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

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. © 2014 Wiley Periodicals, Inc.

On minimizers of interaction functionals with competing attractive and repulsive potentials

“…Recently, a binary nonlocal isoperimetric functional appeared in connection with the modeling of self-assembly of diblock copolymers [43][44][45]. Existence and non-existence results have been presented in [46][47][48]42]. In dimension N = 3, the functional has a Newtonian repulsive component as in (1) with p = −1.…”

“…Existence of a radially symmetric minimizer (i.e. a ball) for m sufficiently small has recently been proved in [46,48]. Whether or not balls are the only minimizers remains open.…”

“…The latter point has an interesting history. Poincaré [49] considered the problem of determining the equilibrium shapes of a fluid body of mass m. In simplified form, this amounts to minimizing the total potential energy of the region of fluid E ⊂ R 3…”

On Equilibrium Shape of Charged Flat Drops

“…On the other hand, there is no minimizer with j j D m as soon as m > m 2 1 , where m 2 m 0 is another universal constant [36]. These conclusions have been extended to the case of higher-dimensional problems and more general Riesz kernels [23,35,37,41]. The basic intuition behind these results is that when the value of m is sufficiently small, the perimeter term dominates the energy, forcing the minimizer to coincide with the one for the classical isoperimetric problem.…”

Large Volume Minimizers of a Nonlocal Isoperimetric Problem: Theoretical and Numerical Approaches

“…Indeed, two terms are competing: the perimeter tends to round things up (and is minimized by balls), whereas the non-local V α term, which can be viewed as an electrostatic energy if n = 3 and α = 2, tends to spread the mass (and is maximized by balls). It was shown in [6] that if the mass m is small enough, then the problem (1.1) admits a unique minimizer (up to translation), namely the ball of volume m (see also [9], [12] and [3] for partial results). On the other hand, for α ∈ (0, 2), it was shown in [12] that for m large enough there is no minimizer of problem (1.1).…”