ACF Almost Monotonicity at Infinity with Applications to Perturbed Global Solutions

We study the large-scale behavior of the coincidence set of perturbations of global solutions to the classical obstacle problem in ${\mathbb{R}}^n\setminus B_1$, with blow-down invariant in the $e_n$ direction. In dimensions $n\ge 3$, we prove that, locally around regular points sufficiently far out, the cross-sections of $\{u=0\}$ perpendicular to $e_n$ are $C^2$ perturbations of ellipsoids. The main ingredient is a new large-scale almost monotonicity formula for the Alt–Caffarelli–Friedman functional. In contrast with the classical small-scale perturbative theory, our argument exploits the stability of the obstacle problem together with the fact that local perturbations vanish under blow-down. The method provides a model mechanism for controlling errors at infinity in stable free boundary problems.