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Entire p-harmonic functions with an isolated critical point and failure of C1-regularity of the natural gradient
Abstract
For every n≥2 and 1<p<∞, we show an existence of a nonconstant homogeneous entire p-harmonic function whose only critical point is the origin. The angular parts of these functions are axially symmetric and even across the equator. To establish the solution we use contraction argument at the pole in conjuction with a shooting argument. It follows that in every dimension n≥3 and for p<2 sufficiently close to $2$, the natural gradient has pointwise Hölder exponent strictly below one at the origin. Such examples settle the remaining scalar case of the conjecture of Balci, Diening, and Weimar by disproving both the C1 assertion and the linear L2 mean oscillation estimate.
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BibTeX
@misc{kauranen2026entire,
title = {{Entire $p$-harmonic functions with an isolated critical point and failure of $C^1$-regularity of the natural gradient}},
author = {Erno Kauranen and Jarkko Siltakoski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29122},
eprint = {2609.29122},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29122},
}
RIS
TY - UNPB
TI - Entire p-harmonic functions with an isolated critical point and failure of C^1-regularity of the natural gradient
AU - Erno Kauranen
AU - Jarkko Siltakoski
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29122
UR - https://arxiv.org/abs/2609.29122
AB - For every n≥2 and 1<p<∞, we show an existence of a nonconstant homogeneous entire p-harmonic function whose only critical point is the origin. The angular parts of these functions are axially symmetric and even across the equator. To establish the solution we use contraction argument at the pole in conjuction with a shooting argument. It follows that in every dimension n≥3 and for p<2 sufficiently close to $2$, the natural gradient has pointwise Hölder exponent strictly below one at the origin. Such examples settle the remaining scalar case of the conjecture of Balci, Diening, and Weimar by disproving both the C^1 assertion and the linear L^2 mean oscillation estimate.
ER -