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Convergence of a class of degenerate Ginzburg-Landau functionals and regularity for a subelliptic harmonic map equation

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Abstract

In this paper, we consider a class of Ginzburg-Landau functionalsE ε associated with a couple of non-commuting vector fields which yield a “degenerate” energy. We study the asymptotic behavior of the minimizers, showing that it does not depend on the topological degree of the boundary datum; and we prove uniqueness and regularity of the minimizer of the limit problem, in spite of the lack of lifting theorems in the natural function spaces for the limit functional.

Résumé

Dans cet article, nous considérons une classe de fonctionnellesE ε du type Ginzburg-Landau associée a un couple de champs de vecteurs définissant une énergie dégénérée. Nous étudions le comportement asymptotique des minimiseurs. Nous démontrons que ce comportement ne dépend pas du degré topologique de la donnée a la frontiere et nous prouvons l’unicité et la régularité du minimiseur du probléme limite, malgré l’absence d’un théorème de lifting dans les espaces de Sobolev naturels pour la même fonctionnelle.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di porta San Donato, 5, 40126, Bologna, Italy

    Bruno Franchi & Elena Serra

Authors
  1. Bruno Franchi
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  2. Elena Serra
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Correspondence to Bruno Franchi.

Additional information

The authors were supported by University of Bologna, funds for selected research topics, and by GNAMPA of the INDAM, Italy, project “Analysis in metric spaces and subelliptic equations.”

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Franchi, B., Serra, E. Convergence of a class of degenerate Ginzburg-Landau functionals and regularity for a subelliptic harmonic map equation. J. Anal. Math. 100, 281–322 (2006). https://doi.org/10.1007/BF02916764

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  • DOI: https://doi.org/10.1007/BF02916764

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