Abstract
This paper provides a quantitative version of the recent result of Knüpfer and Muratov (Commun Pure Appl Math 66:1129–1162, 2013) concerning the solutions of an extension of the classical isoperimetric problem in which a non-local repulsive term involving Riesz potential is present. In that work, it was shown that in two space dimensions the minimizer of the considered problem is either a ball or does not exist, depending on whether or not the volume constraint lies in an explicit interval around zero, provided that the Riesz kernel decays sufficiently slowly. Here, we give an explicit estimate for the exponents of the Riesz kernel for which the result holds.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I. (eds.): Handbook of mathematical functions. National Bureau of Standards (1964)
Acerbi, E., Fusco, N., Morini, M.: Minimality via second variation for a nonlocal isoperimetric problem. Commun. Math. Phys. 322, 515–557 (2013)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, New York (2000)
Ambrosio, L., Paolini, E.: Partial regularity for quasi minimizers of perimeter. Ricerche Mat. 48((supplemnto)), 167–186 (1998)
Bohr, N.: Neutron capture and nuclear constitution. Nature 137, 344–348 (1936)
Bohr, N., Wheeler, J.A.: The mechanism of nuclear fission. Phys. Rev. 56, 426–450 (1939)
Bonacini, M., Cristoferi, R.: Local and global minimality results for a nonlocal isoperimetric problem on \(\mathbb{R}^N\). SIAM J. Math. Anal. (2014, to appear). Preprint: arXiv:1307.5269
Bonnesen, T.: Über das isoperimetrische Defizit ebener Figuren. Math. Ann. 91, 252–268 (1924)
Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer, New York (1988)
Choksi, R.: On global minimizers for a variational problem with long-range interactions. Q. Appl. Math. LXX, 517–537 (2012)
Choksi, R., Peletier, M.A.: Small volume fraction limit of the diblock copolymer problem: II. Diffuse interface functional. SIAM J. Math. Anal. 43, 739–763 (2011)
Cicalese, M., Spadaro, E.: Droplet minimizers of an isoperimetric problem with long-range interactions. Commun. Pure Appl. Math. 66, 1298–1333 (2013)
Figalli, A., Fusco, N., Maggi, F., Millot, V., Morini, M.: Isoperimetry and stability properties of balls with respect to nonlocal energies (2014). Preprint: arXiv:1403.0516
Figalli, A., Maggi, F.: On the shape of liquid drops and crystals in the small mass regime. Arch. Ration. Mech. Anal. 201, 143–207 (2011)
Fuglede, B.: Bonnesen’s inequality for the isoperimetric deficiency of closed curves in the plane. Geom. Dedic. 38, 283–300 (1991)
Gamow, G.: Mass defect curve and nuclear constitution. Proc. R. Soc. Lond. A 126, 632–644 (1930)
Julin, V.: Isoperimetric problem with a Coulombic repulsive term. Indiana Univ. Math. J. (2014, to appear). Preprint: arXiv:1207.0715
Knüpfer, H., Muratov, C.B.: On an isoperimetric problem with a competing non-local term. I. The planar case. Commun. Pure Appl. Math. 66, 1129–1162 (2013)
Knüpfer, H., Muratov, C.B.: On an isoperimetric problem with a competing non-local term. II. The general case. Commun. Pure Appl. Math. (2013, published online)
Lu, J., Otto, F.: Nonexistence of minimizer for Thomas-Fermi-Dirac-von Weizsäcker model. Commun. Pure Appl. Math. (2013, publlished online)
Ohta, T., Kawasaki, K.: Equilibrium morphologies of block copolymer melts. Macromolecules 19, 2621–2632 (1986)
Osserman, R.: Bonnesen-style isoperimetric inequalities. Am. Math. Month. 86, 1–29 (1979)
Sternberg, P., Topaloglu, I.: A note on the global minimizers of the nonlocal isoperimetric problem in two dimensions. Interf. Free Bound. 13, 155–169 (2010)
Tamanini, I.: Regularity results for almost minimal oriented hypersurfaces in \(\mathbb{R}^N\). Quad. Dip. Mat. Univ. Lecce 1, 1–92 (1984)
Topaloglu, I.: On a nonlocal isoperimetric problem on the two-sphere. Commun. Pure Appl. Anal. 12, 597–620 (2013)
Acknowledgments
The work of C. B. M. was supported, in part, by NSF via Grants DMS-0908279, DMS-1119724 and DMS-1313687.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Muratov, C.B., Zaleski, A. On an isoperimetric problem with a competing non-local term: quantitative results. Ann Glob Anal Geom 47, 63–80 (2015). https://doi.org/10.1007/s10455-014-9435-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-014-9435-z