Modeling and Analysis of patterns in multi-constituent systems with long range interaction
Skin pigmentation, animal coats and block copolymers can be considered as multi-constituent inhibitory systems. Exquisitely structured patterns arise as orderly outcomes of the self-organization principle. Analytically, via the sharp interface model, patterns can be studied as nonlocal geometric variational problems. The free energy functional consists of an interface energy and a long range Coulomb-type interaction energy. The admissible class is a collection of Caccioppoli sets with fixed volumes. To overcome the difficulty that the admissible class is not a Hilbert space, we introduce internal variables. Solving the energy functional for stationary sets is recast as a variational problem on a Hilbert space. We prove the existence of a core-shell assembly and the existence of disc assemblies in ternary systems and also a triple-bubble-like stationary solution in a quaternary system. Numerically, via the diffuse interface model, one open question related to the polarity direction of double bubble assemblies is answered. Moreover, it is shown that the average size of bubbles in a single bubble assembly depends on the sum of the minority constituent volumes and the long range interaction coefficients. One further identifies the ranges for volume fractions and the long range interaction coefficients for double bubble assemblies.