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Solutions of the Allen-Cahn equation which are invariant under screw-motion

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Abstract

We study entire solutions of the Allen-Cahn equation which are defined in the 3-dimensional Euclidean space and which are invariant under screw-motion. In particular, we discuss the existence and non existence of nontrivial solutions whose nodal set is a helicoid of \({\mathbb R^{3}}\).

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References

  1. Cabré X., Terra J.: Saddle-shaped solutions of bistable diffusion equations in all of R 2m. J. Eur. Math. Soc. 11(4), 819–843 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cabré, X., Terra, J.: Qualitative properties of saddle-shaped solutions to bistable diffusion equations. Comm. Partial Differ. Equ. 35(11), 1923–1957 (2010). arxiv.org/abs/0907.3008

    Google Scholar 

  3. Chicone C.: The monotonicity of the period function for planar hamiltonian vector fields. J. Differ. Equ. 69, 310–321 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Choe, J., Hoppe, J.: Higher dimensional minimal submanifolds arising from the catenoid and helicoid (in press, 2010)

  5. Dancer, E.N.: New solutions of equations on R n. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30(3–4), 535–563 (2002)

  6. Dang H., Fife PC., Peletier LA.: Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys. 43(6), 984–998 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. del Pino M., Kowalczyk M., Pacard F., Wei J.: Multiple-end solutions to the Allen-Cahn equation in \({\mathbb{R}^2}\). J. Funct. Anal. 258, 458–503 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. del Pino, M., Kowalczyk, M., Pacard, F.: Moduli space theory for some class of solutions to the Allen-Cahn equation in the plane. To appear in Transactions of the AMS

  9. del Pino M., Kowalczyk M., Wei J.: A conjecture by de Giorgi in large dimensions. Ann. Math. 174, 1485–1569 (2011)

    Article  MATH  Google Scholar 

  10. del Pino, M., Kowalczyk, M., Wei, J.: Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature. arXiv:0902.2047v1 [math.AP]

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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

    Manuel del Pino

  2. Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avda. Vicuña Mackenna, 4860, Macul, Chile

    Monica Musso

  3. Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France and Institut Universitaire de France, Paris, France

    Frank Pacard

Authors
  1. Manuel del Pino
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  2. Monica Musso
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  3. Frank Pacard
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Correspondence to Frank Pacard.

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del Pino, M., Musso, M. & Pacard, F. Solutions of the Allen-Cahn equation which are invariant under screw-motion. manuscripta math. 138, 273–286 (2012). https://doi.org/10.1007/s00229-011-0492-3

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  • DOI: https://doi.org/10.1007/s00229-011-0492-3

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