A spiral interface with positive Alt-Caffarelli-Friedman limit at the origin
Analysis of PDEs
2020-01-08 v2
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.
Cite
@article{arxiv.1801.01940,
title = {A spiral interface with positive Alt-Caffarelli-Friedman limit at the origin},
author = {Dennis Kriventsov and Mark Allen},
journal= {arXiv preprint arXiv:1801.01940},
year = {2020}
}