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

      Peihao Zhao Jun Zheng

      Volume 122 Issue 1 February 2012 pp 129-137

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/122/01/0129-0137

    • Keywords

       

      Obstacle problem; Hausdorff measure; stability.

    • Abstract

       

      In this paper, we consider the obstacle problem for the inhomogeneous 𝑝-Laplace equation

      $$\mathrm{div}(\nabla u|^{p-2}\nabla u)=f\cdot p\chi\{u>0\},\quad 1 < p < 2,$$

      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 $2 < p < \infty$.

    • Author Affiliations

       

      Peihao Zhao1 Jun Zheng1

      1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

    • Dates

       
      Published
      February 2012
      Manuscript received
      29 September 2011

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