# American Institute of Mathematical Sciences

August  2009, 24(3): 979-1003. doi: 10.3934/dcds.2009.24.979

## Shell structure as solution to a free boundary problem from block copolymer morphology

 1 Department of Mathematics, The George Washington University, 2115 G Street, Washington, DC 20052

Received  November 2007 Revised  March 2008 Published  April 2009

A shell like structure is sought as a solution of a free boundary problem derived from the Ohta-Kawasaki theory of diblock copolymers. The boundary of the shell satisfies an equation that involves its mean curvature and the location of the entire shell. A variant of Lyapunov-Schmidt reduction process is performed that rigorously reduces the free boundary problem to a finite dimensional problem. The finite dimensional problem is solved numerically. The problem has two parameters: $a$ and $\gamma$. When $a$ is small, there are a lower bound and a sequence such that if $\gamma$ is greater than the lower bound and stays away from the sequence, there is a shell like solution.
Citation: Xiaofeng Ren. Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 979-1003. doi: 10.3934/dcds.2009.24.979
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