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

Mark Allen, Dennis Kriventsov

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)201-214
Number of pages14
JournalAnalysis and PDE
Issue number1
StatePublished - 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Applied Mathematics


  • ACF monotonicity formula
  • Free boundary
  • Monotonicity formula
  • Spiral interface


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