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Ground states of a ternary system including attractive and repulsive Coulomb-type interactions

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Abstract

We study a variational model where two interacting phases are embedded in a third neutral phase. The energy of the system is the sum of a local interfacial contribution and a nonlocal interaction of Coulomb type. Such models are e.g. used to describe systems of copolymer–homopolymer blends or of surfactants in water solutions. We establish existence and regularity properties of global minimizers, together with a full characterization of minimizers in the small mass regime. Furthermore, we prove uniform bounds on the potential of minimizing configurations, which in turn imply some qualitative estimates about the geometry of minimizers in the large mass regime. One key mathematical difficulty in the analysis is related to the fact that two phases have to be minimized simultaneously with both attractive and repulsive interaction present.

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Acknowledgments

MB is member of the INdAM-GNAMPA Project 2015 “Critical Phenomena in the Mechanics of Materials: a Variational Approach”. MB and HK would like to thank the Mathematics Center Heidelberg (MATCH) for support.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany

    Marco Bonacini

  2. Institut für Angewandte Mathematik and Interdisciplinary Center for Scientific Computing (IWR), Universität Heidelberg, 69120, Heidelberg, Germany

    Hans Knüpfer

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  1. Marco Bonacini
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  2. Hans Knüpfer
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Correspondence to Marco Bonacini.

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Communicated by L. Ambrosio.

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Bonacini, M., Knüpfer, H. Ground states of a ternary system including attractive and repulsive Coulomb-type interactions. Calc. Var. 55, 114 (2016). https://doi.org/10.1007/s00526-016-1047-y

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  • DOI: https://doi.org/10.1007/s00526-016-1047-y

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