Stability of Spot and Ring Solutions of the Diblock Copolymer Equation

The Γ-convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions. We determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability. In two dimensions a threshold exists for the stability of the spot solution. It is is stable if the sample size is small and unstable if the sample size is large. The stability of the ring solutions is reduced to a family of finite dimensional eigenvalue problems. In one study no two-interface ring solutions are found by the Γ-convergence method if the sample is small. A stable two-interface ring solution exists if the sample size is increased. It becomes unstable if the sample size is increased further.