

  21. Goldman, M., Novaga, M., Ruffini, B.: On minimizers of an isoperimetric problem with long-range interactions and convexity constraint. Anal. PDE 11(5), 1113–1142, 2018

    Article  MathSciNet  Google Scholar 

  22. Hofmann, S., Mitrea, M., Taylor, M.: Geometric and transformational properties of Lipschitz domains, Semmes–Kenig–Toro domains, and other classes of finite perimeter domains. J. Geom. Anal. 17(4), 593–647, 2007

    Article  MathSciNet  Google Scholar 

  23. Iglesias, J.A., Mercier, G.: Convergence of level sets in total variation denoising through variational curvatures in unbounded domains. SIAM J. Math. Anal. 53(2), 1509–1545, 2021

    Article  MathSciNet  Google Scholar 

  24. Landkof, N.S.: Foundations of Modern Potential Theory. Springer, New York (1972)

    Book  Google Scholar 

  25. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  26. Lu, J., Moroz, V., Muratov, C.: Orbital-free density functional theory of out-of-plane charge screening in graphene. J. Nonlinear Sci. 25(6), 1391–1430, 2015

    Article  ADS  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  28. Mennucci, A.: On perimeters and volumes of fattened sets. Int. J. Math. Math. Sci. 2019, 8283496, 2019

    Article  MathSciNet  Google Scholar 

  29. Muratov, C., Novaga, M.: On well-posedness of variational models of charged drops. Proc. R. Soc. Lond. A 472, 20150808, 2016

    ADS  MathSciNet  MATH  Google Scholar 

  30. Muratov, C., Novaga, M., Ruffini, B.: On equilibrium shapes of charged flat drops. Commun. Pure Appl. Math. 71(6), 1049–1073, 2018

    Article  MathSciNet  Google Scholar 

  31. Novaga, M., Ruffini, B.: Brunn–Minkowski inequality for the \(1\)-Riesz capacity and level set convexity for the \(1/2\)-Laplacian. J. Convex Anal. 22, 1125–1134, 2015

    MathSciNet  MATH  Google Scholar 

  32. Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton (1951)

    MATH  Google Scholar 

  33. Rayleigh, Lord: On the equilibrium of liquid conducting masses charged with electricity. Philos. Mag. 14, 184–186, 1882

    Article  Google Scholar 

  34. Ros-Oton, X.: Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat. 60(1), 3–26, 2016

    Article  MathSciNet  Google Scholar 

  35. Ros-Oton, X., Serra, J.: Boundary regularity estimates for nonlocal elliptic equations in \(C^1\) and \(C^{1,\alpha }\) domains. Ann. Mat. Pura Appl. 196(5), 1637–1668, 2017

    Article  MathSciNet  Google Scholar 

  36. Schmidt, T.: Strict interior approximation of sets of finite perimeter and functions of bounded variation. Proc. Am. Math. Soc. 143(5), 2069–2084, 2015

    Article  MathSciNet  Google Scholar 

  37. Taylor, G.: Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383–397, 1964

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The work of CBM was partially supported by NSF via grants DMS-1614948 and DMS-1908709. MN has been supported by GNAMPA-INdAM and by the University of Pisa via grant PRA 2017-18. BR was partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR) and the GNAMPA-INdAM Project 2019 “Ottimizzazione spettrale non lineare”.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    Cyrill B. Muratov

  2. Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

    Matteo Novaga

  3. Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, place Eugene Bataillon, 34095, Montpellier Cedex 5, France

    Berardo Ruffini

  4. Dipartimento di Matematica, Alma Mater Studiorum – Università di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    Berardo Ruffini

Authors
  1. Cyrill B. Muratov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matteo Novaga
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Berardo Ruffini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cyrill B. Muratov.

Additional information

Communicated by A. Figalli.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Muratov, C.B., Novaga, M. & Ruffini, B. Conducting Flat Drops in a Confining Potential. Arch Rational Mech Anal 243, 1773–1810 (2022). https://doi.org/10.1007/s00205-021-01738-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-021-01738-0

Search

Navigation