Core shells and double bubbles in a weighted nonlocal isoperimetric problem

Abstract: We consider a sharp-interface model of $ABC$ triblock copolymers, for which the surface tension $\sigma_{ij}$ across the interface separating phase $i$ from phase $j$ may depend on the components. We study global minimizers of the associated ternary local isoperimetric problem in $\mathbb{R}^2$, and show how the geometry of minimizers changes with the surface tensions $\sigma_{ij}$, varying from symmetric double-bubbles for equal surface tensions, through asymmetric double bubbles, to core shells as the values of $\sigma_{ij}$ become more disparate. Then we consider the effect of nonlocal interactions in a droplet scaling regime, in which vanishingly small particles of two phases are distributed in a sea of the third phase. We are particularly interested in a degenerate case of $\sigma_{ij}$ in which minimizers exhibit core shell geometry, as this phase configuration is expected on physical grounds in nonlocal ternary systems.