Emergence of nontrivial minimizers for the three-dimensional Ohta–Kawasaki energy

Abstract

This paper is concerned with the diffuse interface Ohta-Kawasaki energy in three space dimensions, in a periodic setting, in the parameter regime corresponding to the onset of non-trivial minimizers. We identify the scaling in which a sharp transition from asymptotically trivial to non-trivial minimizers takes place as the small parameter characterizing the width of the interfaces between the two phases goes to zero, while the volume fraction of the minority phases vanishes at an appropriate rate. The value of the threshold is shown to be related to the optimal binding energy solution of Gamow's liquid drop model of the atomic nucleus. Beyond the threshold the average volume fraction of the minority phase is demonstrated to grow linearly with the distance to the threshold. In addition to these results, we establish a number of properties of the minimizers of the sharp interface screened Ohta-Kawasaki energy in the considered parameter regime. We also establish rather tight upper and lower bounds on the value of the transition threshold.

Uniform bound on the number of partitions for optimal configurations of the Ohta–Kawasaki energy in 3D

“…The proof follows arguments similar to those used to show the sub-addictivity formula (2.1) in [16]. It is well known (see, e.g., [2, 9, 13–15, 20], and the references therein) that there exist , , such that, for all , the minimizer of is given by . Since (resp.…”

Minimisers of a general Riesz-type Problem

“…We observe that the existence result in Proposition 1.2 was previously obtained only in some particular cases, see for instance [13] for N = 3 and g(x) = 1/|x|, and [14] for N = 2 and g(x) = χ [δ,∞) (|x|)/|x| 3 , with δ > 0.…”

Striped patterns for generalized antiferromagnetic functionals with power law kernels of exponent smaller than $d+2$

“…This model is well-studied (see e.g. [1,8,7,27,29,30,22,31,35,38]) but though pattern formation is observed in experiments and numerical simulations ( [4,39]), a rigorous proof is still largely open (see [31] for a proof of striped patterns in thin 2D domains and [38] for an asymptotic limit).…”