Droplet solutions in the diblock copolymer problem with skewed monomer composition

A droplet solution characterizes the lamellar phase of a diblock copolymer when the two composing monomers maintain a skewed ratio. We study the threshold case where the free energy of a droplet solution is comparable to the free energy of the constant solution. Using a Lyapunov-Schmidt reduction approach, adapted to calculus of variations, we prove the existence of a free energy local minimizer with a given number of droplets. Also determined are the free energy, the droplet location, and the droplet size. Supported in part by a Direct Grant from CUHK and an earmarked Grant of RGC of Hong Kong.