On an isoperimetric problem with a competing non-local term. II. The general case

TL;DR

This paper extends the analysis of a modified isoperimetric problem with a non-local repulsive term to general dimensions, establishing existence, non-existence, and shape characterization of minimizers, with implications for nuclear physics models.

Contribution

It generalizes previous work to arbitrary dimensions, providing new existence and non-existence results and characterizing minimizers as balls under certain conditions.

Findings

Existence of minimizers for small masses in all dimensions.

Non-existence of minimizers for large masses in certain kernel ranges.

Minimizers are balls for small masses and low dimensions.

Abstract

This paper is the continuation of [H. Kn\"upfer and C. B. Muratov, Commun. Pure Appl. Math. (2012, to be published)]. We investigate the classical isoperimetric problem modified by an addition of a non-local repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of 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 non-existence 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 may not be attained.