PreprintCC0arXiv
Asymptotic behavior of an integral equation of Landau-Lifschitz type
Abstract
In this paper, we study an integral equation of Landau-Lifschitz-type involving the Riesz potential u(x)=ℓ+C*∫ℝn(u(y)(|∇ u(y)|2+|uk(y)|2)−uk(y)ek)/(|x−y|n−α)dy, where ℓ ∈ ℝk is a constant vector, C* ≠ 0 is a constant, u:ℝn → 𝕊k−1:={u=(u1,u2,⋯,uk) ∈ ℝk;|u|=1}, k ≥ 1, n ≥ 2, and 0<α<n. We provide a decay rate at which |uk| converges to zero under some mild conditions. This argument still works for the Ginzburg-Landau-type integral equation.
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BibTeX
@misc{zhou2026asymptotic,
title = {{Asymptotic behavior of an integral equation of Landau-Lifschitz type}},
author = {Tiantian Zhou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07391},
eprint = {2611.07391},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07391},
note = {CC0},
}
RIS
TY - UNPB
TI - Asymptotic behavior of an integral equation of Landau-Lifschitz type
AU - Tiantian Zhou
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07391
UR - https://arxiv.org/abs/2611.07391
AB - In this paper, we study an integral equation of Landau-Lifschitz-type involving the Riesz potential u(x)=ℓ+C_*∫_ℝ^n(u(y)(|∇ u(y)|^2+|u_k(y)|^2)−u_k(y)e_k)/(|x−y|^n−α)dy, where ℓ ∈ ℝ^k is a constant vector, C_* ≠ 0 is a constant, u:ℝ^n → 𝕊^k−1:={u=(u_1,u_2,⋯,u_k) ∈ ℝ^k;|u|=1}, k ≥ 1, n ≥ 2, and 0<α<n. We provide a decay rate at which |u_k| converges to zero under some mild conditions. This argument still works for the Ginzburg-Landau-type integral equation.
ER -