Nondegenerate Motion of Singular Points in Obstacle Problems with Varying Data.

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.