On the regularity of the free boundary in the p-Laplacian obstacle problem

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Figalli, Alessio¹  Krummel, Brian²  Ros-Oton, Xavier³ 

[1] ETH, Dept Math, Raemistr 101, CH-8092 Zurich, Switzerland

[2] Univ Calif Berkeley, Dept Math, 970 Evans Hall, Berkeley, CA 94720 USA

[3] Univ Texas Austin, Dept Math, 2515 Speedway, Austin, TX 78751 USA

关键词: Obstacle problem;    p-Laplacian;    Free boundary;   

DOI  :  10.1016/j.jde.2017.03.035

来源: Elsevier

【 摘 要 】

We study the regularity of the free boundary in the obstacle for the p-Laplacian, min{-Delta(p)u, u - phi} = 0 in Omega subset of R-n. Here, Delta(p)u = div(vertical bar del u vertical bar(P-2)del u), and p is an element of (1, 2) boolean OR (2, infinity). Near those free boundary points where del phi not equal 0, the operator Delta(p) is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when del phi = 0 then Delta(p) is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where del phi = 0. On the one hand, for every p not equal 2 we construct explicit global 2-homogeneous solutions to the p-Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not C-1 at points where del phi = 0. On the other hand, under the concavity assumption vertical bar del phi vertical bar(2-p), Delta(p)phi < 0, we show the free boundary is countably (n -1)-rectifiable and we prove a nondegeneracy property for it at all free boundary points. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free