Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

Abstract

We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set.Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren's frequency formula, first established by Caffarelli, Salsa, and Silvestre.2000 Mathematics Subject Classification. 35R35.

Properties of a frequency of Almgren type for harmonic functions in Carnot groups

“…The main thrust of this formula is that the derivative of W κ (u, r ) vanishes if and only if u is homogeneous of degree κ. Our main result is Theorem 10.1 below which was inspired to a result that, for the classical Laplacian, was discovered by Petrosyan and the first named author in [23]. Using such result we establish Theorem 10.4 below, which states that if a harmonic function u in G has vanishing discrepancy and constant frequency equal to κ, then u must be a stratified solid harmonic of degree κ.…”

“…The former, Theorem 11.8 below, is the counterpart of Theorem 10.1, except that, interestingly, for the solutions of the Baouendi operators B α we do not have the additional hypothesis of vanishing discrepancy. Our second main result is Theorem 11.14 below, which is inspired to a monotonicity formula for the classical Laplacian in [23].…”

“…In their recent work [23], Petrosyan and the first named author discovered a one-parameter family of Weiss type monotonicity formulas, satisfied by the solutions of the lower-dimensional obstacle problem, and which play a key role in the analysis of the so-called blow-ups at singular points. These authors proved that such family of monotonicity formulas are deeply connected to Almgren's monotonicity of the frequency.…”

“…Our next result represents a generalization to solutions of the operator B α of a monotonicity theorem proved in [23] for solutions of the lower-dimensional obstacle problem for the standard . For the case κ = 2, and in connection with solutions of the classical obstacle problem for , this monotonicity theorem was first proved by Monneau in [34].…”

Higher regularity of the free boundary in the elliptic Signorini problem

“…The proofs can be found in [3,4,14] and in Chapter 9 of [21]. The proofs can be found in [3,4,14] and in Chapter 9 of [21].…”