An Almgren-type formula for planar p-harmonic functions
Abstract
For a nonconstant planar p-harmonic function u, with 1 < p < ∞, and according to previous results, most notably Aronsson's fundamental analysis of the hodograph representation in a neighborhood of an isolated critical point, we introduce the flux-normalized frequency N*(x0,r) = (r∫Br(x0) |D u|p dx)/(∫∂ Br(x0) |D u|p−2(u−u(x0))2 dS). This ratio constitutes the natural counterpart of Almgren's frequency function in the nonlinear setting, as it recovers precisely the degree of homogeneity for every homogeneous p-harmonic profile. We further establish the corresponding gauge-corrected version of Almgren-type monotonicity and reinterpret, in terms of the frequency function, the classical planar unique continuation property. In addition, we derive a quantitative pinching estimate for the flux-normalized frequency and identify the specific obstruction that prevents a straightforward extension of the method to higher dimensions.
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BibTeX
@misc{zhang2026almgrentype,
title = {{An Almgren-type formula for planar $p$-harmonic functions}},
author = {Yi Ru-Ya Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30847},
eprint = {2608.30847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30847},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Almgren-type formula for planar p-harmonic functions
AU - Yi Ru-Ya Zhang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30847
UR - https://arxiv.org/abs/2608.30847
AB - For a nonconstant planar p-harmonic function u, with 1 < p < ∞, and according to previous results, most notably Aronsson's fundamental analysis of the hodograph representation in a neighborhood of an isolated critical point, we introduce the flux-normalized frequency N_*(x_0,r) = (r∫_B_r(x_0) |D u|^p dx)/(∫_∂ B_r(x_0) |D u|^p−2(u−u(x_0))^2 dS). This ratio constitutes the natural counterpart of Almgren's frequency function in the nonlinear setting, as it recovers precisely the degree of homogeneity for every homogeneous p-harmonic profile. We further establish the corresponding gauge-corrected version of Almgren-type monotonicity and reinterpret, in terms of the frequency function, the classical planar unique continuation property. In addition, we derive a quantitative pinching estimate for the flux-normalized frequency and identify the specific obstruction that prevents a straightforward extension of the method to higher dimensions.
ER -