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Remarks on Hausdorff measure and stability for the p-obstacle problem (1 < p < 2)

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Abstract

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

$$ \text {div}\big(|\nabla u|^{p-2} \nabla u\big)=f\cdot \chi_{ \{u>0\},}<p<2, $$

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 < ∞.

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

    PEIHAO ZHAO & JUN ZHENG

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  1. PEIHAO ZHAO
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  2. JUN ZHENG
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Correspondence to JUN ZHENG.

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ZHAO, P., ZHENG, J. Remarks on Hausdorff measure and stability for the p-obstacle problem (1 < p < 2). Proc Math Sci 122, 129–137 (2012). https://doi.org/10.1007/s12044-012-0053-z

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