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arXiv · 1801.01940

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

Dennis Kriventsov·Mark Allen
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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 appearing in the literature (see \cite{cs05}) 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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Dennis Kriventsov, Mark Allen. 2018-09-13. A spiral interface with positive Alt-Caffarelli-Friedman limit at the origin. https://doi.org/10.2140/apde.2020.13.201

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