Parameter-uniform Robin uniqueness on large dilations
Berestycki and Graham proved large-dilation uniqueness for bounded positive solutions of $-\Delta u=f(u)$ in $\kappa\Omega$, with $u+\alpha\partial_\nu u=0$ on $\partial(\kappa\Omega)$, when $\alpha$ is fixed, and remarked that the dilation threshold should not depend on $\alpha$. We show that it does not, including at the Dirichlet and Neumann endpoints, for possibly unbounded uniformly $C^{2,\gamma}$ domains. The half-space linearizations retain a common positive spectral gap across the compactified boundary parameter. The remaining endpoint difficulty is nonlinear compactness at the Dirichlet end, where the Robin coefficient diverges. Rescaling at its reciprocal scale makes the limiting equation harmonic. Any trace that persisted in this limit would give a bounded harmonic half-space solution of $\partial_\nu v+v=0$, which a Liouville lemma rules out. Combining this endpoint compactness with the half-space gap and localization gives uniform large-dilation uniqueness. When $\partial\Omega\neq\varnothing$, we further obtain convergence of the spectral bottom to its half-space value with error $O(\kappa^{-1/2})$. For bounded $C^{4,\gamma}$ domains the boundary layer also has a first mean-curvature correction, with remainder $O(\kappa^{-1-\gamma}+\kappa^{-2})$ on each fixed boundary strip.