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A counterexample to the pointwise validity of an asymptotic mean value property for p-harmonic functions

  • Ángel Arroyo
  • Jeff Calder
  • Mikko Parviainen
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Abstract

We give a counterexample showing that an asymptotic mean value property for p-harmonic functions in ℝd does not necessarily hold in the pointwise sense for d≥ 3 and p>2 near $2$. The proof uses a perturbation argument from p=2 and the Implicit Function Theorem to establish the counterexample. We also use a computer assisted proof to give an explicit range of values of p for which the counterexample holds. Finally, we also establish a counterexample for the pointwise asymptotic variational mean value property.

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Cite

BibTeXRIS
BibTeX
@misc{arroyo2026counterexample,
  title = {{A counterexample to the pointwise validity of an asymptotic mean value property for $p$-harmonic functions}},
  author = {Ángel Arroyo and Jeff Calder and Mikko Parviainen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28069},
  eprint = {2609.28069},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28069},
}
RIS
TY  - UNPB
TI  - A counterexample to the pointwise validity of an asymptotic mean value property for p-harmonic functions
AU  - Ángel Arroyo
AU  - Jeff Calder
AU  - Mikko Parviainen
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28069
UR  - https://arxiv.org/abs/2609.28069
AB  - We give a counterexample showing that an asymptotic mean value property for p-harmonic functions in ℝ^d does not necessarily hold in the pointwise sense for d≥ 3 and p>2 near $2$. The proof uses a perturbation argument from p=2 and the Implicit Function Theorem to establish the counterexample. We also use a computer assisted proof to give an explicit range of values of p for which the counterexample holds. Finally, we also establish a counterexample for the pointwise asymptotic variational mean value property.
ER  -