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The hedgehog uniquely minimises every p-energy

  • Siran Li
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Abstract

We prove that for every dimension n ≥ 2 and parameter p ∈ ]1,n[, the hedgehog u0(x) = x/|x| is the unique minimiser of the p-energy ℰp [u]:=∫Bn|∇ u|p dx in the class of W1,p-mappings from the n-dimensional unit ball Bn to the (n−1)-dimensional unit sphere Sn−1=∂Bn whose trace equal to the identity on Sn−1. This answers in the affirmative an open question on p.29 in [Robert M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 15--34].

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Cite

BibTeXRIS
BibTeX
@misc{li2026hedgehog,
  title = {{The hedgehog uniquely minimises every $p$-energy}},
  author = {Siran Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13979},
  eprint = {2609.13979},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13979},
}
RIS
TY  - UNPB
TI  - The hedgehog uniquely minimises every p-energy
AU  - Siran Li
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.13979
UR  - https://arxiv.org/abs/2609.13979
AB  - We prove that for every dimension n ≥ 2 and parameter p ∈ ]1,n[, the hedgehog u_0(x) = x/|x| is the unique minimiser of the p-energy ℰ_p [u]:=∫_B^n|∇ u|^p dx in the class of W^1,p-mappings from the n-dimensional unit ball B^n to the (n−1)-dimensional unit sphere S^n−1=∂B^n whose trace equal to the identity on S^n−1. This answers in the affirmative an open question on p.29 in [Robert M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 15--34].
ER  -