Some New Monotonicity Formulas and the Singular Set in the Lower Dimensional Obstacle Problem

TL;DR

This paper introduces new monotonicity formulas for analyzing free boundary points in the lower dimensional obstacle problem, leading to a structural understanding of the singular set and applicable to both zero and smooth obstacles.

Contribution

It develops two novel families of monotonicity formulas tailored for singular and general free boundary points, extending previous methods.

Findings

Proves uniqueness and continuous dependence of blowups at singular points.

Establishes a structural theorem for the singular set.

Generalizes Almgren's frequency formula for smooth obstacles.

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.