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Explosion versus decay for boundary derivatives of p-harmonic functions as p tends to 1: nonlocality

  • Yuval Peres
  • Han Wang
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Abstract

We consider the Dirichlet problem for the p-Laplacian on a bounded Lipschitz domain Ω⊂ℝd, with boundary data given by the indicator of a closed set. At a smooth boundary point where the data vanish, we study the upper and lower inward normal Dini derivatives as p↓1. We give sufficient geometric conditions for these derivatives to be of order (p−1)−1, and other conditions under which they decay exponentially in 1/(p−1). Whether explosion or decay occurs is not determined locally: on one fixed planar domain with real-analytic boundary, changing the data on a set of arbitrarily small total length, uniformly separated from the observation point, changes explosion into exponential decay. We also exhibit a critical example of a cylinder in ℝd+1 where the inward normal derivative is of order √(d/(p−1)).

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Cite

BibTeXRIS
BibTeX
@misc{peres2026explosion,
  title = {{Explosion versus decay for boundary derivatives of $p$-harmonic functions as $p$ tends to 1: nonlocality}},
  author = {Yuval Peres and Han Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.03322},
  eprint = {2605.03322},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.03322},
}
RIS
TY  - UNPB
TI  - Explosion versus decay for boundary derivatives of p-harmonic functions as p tends to 1: nonlocality
AU  - Yuval Peres
AU  - Han Wang
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2605.03322
UR  - https://arxiv.org/abs/2605.03322
AB  - We consider the Dirichlet problem for the p-Laplacian on a bounded Lipschitz domain Ω⊂ℝ^d, with boundary data given by the indicator of a closed set. At a smooth boundary point where the data vanish, we study the upper and lower inward normal Dini derivatives as p↓1. We give sufficient geometric conditions for these derivatives to be of order (p−1)^−1, and other conditions under which they decay exponentially in 1/(p−1). Whether explosion or decay occurs is not determined locally: on one fixed planar domain with real-analytic boundary, changing the data on a set of arbitrarily small total length, uniformly separated from the observation point, changes explosion into exponential decay. We also exhibit a critical example of a cylinder in ℝ^d+1 where the inward normal derivative is of order √(d/(p−1)).
ER  -