Generic regularity of free boundaries for the obstacle problem

被引：26
|
作者：
Figalli, Alessio [1 ]
Ros-Oton, Xavier [2 ,3 ,4 ]
Serra, Joaquim [1 ]
机构：
[1] Swiss Fed Inst Technol, Dept Math, Raemistr 101, CH-8092 Zurich, Switzerland
[2] Univ Zurich, Inst Math, Winterthurerstr 190, CH-8057 Zurich, Switzerland
[3] ICREA, Pg Lluis Co 23, Barcelona 08010, Spain
[4] Univ Barcelona, Dept Matemat & Informat, Gran Via Corts Catalanes 585, Barcelona 08007, Spain
来源：
PUBLICATIONS MATHEMATIQUES DE L IHES | 2020年 / 132卷 / 01期
基金：
瑞士国家科学基金会; 欧洲研究理事会;
关键词：
D O I：
10.1007/s10240-020-00119-9
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in R-n. By classical results of Caffarelli, the free boundary is C-infinity outside a set of singular points. Explicit examples show that the singular set could be in general(n-1)-dimensional-that is, as large as the regular set. Our main result establishes that, generically, the singular set has zeroHn-4 measure (in particular, it has codimension 3 inside the free boundary). Thus, for n <= 4, the free boundary is generically a C-infinity manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions n <= 4.
引用
收藏
页码：181 / 292
页数：112
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