Remarks on Hausdorff Measure and Stability for the 𝑝-Obstacle Problem (1 < 𝑝 < 2)

Proceedings Mathematical Sciences

Peihao Zhao²  Jun Zheng¹ 

[1] $$;School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 0000, China$$

关键词: Obstacle problem;    Hausdorff measure;    stability.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this paper, we consider the obstacle problem for the inhomogeneous 𝑝-Laplace equation$$mathrm{div}(abla u|^{p-2}abla u)=fcdot p𝜒{u>0},quad 1 < p < 2,$$where 𝑓 is a positive, Lipschitz function. We prove that the free boundary has finite $(N-1)$-Hausdorff measure and stability property, which completes previous works by Caffarelli (J. Fourier Anal. Appl. 4(4--5) (1998) 383--402) for 𝑝=2, and Lee and Shahgholian (J. Differ. Equ. 195 (2003) 14--24) for 2 < 𝑝 < ∞.

【 授权许可】

Unknown