Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 5 (2018), 1113-1142.

On minimizers of an isoperimetric problem with long-range interactions under a convexity constraint

Michael Goldman, Matteo Novaga, and Berardo Ruffini

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We study a variational problem modeling the behavior at equilibrium of charged liquid drops under a convexity constraint. After proving the well-posedness of the model, we show C 1 , 1 -regularity of minimizers for the Coulombic interaction in dimension two. As a by-product we obtain that balls are the unique minimizers for small charge. Eventually, we study the asymptotic behavior of minimizers, as the charge goes to infinity.

Article information

Anal. PDE, Volume 11, Number 5 (2018), 1113-1142.

Received: 8 November 2016
Revised: 6 July 2017
Accepted: 2 January 2018
First available in Project Euclid: 17 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49J30: Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 49J45: Methods involving semicontinuity and convergence; relaxation 49S05: Variational principles of physics (should also be assigned at least one other classification number in section 49)

nonlocal isoperimetric problem convexity constraint


Goldman, Michael; Novaga, Matteo; Ruffini, Berardo. On minimizers of an isoperimetric problem with long-range interactions under a convexity constraint. Anal. PDE 11 (2018), no. 5, 1113--1142. doi:10.2140/apde.2018.11.1113.

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  • H. W. Alt and L. A. Caffarelli, “Existence and regularity for a minimum problem with free boundary”, J. Reine Angew. Math. 1981:325 (1981), 105–144.
  • L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Clarendon, New York, 2000.
  • H. Brunn, Über Ovale und Eiflächen, dissertation, Ludwig Maximilian University of Munich, 1887.
  • G. Crasta, I. Fragalà, and F. Gazzola, “On a long-standing conjecture by Pólya–Szegö and related topics”, Z. Angew. Math. Phys. 56:5 (2005), 763–782.
  • B. E. J. Dahlberg, “Estimates of harmonic measure”, Arch. Rational Mech. Anal. 65:3 (1977), 275–288.
  • L. Esposito and N. Fusco, “A remark on a free interface problem with volume constraint”, J. Convex Anal. 18:2 (2011), 417–426.
  • M. A. Fontelos and A. Friedman, “Symmetry-breaking bifurcations of charged drops”, Arch. Ration. Mech. Anal. 172:2 (2004), 267–294.
  • N. Fusco, F. Maggi, and A. Pratelli, “The sharp quantitative isoperimetric inequality”, Ann. of Math. $(2)$ 168:3 (2008), 941–980.
  • J. B. Garnett and D. E. Marshall, Harmonic measure, New Mathematical Monographs 2, Cambridge University Press, 2005.
  • M. Goldman and M. Novaga, “Volume-constrained minimizers for the prescribed curvature problem in periodic media”, Calc. Var. Partial Differential Equations 44:3-4 (2012), 297–318.
  • M. Goldman, M. Novaga, and B. Ruffini, “Existence and stability for a non-local isoperimetric model of charged liquid drops”, Arch. Ration. Mech. Anal. 217:1 (2015), 1–36.
  • D. P. Hardin and E. B. Saff, “Minimal Riesz energy point configurations for rectifiable $d$-dimensional manifolds”, Adv. Math. 193:1 (2005), 174–204.
  • D. Jerison, “A Minkowski problem for electrostatic capacity”, Acta Math. 176:1 (1996), 1–47.
  • D. S. Jerison and C. E. Kenig, “Boundary behavior of harmonic functions in nontangentially accessible domains”, Adv. in Math. 46:1 (1982), 80–147.
  • F. John, “Extremum problems with inequalities as subsidiary conditions”, pp. 187–204 in Studies and essays presented to R. Courant on his 60th birthday, January 8, 1948, Interscience, New York, 1948.
  • C. E. Kenig and T. Toro, “Harmonic measure on locally flat domains”, Duke Math. J. 87:3 (1997), 509–551.
  • C. E. Kenig and T. Toro, “Free boundary regularity for harmonic measures and Poisson kernels”, Ann. of Math. $(2)$ 150:2 (1999), 369–454.
  • J. Lamboley, A. Novruzi, and M. Pierre, “Regularity and singularities of optimal convex shapes in the plane”, Arch. Ration. Mech. Anal. 205:1 (2012), 311–343.
  • J. Lamboley, A. Novruzi, and M. Pierre, “Estimates of first and second order shape derivatives in nonsmooth multidimensional domains and applications”, J. Funct. Anal. 270:7 (2016), 2616–2652.
  • N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der Mathematischen Wissenschaften 180, Springer, 1972.
  • A. Martínez-Finkelshtein, V. Maymeskul, E. A. Rakhmanov, and E. B. Saff, “Asymptotics for minimal discrete Riesz energy on curves in $\mathbb{R}^d$”, Canad. J. Math. 56:3 (2004), 529–552.
  • C. Maxwell, “On the electrical capacity of a long narrow cylinder, and of a disk of sensible thickness”, Proc. Lond. Math. Soc. 9 (1877), 94–99.
  • C. B. Muratov and M. Novaga, “On well-posedness of variational models of charged drops”, Proc. A. 472:2187 (2016), art. id. 20150808.
  • C. B. Muratov, M. Novaga, and B. Ruffini, “On equilibrium shapes of charged flat drops”, preprint, 2016. To appear in Comm. Pure Appl. Math.
  • M. Novaga and B. Ruffini, “Brunn–Minkowski inequality for the 1-Riesz capacity and level set convexity for the $1/2$-Laplacian”, J. Convex Anal. 22:4 (2015), 1125–1134.
  • C. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften 299, Springer, 1992.
  • E. B. Saff and V. Totik, Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften 316, Springer, 1997.
  • J. W. Strutt (Lord Rayleigh), “On the equilibrium of liquid conducting masses charged with electricity”, Phil. Mag. 14 (1882), 184–186.
  • G. Taylor, “Disintegration of water drops in an electric field”, Proc. Roy. Soc. Lond. A 280:1382 (1964), 383–397.
  • S. E. Warschawski and G. E. Schober, “On conformal mapping of certain classes of Jordan domains”, Arch. Rational Mech. Anal. 22 (1966), 201–209.
  • J. Zeleny, “Instability of electricfied liquid surfaces”, Phys. Rev. 10:1 (1917), 1–6.