On energy minimizers of the diblock copolymer problem

The original formula in [12] is given for the whole space. The expression here on a bounded domain first appeared in Nishiura and Ohnishi [10]. The two unlike monomer units are represented byu = 1 and u = 1 respectively. The connectivity of the monomers in a chain leads to the long range interaction (/ 2)|( ) 1/2 (u m)| 2 in the free energy. Here is viewed as a positive operator, and ( ) 1/2 is the square root of its inverse. The parameter is proportional to the inverse of the square root of the total chain length of the copolymer. ( 2 /2)|ru| 2 represents the interfacial energy density at bonding points. The parameter is proportional to the thickness of interfaces between the two monomers. m stands for the mass ratio of the two monomer units.