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Multiphase Solutions to the Vector Allen–Cahn Equation: Crystalline and Other Complex Symmetric Structures

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Abstract

We present a systematic study of entire symmetric solutions \({u : \mathbb{R}^n \rightarrow\mathbb{R}^m}\) of the vector Allen–Cahn equation

$$\Delta u - W_u(u) = 0 \quad\text{for all}\quad x \in \mathbb{R}^n,$$

where \({W:\mathbb{R}^m \rightarrow \mathbb{R}}\) is smooth, symmetric, nonnegative with a finite number of zeros, and where \({ W_u= (\partial W / \partial u_1,\dots,\partial W / \partial u_m)^{\top}}\). We introduce a general notion of equivariance with respect to a homomorphism \({f:G\rightarrow\Gamma}\) (\({G, \Gamma}\) reflection groups) and prove two abstract results, concerning the cases of G finite and G discrete, for the existence of equivariant solutions. Our approach is variational and based on a mapping property of the parabolic vector Allen–Cahn equation and on a pointwise estimate for vector minimizers.

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Peter W. Bates

  2. Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, Via Vetoio, 67010, Coppito, L’Aquila, Italy

    Giorgio Fusco

  3. Department of Mathematics, University of Athens, Panepistemiopolis, 15784, Athens, Greece

    Panayotis Smyrnelis

Authors
  1. Peter W. Bates
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  2. Giorgio Fusco
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  3. Panayotis Smyrnelis
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Corresponding author

Correspondence to Peter W. Bates.

Additional information

Communicated by D. Kinderlehrer

PWB was supported in part by NSF DMS-0908348, DMS-1413060, and the IMA. GF was partially supported by the IMA. PS was partially supported through the project PDEGE Partial Differential Equations Motivated by Geometric Evolution, co-financed by the European Union European Social Fund (ESF) and national resources, in the framework of the program Aristeia of the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF).

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Bates, P.W., Fusco, G. & Smyrnelis, P. Multiphase Solutions to the Vector Allen–Cahn Equation: Crystalline and Other Complex Symmetric Structures. Arch Rational Mech Anal 225, 685–715 (2017). https://doi.org/10.1007/s00205-017-1112-5

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  • DOI: https://doi.org/10.1007/s00205-017-1112-5

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