We give an example of a pair of nonnegative subharmonic functions with disjoint
support for which the Alt–Caffarelli–Friedman monotonicity formula has strictly
positive limit at the origin, and yet the interface between their supports lacks a
(unique) tangent there. This clarifies a remark of Caffarelli and Salsa (A geometricapproach to free boundary problems, 2005) that the positivity of the limit of the
ACF formula implies unique tangents; this is true under some additional
assumptions, but false in general. In our example, blow-ups converge to the
expected piecewise linear two-plane function along subsequences, but the
limiting function depends on the subsequence due to the spiraling nature of the
interface.
Keywords
ACF monotonicity formula, spiral interface, free boundary,
monotonicity formula
