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Zeitschrift für Analysis und ihre Anwendungen


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Volume 31, Issue 4, 2012, pp. 427–439
DOI: 10.4171/ZAA/1467

Published online: 2012-10-10

A Remark on Hausdorff Measure in Obstacle Problems

Jun Zheng[1] and Peihao Zhao[2]

(1) Lanzhou University, Lanzhou (Gansu), China
(2) Lanzhou University, Lanzhou (Gansu), China

In this paper, we consider the identical zero obstacle problem for the second order elliptic equation $$-\text { div}\ a(\nabla u)=-1\quad \text{in}\ \mathcal {D}'(\Omega),$$ where $\Omega$ is an open bounded domain of $\mathbb{R}^{N},N\geq 2$. We prove that the free boundary has finite $(N-1)$-Hausdorff measure, which extends the previous works by Caffarelli, Lee and Shahgholian for $p$-Laplacian equations with $p=2,p>2$ respectively and contains the singular case of $1< p <2$.

Keywords: Elliptic equation, obstacle problem, free boundary, Hausdorff measure

Zheng Jun, Zhao Peihao: A Remark on Hausdorff Measure in Obstacle Problems. Z. Anal. Anwend. 31 (2012), 427-439. doi: 10.4171/ZAA/1467