Parameter-uniform Robin uniqueness on large dilations

Abstract

Berestycki and Graham proved large-dilation uniqueness for bounded positive solutions of \[ -\Delta u=f(u)\quad\hbox{in }\kappa\Omega, \qquad u+\alpha\partial_\nu u=0\quad\hbox{on }\partial(\kappa\Omega),\] 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 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 $\partial\Omega\neq\varnothing$, 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.