On the fine structure of the free boundary for the classical obstacle problem

• 成果类型: Article
• 署名作者: Figalli, Alessio; Serra, Joaquim
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0827-8
• 发表日期: 2019
• 页码: 311-366
• 关键词: regularity

摘要

In the classical obstacle problem, the free boundary can be decomposed into regular and singular points. As shown by Caffarelli in his seminal papers (Caffarelli in Acta Math 139:155-184, 1977; J Fourier Anal Appl 4:383-402, 1998), regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of C1 manifolds of varying dimension. In two dimensions, this C1 result has been improved to C1, by Weiss (Invent Math 138:23-50, 1999). In this paper we prove that, for n=2 singular points are locally contained in a C2 curve. In higher dimension n3, we show that the same result holds with C1,1 manifolds (or with countably many C2 manifolds), up to the presence of some anomalous points of higher codimension. In addition, we prove that the higher dimensional stratum is always contained in a C1, manifold, thus extending to every dimension the result in Weiss (1999). We note that, in terms of density decay estimates for the contact set, our result is optimal. In addition, for n3 we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.