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End-to-end construction for the Allen–Cahn equation in the plane

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Abstract

In this paper, we construct a wealth of bounded, entire solutions of the Allen–Cahn equation in the plane. The asymptotic behavior at infinity of these solutions is determined by \(2L\) half affine lines, in the sense that, along each of these half affine lines, the solution is close to a suitable translated and rotated copy of a one dimensional heteroclinic solution. The solutions we construct belong to a smooth \(2L\)-dimensional family of bounded, entire solutions of the Allen–Cahn equation, in agreement with the result of del Pino (Trans Am Math Soc 365(2):721–766, 2010) and, in some sense, they provide a description of a collar neighborhood of part of the compactification of the moduli space of \(2L\)-ended solutions for the Allen–Cahn equation. Our construction is inspired by a construction of minimal surfaces by Traizet [Ann. Inst. Fourier (Grenoble) 46(5), 1385–1442, 1996].

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Acknowledgments

Y. Liu is partially supported by NSFC grant 11101141 and the project sponsored by SRF for ROCS, SEM, M. Kowalczyk is partially supported by Chilean research grants Fondecyt 1090103, Fondo Basal CMM-Chile, Project Anillo ACT-125 CAPDE, and a MathAmSud project NAPDE, F. Pacard is partially supported by ANR-11-IS01-0002 grant and a MathAmSud project NAPDE, and J. Wei is partially supported by General Research Fund from RGC of Hong Kong.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Michał Kowalczyk

  2. Department of Mathematics and Physics, North China Electric Power University, Beijing, China

    Yong Liu

  3. Centre de Mathématiques Laurent Schwartz UMR-CNRS 7640, École Polytechnique, 91128 , Palaiseau, France

    Frank Pacard

  4. Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, China

    Juncheng Wei

  5. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Juncheng Wei

Authors
  1. Michał Kowalczyk
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  2. Yong Liu
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  3. Frank Pacard
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  4. Juncheng Wei
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Corresponding author

Correspondence to Michał Kowalczyk.

Additional information

Communicated by O. Savin.

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Kowalczyk, M., Liu, Y., Pacard, F. et al. End-to-end construction for the Allen–Cahn equation in the plane. Calc. Var. 52, 281–302 (2015). https://doi.org/10.1007/s00526-014-0712-2

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  • DOI: https://doi.org/10.1007/s00526-014-0712-2

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