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2020 A spiral interface with positive Alt–Caffarelli–Friedman limit at the origin
Mark Allen, Dennis Kriventsov
Anal. PDE 13(1): 201-214 (2020). 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.

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Mark Allen. Dennis Kriventsov. "A spiral interface with positive Alt–Caffarelli–Friedman limit at the origin." Anal. PDE 13 (1) 201 - 214, 2020. https://doi.org/10.2140/apde.2020.13.201

Information

Received: 26 February 2018; Revised: 13 September 2018; Accepted: 19 December 2018; Published: 2020
First available in Project Euclid: 16 January 2020

zbMATH: 1430.35275
MathSciNet: MR4047645
Digital Object Identifier: 10.2140/apde.2020.13.201

Subjects:
Primary: 35J05 , 35R35

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

Rights: Copyright © 2020 Mathematical Sciences Publishers

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