Vol. 2, No. 1, 2020

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

Hans Knüpfer, Cyrill B. Muratov and Matteo Novaga

Vol. 2 (2020), No. 1, 1–21
DOI: 10.2140/paa.2020.2.1
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 nontrivial minimizers. We identify the scaling in which a sharp transition from asymptotically trivial to nontrivial 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.

Keywords
energy driven pattern formation, Gamma convergence
Mathematical Subject Classification 2010
Primary: 34A34, 35A15, 35J50
Secondary: 49Q20, 51P05
Milestones
Received: 6 December 2018
Revised: 8 June 2019
Accepted: 16 August 2019
Published: 9 November 2019
Authors
Hans Knüpfer
Institut für Angewandte Mathematik
Universität Heidelberg
Heidelberg
Germany
Cyrill B. Muratov
Department of Mathematical Sciences
New Jersey Institute of Technology
Newark, NJ
United States
Matteo Novaga
Dipartimento di Matematica
Università di Pisa
Pisa
Italy