Local and Global Minimality Results for a Nonlocal Isoperimetric Problem on $\mathbb{R}^N$

Abstract

We consider a nonlocal isoperimetric problem defined in the whole space R N , whose nonlocal part is given by a Riesz potential with exponent α ∈ (0, N − 1) . We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L 1 -norm. This criterion provides the existence of an (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer, and it allows us to address several global minimality issues.

Low Density Phases in a Uniformly Charged Liquid

“…Recalling again Corollary 3.4, we obtain 44) for some universal C > 0, provided that ε is sufficiently small. We note thatũ ε (x) ≤ u ε (x) for every x ∈ R 3 .…”

“…Furthermore, the physical picture of identical radial droplets in the limit is expected for sufficiently long-ranged kernels, i.e., those kernels that satisfy G(x, y) ∼ |x − y| −α for |x − y| 1, with 0 < α 1 [36,47]. Note that although a similar characterization of the minimizers for long-ranged kernels exists in higher dimensions as well [44,46], these results are still not sufficient to be used to characterize the limit droplets, since they do not give an explicit interval of existence of the minimizers of the self-interaction problem. Also, since the non-existence result for the self-energy with such kernels is available only for α < 2 [37], our results may not extend to the case of α ≥ 2 in dimensions three and above.…”

“…with a universal C > 0, for any R ≥ 1 and x ∈ T , provided that 1 independently of x. Recalling (6.40) and arguing as in Lemma 4.2, for all 1 there exists x 0 ∈ F 0 such that 44) for some universal c, C > 0. The claim then follows by choosing a universal R that is sufficiently large.…”

“…Importantly, the model in (1.1) is a paradigm for the energy-driven pattern forming systems in which spatial patterns (global or local energy minimizers) form as a result of the competition of short-range attractive and long-range repulsive forces. This is why this model and its generalizations attracted considerable attention of mathematicians in recent years (see, e.g., [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], this list is certainly not exhaustive). In particular, the volume-constrained global minimization problem for (1.1) in the whole space with no neutralizing background, which we will also refer to as the "self-energy problem", has been investigated in [34,37,45,49].…”

“…The quantity m c1 is the maximum value of m for which a ball of mass m has less energy than twice the energy of a ball with mass 1 2 m. However, such a result is not available at present and remains an important challenge for the considered class of variational problems (for several related results see [36,44,47]). …”