Core shells and double bubbles in a weighted nonlocal isoperimetric
problem
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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.