Asymmetric And Symmetric Double Bubbles In A Ternary Inhibitory System

A ternary inhibitory system contains two terms in its free energy: the interface energy that favors microdomain growth and the longer ranging confinement energy that prevents unlimited spreading. In a parameter regime where two constituents are small in size compared to the third constituent and the longer ranging energy does not dominate, there is a double-bubble-like stable stationary point of the energy functional. The two minority constituents occupy the two bubbles of the double bubble, respectively, and the majority constituent fills the background. A special way of perturbing an exact double bubble leads to a restricted class of perturbed double bubbles that can be described by internal variables which are elements in a Hilbert space. The exact double bubble is nondegenerate in this class and nearby there is a perturbed double bubble that locally minimizes the free energy within the restricted class. This perturbed double bubble satisfies three of the four equations for stationary points of the free energy, namely, the three equations involving the curvature and the inhibitor variables on its three boundary curves. However it does not satisfy the 120 degree angle condition at its triple points. By translating and rotating the entire restricted class of perturbed double bubbles, one finds a particular direction and location in the domain of the problem where the locally minimizing perturbed double bubble in this specific restricted class also satisfies the 120 degree condition. This approach can handle both asymmetric and symmetric double bubbles.