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.

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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