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Spontaneous Breaking of Rotational Symmetry with Arbitrary Defects and a Rigidity Estimate

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Abstract

The goal of this paper is twofold. First we prove a rigidity estimate, which generalises the theorem on geometric rigidity of Friesecke, James and Müller to 1-forms with non-vanishing exterior derivative. Second we use this estimate to prove a kind of spontaneous breaking of rotational symmetry for some models of crystals, which allow almost all kinds of defects, including unbounded defects as well as edge, screw and mixed dislocations, i.e. defects with Burgers vectors.

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Notes

  1. A \(\partial \)-manifold is a complete manifold with boundary equipped with an oriented smooth atlas, see [7, Definition 1.1.2].

References

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Acknowledgments

The author is grateful to Franz Merkl for stimulating discussions and helpful remarks. This research was supported by a scholarship of the Cusanuswerk, one of the German national academic foundations.

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  1. Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333, Munich, Germany

    Simon Aumann

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  1. Simon Aumann
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Correspondence to Simon Aumann.

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Aumann, S. Spontaneous Breaking of Rotational Symmetry with Arbitrary Defects and a Rigidity Estimate. J Stat Phys 160, 168–208 (2015). https://doi.org/10.1007/s10955-015-1234-9

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  • DOI: https://doi.org/10.1007/s10955-015-1234-9

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