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Large solutions for harmonic maps in two dimensions

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Abstract

We seek critical points of the functionalE(u)=\(\mathop \smallint \limits_\Omega\)|βu|2, where Ω is the unit disk in ℝ2 andu:Ω→S 2 satisfies the boundary conditionu=γ on ∂Ω. We prove that if γ is not a constant, thenE has a local minimum which is different from the absolute minimum. We discuss in more details the case where γ(x, y)=(R x,R y,\(\sqrt {1 - R^2 }\)) andR<1.

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris VI, 4, Place Jussieu, F-75230, Paris Cedex 05, France

    Haim Brezis

  2. Département de Mathématiques, Ecole Polytechnique, F-91128, Palaiseau Cedex, France

    Jean-Michel Coron

Authors
  1. Haim Brezis
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  2. Jean-Michel Coron
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Additional information

Communicated by A. Jaffe

Work partially supported by US National Science Foundation grant PHY-8116101-A01

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Brezis, H., Coron, JM. Large solutions for harmonic maps in two dimensions. Commun.Math. Phys. 92, 203–215 (1983). https://doi.org/10.1007/BF01210846

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  • DOI: https://doi.org/10.1007/BF01210846

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