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The minimality of the map \(\frac{x}{\|{x}\|}\) for weighted energy

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Abstract

In this paper, we investigate the minimality of the map \(\frac{x}{\|{x}\|}\) from the Euclidean unit ball Bn to its boundary 핊n−1 for weighted energy functionals of the type Ep,f = ∫ B n f(r)‖∇ u‖p dx, where f is a non-negative function. We prove that in each of the two following cases:

  1. i)

    p = 1 and f is non-decreasing,

  2. ii)

    p is integer, p ≤ n−1 and f = rα with α ≥ 0, the map \(\frac{x}{\|{x}\|}\) minimizes Ep,f among the maps in W1,p(Bn, 핊n−1) which coincide with \(\frac{x}{\|{x}\|}\) on ∂ Bn. We also study the case where f(r) = rα with −n+2 < α < 0 and prove that \(\frac{x}{\|{x}\|}\) does not minimize Ep,f for α close to −n+2 and when n ≥ 6, for α close to 4−n.

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  1. Laboratoire de Mathematiques et Physiques Théorique, Université de Tours, Parc Grandmont, 37200, Tours, France

    Jean-Christophe Bourgoin

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  1. Jean-Christophe Bourgoin
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Correspondence to Jean-Christophe Bourgoin.

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Mathematics Subject Classification (2000) 58E20; 53C43

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Bourgoin, JC. The minimality of the map \(\frac{x}{\|{x}\|}\) for weighted energy. Calc. Var. 25, 469–489 (2006). https://doi.org/10.1007/s00526-005-0350-9

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  • DOI: https://doi.org/10.1007/s00526-005-0350-9

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