Analysis & PDE
- Anal. PDE
- Volume 13, Number 1 (2020), 201-214.
A spiral interface with positive Alt–Caffarelli–Friedman limit at the origin
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 geometric approach 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.
Anal. PDE, Volume 13, Number 1 (2020), 201-214.
Received: 26 February 2018
Revised: 13 September 2018
Accepted: 19 December 2018
First available in Project Euclid: 16 January 2020
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Allen, Mark; Kriventsov, Dennis. A spiral interface with positive Alt–Caffarelli–Friedman limit at the origin. Anal. PDE 13 (2020), no. 1, 201--214. doi:10.2140/apde.2020.13.201. https://projecteuclid.org/euclid.apde/1579143671