Abstract
The \(\Gamma \) -convergence theory deals with a singular limit phenomenon in the calculus of variations. It provides a rigorous notion for a family of functionals to converge to a functional of seemingly a different type, while still retaining vital properties in the limiting functional. For instance, global minimizers of the functionals in the family converge to a global minimizer of the limiting functional. Near an isolated local minimizer of the limiting functional, there exist local minimizers of the functionals in the converging family that are sufficiently close to the limiting functional. This theory has found a surprising application in the study of block copolymer morphology. Block copolymers are soft materials characterized by fluid-like disorder on the molecular scale and a high degree of order at a longer length scale. This chapter presents a description of the Ohta–Kawasaki theory density theory for block copolymer morphology and applies the \(\Gamma \) -convergence theory to reduce the Ohta–Kawasaki theory to a geometric problem containing perimeter minimization and nonlocal interaction. As an application of \(\Gamma \) -convergence, one determines all the global and local minimizers in one dimension. Consequently global and local minimizers are also characterized for the Ohta–Kawasaki theory.
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Supported in part by NSF grant DMS-0907777.
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Ren, X. (2013). On the \(\Gamma \)-Convergence Theory and Its Application to Block Copolymer Morphology. In: Toni, B. (eds) Advances in Interdisciplinary Mathematical Research. Springer Proceedings in Mathematics & Statistics, vol 37. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6345-0_2
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DOI: https://doi.org/10.1007/978-1-4614-6345-0_2
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