Pattern formation for a local/nonlocal interaction functional arising in colloidal systems.

*(English)*Zbl 07245265The ability of matter to arrange itself in periodic structures
is often referred to as spontaneous pattern formation. This phenomenon is of fundamental
importance in science, technology, engineering, and mathematics, and it
is often caused by the interaction between local attractive and nonlocal repulsive
forces.
In this paper, the authors study pattern formation for a physical local/nonlocal interaction
functional where the local attractive term is given by the 1-perimeter and the nonlocal repulsive
term is the Yukawa (or screened Coulomb) potential [B. Derjaguin and L. Landau, “Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes”, Acta Phys. Chem. USSR, 14, 633–662 (1941)]. This model is physically interesting as it is
the \(\Gamma\)-limit of a double Yukawa model used to explain and simulate pattern formation in colloidal
systems. The authors prove that in a suitable regime minimizers are periodic stripes in any
space dimension.

Reviewer: Nasir N. Ganikhodjaev (Tashkent)

##### MSC:

82B21 | Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics |

49N20 | Periodic optimal control problems |

49S05 | Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX) |

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\textit{S. Daneri} and \textit{E. Runa}, SIAM J. Math. Anal. 52, No. 3, 2531--2560 (2020; Zbl 07245265)

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