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.

44 pages, 2 figures

References in corpus (2)

• An extension problem related to the fractional Laplacian
• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian

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• Free boundary partial regularity in the thin obstacle problem
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• Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory