A spiral interface with positive Alt-Caffarelli-Friedman limit at the origin

TL;DR

This paper constructs a counterexample showing that a positive limit of the Alt-Caffarelli-Friedman monotonicity formula does not necessarily imply a unique tangent at the interface, challenging previous assumptions.

Contribution

It provides the first explicit example where positive ACF limit coexists with a non-unique tangent interface, clarifying limitations of previous theoretical results.

Findings

Counterexample with spiraling interface

Positive ACF limit without unique tangent

Blow-up limits depend on subsequence

Abstract

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 appearing in the literature (see \cite{cs05}) 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.