On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case

Cyrill Muratov, Hans Knüpfer

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Abstract

This paper is the continuation of a previous paper (H. Knüpfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129-1162). We investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of a general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel, we also obtain nonexistence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy cannot be attained.

Original languageEnglish (US)
Pages (from-to)1974-1994
Number of pages21
JournalCommunications on Pure and Applied Mathematics
Volume67
Issue number12
DOIs
StatePublished - Dec 1 2014

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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