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 ([17]), 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 IR n 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.

A Uniqueness Theorem for Mean Value Sets for Elliptic Divergence Form Operators

“…Combining the vocabular of Sakai that we have defined above, together with the results found within [2], [3], and [4], and with the agreement that we take the mean value sets given as the positivity set of the height function, i.e. we take Ω(t) := D t (x 0 ) and not Ω(t) := [D t (x 0 )], we have the following corollary:…”

“…In fact, the theorem cannot be found there (or in other similar texts), so in [5] the second author of this paper and Z. Hao proved the theorem in detail. Of course, even after stating what "comparable to B r (x 0 )" means precisely, it is still clear that the more that is known about the collection of these D r (x 0 )'s, the more useful this theorem becomes, and since making this observation, both authors of the current paper have been studying the properties of these sets [1,2,3,4]. Of course, there is a natural definition to make: Definition 1.1 (Mean value sets).…”

An interpolating inequality for solutions of uniformly elliptic equations

“…Issues related to this theorem and the study of the geometric properties of the sets D r (x 0 ) have been recently studied by I. Blank and his collaborators in [4,5,6].…”

A Nonlinear Mean Value Property for Monge-Ampère

“…In [17] (see also [10,20]), a simpler statement in terms of mean value sets was pointed out (a set D is a mean value set for the point x ∈ D and the operator L if the mean value property (1.1) holds with D replacing B ε (x) for every u such that Lu = 0). It is worth mentioning that mean value sets and their properties are yet to be fully understood, see [2,3,6,8,29]. For mean value properties in the sub-Riemannian setting we refer to [12].…”