Parameter-uniform Robin uniqueness on large dilations
Abstract
Berestycki and Graham proved large-dilation uniqueness for bounded positive solutions of −Δu=f(u) in κΩ, u+α∂νu=0 on ∂(κΩ), when α is fixed, and remarked that the dilation threshold should not depend on α. We show that it does not, including at the Dirichlet and Neumann endpoints, for possibly unbounded uniformly C2,γ domains. The half-space linearizations have a common positive spectral gap over the compactified boundary parameter. The Dirichlet end requires a separate compactness argument because the Robin coefficient diverges there. After rescaling by its inverse, the equation has a harmonic half-space limit. A Liouville lemma rules out a nonzero limiting trace, and the resulting endpoint compactness, together with the half-space gap and localization, yields the uniform uniqueness statement. When ∂Ω≠∅, we further obtain convergence of the spectral bottom to its half-space value with error O(κ−1/2). For bounded C4,γ domains the boundary layer also has a first mean-curvature correction, with remainder O(κ−1−γ+κ−2) on each fixed boundary strip.
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BibTeX
@misc{sun2026parameteruniform,
title = {{Parameter-uniform Robin uniqueness on large dilations}},
author = {Sophie Sun and Xuanrui Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30211},
eprint = {2608.30211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30211},
}
RIS
TY - UNPB
TI - Parameter-uniform Robin uniqueness on large dilations
AU - Sophie Sun
AU - Xuanrui Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2608.30211
UR - https://arxiv.org/abs/2608.30211
AB - Berestycki and Graham proved large-dilation uniqueness for bounded positive solutions of −Δu=f(u) in κΩ, u+α∂_νu=0 on ∂(κΩ), when α is fixed, and remarked that the dilation threshold should not depend on α. We show that it does not, including at the Dirichlet and Neumann endpoints, for possibly unbounded uniformly C^2,γ domains. The half-space linearizations have a common positive spectral gap over the compactified boundary parameter. The Dirichlet end requires a separate compactness argument because the Robin coefficient diverges there. After rescaling by its inverse, the equation has a harmonic half-space limit. A Liouville lemma rules out a nonzero limiting trace, and the resulting endpoint compactness, together with the half-space gap and localization, yields the uniform uniqueness statement. When ∂Ω≠∅, we further obtain convergence of the spectral bottom to its half-space value with error O(κ^−1/2). For bounded C^4,γ domains the boundary layer also has a first mean-curvature correction, with remainder O(κ^−1−γ+κ^−2) on each fixed boundary strip.
ER -