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

Allen, Mark; Kriventsov, Dennis

doi:10.2140/apde.2020.13.201

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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

DOI: 10.2140/apde.2020.13.201

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 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.

Keywords

ACF monotonicity formula, spiral interface, free boundary, monotonicity formula

Mathematical Subject Classification 2010

Primary: 35R35, 35J05

Milestones

Received: 26 February 2018

Revised: 13 September 2018

Accepted: 19 December 2018

Published: 6 January 2020

Authors

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

Vol. 13, No. 1, 2020