Nondegenerate Motion of Singular Points in Obstacle Problems with Varying Data

TL;DR

This paper investigates how singular free boundaries in obstacle problems respond immediately to changes in data, providing new insights into their motion and applications to uniqueness of mean value sets.

Contribution

It demonstrates that singular free boundaries in certain obstacle problems move instantly with data variations, and establishes a uniqueness result for mean value sets centered at the same point.

Findings

Singular free boundaries respond immediately to data changes.

A uniqueness result for non-intersecting mean value sets is proved.

Applications include insights into obstacle problem boundary behavior.

Abstract

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.