Boundary Vanishing Orders and Asymptotic Expansions for Planar Harmonic Functions with a Sublinear Robin Condition
Abstract
We study boundary vanishing orders and asymptotic expansions of planar harmonic functions satisfying the sublinear Robin condition ∂νu=λu−|u|p−1u, with λ≥0 and 0<p<1. For nontrivial global solutions on bounded connected Lipschitz domains, we bound the number of nodal domains by the number of Steklov eigenvalues strictly below λ and establish finite vanishing order at every boundary point. The boundary scaling gives the critical exponent 1/(1−p). On C1 boundaries, every positive subcritical vanishing order is an integer, and the leading term is a unique homogeneous Neumann profile in conformal coordinates. At a noninteger critical order, the rescaled solution converges in normalized graph coordinates to an explicit homogeneous profile determined by p up to sign. Every supercritical bulk vanishing order m is an integer, with a unique homogeneous Dirichlet profile in conformal coordinates. The boundary trace has vanishing order (m−1)/p>m. Integer critical vanishing orders are excluded if the boundary is C1 with Dini mean oscillation.
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BibTeX
@misc{liu2026boundary,
title = {{Boundary Vanishing Orders and Asymptotic Expansions for Planar Harmonic Functions with a Sublinear Robin Condition}},
author = {Hairong Liu and Long Tian and Xiaoping Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15465},
eprint = {2610.15465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15465},
}
RIS
TY - UNPB
TI - Boundary Vanishing Orders and Asymptotic Expansions for Planar Harmonic Functions with a Sublinear Robin Condition
AU - Hairong Liu
AU - Long Tian
AU - Xiaoping Yang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15465
UR - https://arxiv.org/abs/2610.15465
AB - We study boundary vanishing orders and asymptotic expansions of planar harmonic functions satisfying the sublinear Robin condition ∂_νu=λu−|u|^p−1u, with λ≥0 and 0<p<1. For nontrivial global solutions on bounded connected Lipschitz domains, we bound the number of nodal domains by the number of Steklov eigenvalues strictly below λ and establish finite vanishing order at every boundary point. The boundary scaling gives the critical exponent 1/(1−p). On C^1 boundaries, every positive subcritical vanishing order is an integer, and the leading term is a unique homogeneous Neumann profile in conformal coordinates. At a noninteger critical order, the rescaled solution converges in normalized graph coordinates to an explicit homogeneous profile determined by p up to sign. Every supercritical bulk vanishing order m is an integer, with a unique homogeneous Dirichlet profile in conformal coordinates. The boundary trace has vanishing order (m−1)/p>m. Integer critical vanishing orders are excluded if the boundary is C^1 with Dini mean oscillation.
ER -