Nondegenerate motion of singular points in obstacle problems with varying data

Abstract

Recent work by Serfaty and Serra give a formula for the velocity of the free boundary of the obstacle problem at regular points ([19]), and much older work by King, Lacey, and Vázquez gives an example of a singular free boundary point (in the Hele-Shaw flow) that remains stationary for a positive amount of time ([13]). 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 Rn and for the Laplace-Beltrami operator on a smooth manifold, the boundaries of distinct mean value sets (of the type found in [7] and [5]) which are centered at the same point do not intersect.