  Recent work by Serfaty and Serra give a formula for the velocity of the free
boundary of the obstacle problem at regular points [Serfaty-Serra 2018], and
much older work by King, Lacey, and Vazquez gives an example of a singular free
boundary point (in the Hele-Shaw flow) that remains stationary for a positive
amount of time [King-Lacey-Vazquez 1995]. The authors show how singular free
boundaries in the obstacle problem in some settings move immediately in
response to varying data. Three applications of this result are given, and in
particular, the authors show a uniqueness result: For sufficiently smooth
elliptic divergence form operators on domains in $\mathrm{I \! R}^n$ and for
the Laplace-Beltrami operator on a smooth manifold, the boundaries of distinct
mean value sets (of the type found in [Blank-Hao 2015] and
[Benson-Blank-LeCrone 2018]) which are centered at the same point do not
intersect.
