Vol. 13, No. 1, 2020

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

Mark Allen and Dennis Kriventsov

Vol. 13 (2020), No. 1, 201–214

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.

ACF monotonicity formula, spiral interface, free boundary, monotonicity formula
Mathematical Subject Classification 2010
Primary: 35R35, 35J05
Received: 26 February 2018
Revised: 13 September 2018
Accepted: 19 December 2018
Published: 6 January 2020
Mark Allen
Department of Mathematics
Brigham Young University
Provo, UT
United States
Dennis Kriventsov
Courant Institute of Mathematical Sciences
New York University
New York, NY
United States