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

• 成果类型: Article
• 署名作者: Garofalo, Nicola; Petrosyan, Arshak
• 署名单位: Purdue University System; Purdue University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-009-0188-4
• 发表日期: 2009
• 页码: 415-461
• 关键词: fractional laplacian; unique continuation; free boundaries; REGULARITY

摘要

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.