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Mathematics Behind Microstructures: A Lead to Generalizations of Convexity

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Bridging Mathematics, Statistics, Engineering and Technology

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 24))

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Abstract

We consider a mathematical model aimed at explaining pattern formation in microstructures. Usually such models are also useful for understanding problems of solid–solid phase transitions in material science. Our goal is to analyze the limiting behavior of certain non-linear energy type functionals, with restrictions, from a variational point of view. In order to better understand this problem we develop some generalizations for the notions of rank-one convexity and quasi-convexity and demonstrate their relevance in the context of energy minimizing sequences.

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Authors and Affiliations

  1. Department of Mathematics, Christopher Newport University, Newport News, VA, 23606, USA

    Daniel Vasiliu

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  1. Daniel Vasiliu
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Correspondence to Daniel Vasiliu .

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Editors and Affiliations

  1. , Dept of Mathematics & Computer Sciences, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Bourama Toni

  2. , Department of Technology, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Keith Williamson

  3. , Department of Engineering, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Nasser Ghariban

  4. , Department of Mathematics & Computer Sci, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Dawit Haile

  5. , Department of Mathematics & Computer Sci, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Zhifu Xie

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Vasiliu, D. (2012). Mathematics Behind Microstructures: A Lead to Generalizations of Convexity. In: Toni, B., Williamson, K., Ghariban, N., Haile, D., Xie, Z. (eds) Bridging Mathematics, Statistics, Engineering and Technology. Springer Proceedings in Mathematics & Statistics, vol 24. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4559-3_8

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